Calculus Illustrated. Volume 5: Differential Equations [5]

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Calculus Illustrated. Volume 5: Differential Equations [5]

Table of contents :
Preface
Ordinary differential equations
Incremental motion
Discrete models: how to set up ODEs
Discrete forms
Differential forms
Solution sets of ODEs
Separation of variables in ODEs
The method of integrating factors
Change of variables in ODEs
Euler's method: back to the discrete
How large is the difference between the discrete and the continuous?
Qualitative analysis of ODEs
Linearization of ODEs
Motion under forces: the acceleration
Discrete models: how to set up ODEs of second order
Discrete forms, continued
Vector variables
Where matrices come from
Transformations of the plane
Linear operators
Examining and building linear operators
The determinant of a matrix
It's a stretch: eigenvalues and eigenvectors
The significance of eigenvectors
Bases
Classification of linear operators according to their eigenvalues
Vector and complex variables
Algebra of linear operators and matrices
Compositions of linear operators
How complex numbers emerge
Classification of quadratic polynomials
The complex plane C is the Euclidean space R2
Multiplication of complex numbers: C isn't just R2
Complex functions
Complex linear operators
Linear operators with complex eigenvalues
Complex calculus
Series and power series
Solving ODEs with power series
Systems of ODEs
Parametric curves
The predator-prey model
Qualitative analysis of the predator-prey model
Solving the Lotka–Volterra equations
Vector fields and systems of ODEs
Discrete systems of ODEs
Qualitative analysis of systems of ODEs
The vector notation and linear systems
Classification of linear systems
Classification of linear systems, continued
Applications of ODEs
Vector-valued forms
The pursuit curves
ODEs of second order as systems
Vector ODEs of second order: a double spring
A pendulum
Planetary motion
The two- and three-body problems
A cannon is fired...
Boundary value problems
Partial differential equations
Heat transfer between adjacent objects
Heat transfer depends on permeability
Heat transfer is caused by temperature differences
Heat transfer depends on the geometry
The heat PDE
Cells and forms in higher dimensions
Heat transfer in dimension 2: a plate
The heat PDE for dimension 2
Wave propagation in dimension 1: springs and strings
The wave PDE
Wave propagation in dimension 2: a membrane
Exercises
Exercises: Basics
Exercises: Analytical methods
Exercises: Euler's method
Exercises: Generalities
Exercises: Models and setting up ODEs
Exercises: Qualitative analysis
Exercises: Systems
Exercises: Second order
Exercises: Advanced
Exercises: PDEs
Exercises: Computing
Index

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❚♦ t❤❡ st✉❞❡♥t✳✳✳

❚♦ t❤❡ st✉❞❡♥t ▼❛t❤❡♠❛t✐❝s ✐s ❛ s❝✐❡♥❝❡✳ ❏✉st ❛s t❤❡ r❡st ♦❢ t❤❡ s❝✐❡♥t✐sts✱ ♠❛t❤❡♠❛t✐❝✐❛♥s ❛r❡ tr②✐♥❣ t♦ ✉♥❞❡rst❛♥❞ ❤♦✇ t❤❡ ❯♥✐✈❡rs❡ ♦♣❡r❛t❡s ❛♥❞ ❞✐s❝♦✈❡r ✐ts ❧❛✇s✳

❲❤❡♥ s✉❝❝❡ss❢✉❧✱ t❤❡② ✇r✐t❡ t❤❡s❡ ❧❛✇s ❛s s❤♦rt st❛t❡♠❡♥ts

❝❛❧❧❡❞ ✏t❤❡♦r❡♠s✑✳ ■♥ ♦r❞❡r t♦ ♣r❡s❡♥t t❤❡s❡ ❧❛✇s ❝♦♥❝❧✉s✐✈❡❧② ❛♥❞ ♣r❡❝✐s❡❧②✱ ❛ ❞✐❝t✐♦♥❛r② ♦❢ t❤❡ ♥❡✇ ❝♦♥❝❡♣ts ✐s ❛❧s♦ ❞❡✈❡❧♦♣❡❞❀ ✐ts ❡♥tr✐❡s ❛r❡ ❝❛❧❧❡❞ ✏❞❡✜♥✐t✐♦♥s✑✳ ❚❤❡s❡ t✇♦ ♠❛❦❡ ✉♣ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ♣❛rt ♦❢ ❛♥② ♠❛t❤❡♠❛t✐❝s ❜♦♦❦✳ ❚❤✐s ✐s ❤♦✇ ❞❡✜♥✐t✐♦♥s✱ t❤❡♦r❡♠s✱ ❛♥❞ s♦♠❡ ♦t❤❡r ✐t❡♠s ❛r❡ ✉s❡❞ ❛s ❜✉✐❧❞✐♥❣ ❜❧♦❝❦s ♦❢ t❤❡ s❝✐❡♥t✐✜❝ t❤❡♦r② ✇❡ ♣r❡s❡♥t ✐♥ t❤✐s t❡①t✳ ❊✈❡r② ♥❡✇ ❝♦♥❝❡♣t ✐s ✐♥tr♦❞✉❝❡❞ ✇✐t❤ ✉t♠♦st s♣❡❝✐✜❝✐t②✳

❉❡✜♥✐t✐♦♥ ✵✳✵✳✶✿ sq✉❛r❡ r♦♦t ❙✉♣♣♦s❡

x✱

a

✐s ❛ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ❚❤❡♥ t❤❡ sq✉❛r❡ r♦♦t ♦❢ x2 = a✳

a

✐s ❛ ♣♦s✐t✐✈❡ ♥✉♠❜❡r

s✉❝❤ t❤❛t

❚❤❡ t❡r♠ ❜❡✐♥❣ ✐♥tr♦❞✉❝❡❞ ✐s ❣✐✈❡♥ ✐♥ ✐t❛❧✐❝s✳ ❚❤❡ ❞❡✜♥✐t✐♦♥s ❛r❡ t❤❡♥ ❝♦♥st❛♥t❧② r❡❢❡rr❡❞ t♦ t❤r♦✉❣❤♦✉t t❤❡ t❡①t✳ ◆❡✇ s②♠❜♦❧✐s♠ ♠❛② ❛❧s♦ ❜❡ ✐♥tr♦❞✉❝❡❞✳

❙q✉❛r❡ r♦♦t √

a

❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ♥♦t❛t✐♦♥ ✐s ❢r❡❡❧② ✉s❡❞ t❤r♦✉❣❤♦✉t t❤❡ t❡①t✳ ❲❡ ♠❛② ❝♦♥s✐❞❡r ❛ s♣❡❝✐✜❝ ✐♥st❛♥❝❡ ♦❢ ❛ ♥❡✇ ❝♦♥❝❡♣t ❡✐t❤❡r ❜❡❢♦r❡ ♦r ❛❢t❡r ✐t ✐s ❡①♣❧✐❝✐t❧② ❞❡✜♥❡❞✳

❊①❛♠♣❧❡ ✵✳✵✳✷✿ ❧❡♥❣t❤ ♦❢ ❞✐❛❣♦♥❛❧ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ ❛

1 × 1 sq✉❛r❡❄ ❚❤❡ sq✉❛r❡ ✐s ♠❛❞❡ ♦❢ t✇♦ r✐❣❤t tr✐❛♥❣❧❡s ❛♥❞ t❤❡ a✳ ❚❤❡♥✱ ❜② t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠ ✱ t❤❡ sq✉❛r❡ ♦❢

❞✐❛❣♦♥❛❧ ✐s t❤❡✐r s❤❛r❡❞ ❤②♣♦t❡♥✉s❡✳ ▲❡t✬s ❝❛❧❧ ✐t a ✐s 12 + 12 = 2✳ ❈♦♥s❡q✉❡♥t❧②✱ ✇❡ ❤❛✈❡✿

a2 = 2 . ❲❡ ✐♠♠❡❞✐❛t❡❧② s❡❡ t❤❡ ♥❡❡❞ ❢♦r t❤❡ sq✉❛r❡ r♦♦t✦ ❚❤❡ ❧❡♥❣t❤ ✐s✱ t❤❡r❡❢♦r❡✱

a=



2✳

❨♦✉ ❝❛♥ s❦✐♣ s♦♠❡ ♦❢ t❤❡ ❡①❛♠♣❧❡s ✇✐t❤♦✉t ✈✐♦❧❛t✐♥❣ t❤❡ ✢♦✇ ♦❢ ✐❞❡❛s✱ ❛t ②♦✉r ♦✇♥ r✐s❦✳ ❆❧❧ ♥❡✇ ♠❛t❡r✐❛❧ ✐s ❢♦❧❧♦✇❡❞ ❜② ❛ ❢❡✇ ❧✐tt❧❡ t❛s❦s✱ ♦r q✉❡st✐♦♥s✱ ❧✐❦❡ t❤✐s✳

❊①❡r❝✐s❡ ✵✳✵✳✸ ❋✐♥❞ t❤❡ ❤❡✐❣❤t ♦❢ ❛♥ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s✐❞❡ ♦❢ ✇❤✐❝❤ ✐s

1✳

❚❤❡ ❡①❡r❝✐s❡s ❛r❡ t♦ ❜❡ ❛tt❡♠♣t❡❞ ✭♦r ❛t ❧❡❛st ❝♦♥s✐❞❡r❡❞✮ ✐♠♠❡❞✐❛t❡❧②✳ ▼♦st ♦❢ t❤❡ ✐♥✲t❡①t ❡①❡r❝✐s❡s ❛r❡ ♥♦t ❡❧❛❜♦r❛t❡✳

❚❤❡② ❛r❡♥✬t✱ ❤♦✇❡✈❡r✱ ❡♥t✐r❡❧② r♦✉t✐♥❡ ❛s t❤❡② r❡q✉✐r❡

✉♥❞❡rst❛♥❞✐♥❣ ♦❢✱ ❛t ❧❡❛st✱ t❤❡ ❝♦♥❝❡♣ts t❤❛t ❤❛✈❡ ❥✉st ❜❡❡♥ ✐♥tr♦❞✉❝❡❞✳ ❆❞❞✐t✐♦♥❛❧ ❡①❡r❝✐s❡ s❡ts ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ❛♣♣❡♥❞✐① ❛s ✇❡❧❧ ❛s ❛t t❤❡ ❜♦♦❦✬s ✇❡❜s✐t❡✿ ❝❛❧❝✉❧✉s✶✷✸✳❝♦♠✳ ❉♦ ♥♦t st❛rt ②♦✉r st✉❞② ✇✐t❤ t❤❡ ❡①❡r❝✐s❡s✦ ❑❡❡♣ ✐♥ ♠✐♥❞ t❤❛t t❤❡ ❡①❡r❝✐s❡s ❛r❡ ♠❡❛♥t t♦ t❡st ✕ ✐♥❞✐r❡❝t❧② ❛♥❞ ✐♠♣❡r❢❡❝t❧② ✕ ❤♦✇ ✇❡❧❧ t❤❡ ❝♦♥❝❡♣ts ❤❛✈❡ ❜❡❡♥ ❧❡❛r♥❡❞✳ ❚❤❡r❡ ❛r❡ s♦♠❡t✐♠❡s ✇♦r❞s ♦❢ ❝❛✉t✐♦♥ ❛❜♦✉t ❝♦♠♠♦♥ ♠✐st❛❦❡s ♠❛❞❡ ❜② t❤❡ st✉❞❡♥ts✳

❚♦ t❤❡ st✉❞❡♥t✳✳✳



❲❛r♥✐♥❣✦ 2 √ (−1) = 1✱ 1✱ 1 = 1✳

■♥ s♣✐t❡ ♦❢ t❤❡ ❢❛❝t t❤❛t ♦♥❡ sq✉❛r❡ r♦♦t ♦❢

t❤❡r❡ ✐s ♦♥❧②

❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❢❛❝ts ❛❜♦✉t t❤❡ ♥❡✇ ❝♦♥❝❡♣ts ❛r❡ ♣✉t ❢♦r✇❛r❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛♥♥❡r✳

❚❤❡♦r❡♠ ✵✳✵✳✹✿ Pr♦❞✉❝t ♦❢ ❘♦♦ts ❋♦r ❛♥② t✇♦ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs

a

b✱

❛♥❞







✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t②✿

b=



a·b

❚❤❡ t❤❡♦r❡♠s ❛r❡ ❝♦♥st❛♥t❧② r❡❢❡rr❡❞ t♦ t❤r♦✉❣❤♦✉t t❤❡ t❡①t✳ ❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡♦r❡♠s ♠❛② ❝♦♥t❛✐♥ ❢♦r♠✉❧❛s❀ ❛ t❤❡♦r❡♠ s✉♣♣❧✐❡s ❧✐♠✐t❛t✐♦♥s ♦♥ t❤❡ ❛♣♣❧✐❝❛❜✐❧✐t② ♦❢ t❤❡ ❢♦r♠✉❧❛ ✐t ❝♦♥t❛✐♥s✳

❋✉rt❤❡r♠♦r❡✱ ❡✈❡r② ❢♦r♠✉❧❛ ✐s ❛ ♣❛rt ♦❢ ❛ t❤❡♦r❡♠✱ ❛♥❞ ✉s✐♥❣ t❤❡ ❢♦r♠❡r ✇✐t❤♦✉t

❦♥♦✇✐♥❣ t❤❡ ❧❛tt❡r ✐s ♣❡r✐❧♦✉s✳ ❚❤❡r❡ ✐s ♥♦ ♥❡❡❞ t♦ ♠❡♠♦r✐③❡ ❞❡✜♥✐t✐♦♥s ♦r t❤❡♦r❡♠s ✭❛♥❞ ❢♦r♠✉❧❛s✮✱ ✐♥✐t✐❛❧❧②✳ ❲✐t❤ ❡♥♦✉❣❤ t✐♠❡ s♣❡♥t ✇✐t❤ t❤❡ ♠❛t❡r✐❛❧✱ t❤❡ ♠❛✐♥ ♦♥❡s ✇✐❧❧ ❡✈❡♥t✉❛❧❧② ❜❡❝♦♠❡ ❢❛♠✐❧✐❛r ❛s t❤❡② ❝♦♥t✐♥✉❡ t♦ r❡❛♣♣❡❛r ✐♥ t❤❡ t❡①t✳ ❲❛t❝❤ ❢♦r ✇♦r❞s ✏✐♠♣♦rt❛♥t✑✱ ✏❝r✉❝✐❛❧✑✱ ❡t❝✳ ❚❤♦s❡ ♥❡✇ ❝♦♥❝❡♣ts t❤❛t ❞♦ ♥♦t r❡❛♣♣❡❛r ✐♥ t❤✐s t❡①t ❛r❡ ❧✐❦❡❧② t♦ ❜❡ s❡❡♥ ✐♥ t❤❡ ♥❡①t ♠❛t❤❡♠❛t✐❝s ❜♦♦❦ t❤❛t ②♦✉ r❡❛❞✳ ❨♦✉ ♥❡❡❞ t♦✱ ❤♦✇❡✈❡r✱ ❜❡ ❛✇❛r❡ ♦❢ ❛❧❧ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥s ❛♥❞ t❤❡♦r❡♠s ❛♥❞ ❜❡ ❛❜❧❡ t♦ ✜♥❞ t❤❡ r✐❣❤t ♦♥❡ ✇❤❡♥ ♥❡❝❡ss❛r②✳ ❖❢t❡♥✱ ❜✉t ♥♦t ❛❧✇❛②s✱ ❛ t❤❡♦r❡♠ ✐s ❢♦❧❧♦✇❡❞ ❜② ❛ t❤♦r♦✉❣❤ ❛r❣✉♠❡♥t ❛s ❛ ❥✉st✐✜❝❛t✐♦♥✳

Pr♦♦❢✳ ❙✉♣♣♦s❡

A=



a

❛♥❞

B=



b✳

❚❤❡♥✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡✜♥✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

a = A2

❛♥❞

b = B2 .

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

❍❡♥❝❡✱



a · b = A2 · B 2 = A · A · B · B = (A · B) · (A · B) = (AB)2 . ab = A · B ✱

❛❣❛✐♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡✜♥✐t✐♦♥✳

❙♦♠❡ ♣r♦♦❢s ❝❛♥ ❜❡ s❦✐♣♣❡❞ ❛t ✜rst r❡❛❞✐♥❣✳ ■ts ❤✐❣❤❧② ❞❡t❛✐❧❡❞ ❡①♣♦s✐t✐♦♥ ♠❛❦❡s t❤❡ ❜♦♦❦ ❛ ❣♦♦❞ ❝❤♦✐❝❡ ❢♦r s❡❧❢✲st✉❞②✳ ■❢ t❤✐s ✐s ②♦✉r ❝❛s❡✱ t❤❡s❡ ❛r❡ ♠② s✉❣❣❡st✐♦♥s✳ ❲❤✐❧❡ r❡❛❞✐♥❣ t❤❡ ❜♦♦❦✱ tr② t♦ ♠❛❦❡ s✉r❡ t❤❛t ②♦✉ ✉♥❞❡rst❛♥❞ ♥❡✇ ❝♦♥❝❡♣ts ❛♥❞ ✐❞❡❛s✳ ❤♦✇❡✈❡r✱ t❤❛t s♦♠❡ ❛r❡ ♠♦r❡ ✐♠♣♦rt❛♥t t❤❛t ♦t❤❡rs❀ t❤❡② ❛r❡ ♠❛r❦❡❞ ❛❝❝♦r❞✐♥❣❧②✳

❑❡❡♣ ✐♥ ♠✐♥❞✱

❈♦♠❡ ❜❛❝❦ ✭♦r ❥✉♠♣

❢♦r✇❛r❞✮ ❛s ♥❡❡❞❡❞✳ ❈♦♥t❡♠♣❧❛t❡✳ ❋✐♥❞ ♦t❤❡r s♦✉r❝❡s ✐❢ ♥❡❝❡ss❛r②✳ ❨♦✉ s❤♦✉❧❞ ♥♦t t✉r♥ t♦ t❤❡ ❡①❡r❝✐s❡ s❡ts ✉♥t✐❧ ②♦✉ ❤❛✈❡ ❜❡❝♦♠❡ ❝♦♠❢♦rt❛❜❧❡ ✇✐t❤ t❤❡ ♠❛t❡r✐❛❧✳ ❲❤❛t t♦ ❞♦ ❛❜♦✉t ❡①❡r❝✐s❡s ✇❤❡♥ s♦❧✉t✐♦♥s ❛r❡♥✬t ♣r♦✈✐❞❡❞❄ ❋✐rst✱ ✉s❡ t❤❡ ❡①❛♠♣❧❡s✳ ▼❛♥② ♦❢ t❤❡♠ ❝♦♥t❛✐♥ ❛ ♣r♦❜❧❡♠ ✕ ✇✐t❤ ❛ s♦❧✉t✐♦♥✳ ❚r② t♦ s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠ ✕ ❜❡❢♦r❡ ♦r ❛❢t❡r r❡❛❞✐♥❣ t❤❡ s♦❧✉t✐♦♥✳ ❨♦✉ ❝❛♥ ❛❧s♦ ✜♥❞ ❡①❡r❝✐s❡s ♦♥❧✐♥❡ ♦r ♠❛❦❡ ✉♣ ②♦✉r ♦✇♥ ♣r♦❜❧❡♠s ❛♥❞ s♦❧✈❡ t❤❡♠✦ ■ str♦♥❣❧② s✉❣❣❡st t❤❛t ②♦✉r s♦❧✉t✐♦♥ s❤♦✉❧❞ ❜❡ t❤♦r♦✉❣❤❧② ✇r✐tt❡♥✳ ❨♦✉ s❤♦✉❧❞ ✇r✐t❡ ✐♥ ❝♦♠♣❧❡t❡ s❡♥t❡♥❝❡s✱ ✐♥❝❧✉❞✐♥❣ ❛❧❧ t❤❡ ❛❧❣❡❜r❛✳ ❋♦r ❡①❛♠♣❧❡✱ ②♦✉ s❤♦✉❧❞ ❛♣♣r❡❝✐❛t❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦✿

❲r♦♥❣✿

1+1 2

❘✐❣❤t✿

1+1 =2



❚♦ t❤❡ st✉❞❡♥t✳✳✳

❚❤❡ ❧❛tt❡r r❡❛❞s ✏♦♥❡ ❛❞❞❡❞ t♦ ♦♥❡ ✐s t✇♦✑✱ ✇❤✐❧❡ t❤❡ ❢♦r♠❡r ❝❛♥♥♦t ❜❡ r❡❛❞✳ ❨♦✉ s❤♦✉❧❞ ❛❧s♦ ❥✉st✐❢② ❛❧❧ ②♦✉r st❡♣s ❛♥❞ ❝♦♥❝❧✉s✐♦♥s✱ ✐♥❝❧✉❞✐♥❣ ❛❧❧ t❤❡ ❛❧❣❡❜r❛✳ ❋♦r ❡①❛♠♣❧❡✱ ②♦✉ s❤♦✉❧❞ ❛♣♣r❡❝✐❛t❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦✿ ❲r♦♥❣✿

2x = 4 x=2

❘✐❣❤t✿

2x = 4 ; x = 2.

t❤❡r❡❢♦r❡✱

❚❤❡ st❛♥❞❛r❞s ♦❢ t❤♦r♦✉❣❤♥❡ss ❛r❡ ♣r♦✈✐❞❡❞ ❜② t❤❡ ❡①❛♠♣❧❡s ✐♥ t❤❡ ❜♦♦❦✳ ◆❡①t✱ ②♦✉r s♦❧✉t✐♦♥ s❤♦✉❧❞ ❜❡ t❤♦r♦✉❣❤❧② r❡❛❞✳ ❚❤✐s ✐s t❤❡ t✐♠❡ ❢♦r s❡❧❢✲❝r✐t✐❝✐s♠✿ ▲♦♦❦ ❢♦r ❡rr♦rs ❛♥❞ ✇❡❛❦ s♣♦ts✳ ■t s❤♦✉❧❞ ❜❡ r❡✲r❡❛❞ ❛♥❞ t❤❡♥ r❡✇r✐tt❡♥✳ ❖♥❝❡ ②♦✉ ❛r❡ ❝♦♥✈✐♥❝❡❞ t❤❛t t❤❡ s♦❧✉t✐♦♥ ✐s ❝♦rr❡❝t ❛♥❞ t❤❡ ♣r❡s❡♥t❛t✐♦♥ ✐s s♦❧✐❞✱ ②♦✉ ♠❛② s❤♦✇ ✐t t♦ ❛ ❦♥♦✇❧❡❞❣❡❛❜❧❡ ♣❡rs♦♥ ❢♦r ❛ ♦♥❝❡✲♦✈❡r✳ ◆❡①t✱ ②♦✉ ♠❛② t✉r♥ t♦ ♠♦❞❡❧✐♥❣ ♣r♦❥❡❝ts✳ ❙♣r❡❛❞s❤❡❡ts ✭▼✐❝r♦s♦❢t ❊①❝❡❧ ♦r s✐♠✐❧❛r✮ ❛r❡ ❝❤♦s❡♥ t♦ ❜❡ ✉s❡❞ ❢♦r ❣r❛♣❤✐♥❣ ❛♥❞ ♠♦❞❡❧✐♥❣✳ ❖♥❡ ❝❛♥ ❛❝❤✐❡✈❡ ❛s ❣♦♦❞ r❡s✉❧ts ✇✐t❤ ♣❛❝❦❛❣❡s s♣❡❝✐✜❝❛❧❧② ❞❡s✐❣♥❡❞ ❢♦r t❤❡s❡ ♣✉r♣♦s❡s✱ ❜✉t s♣r❡❛❞s❤❡❡ts ♣r♦✈✐❞❡ ❛ t♦♦❧ ✇✐t❤ ❛ ✇✐❞❡r s❝♦♣❡ ♦❢ ❛♣♣❧✐❝❛t✐♦♥s✳ ♦♣t✐♦♥✳ ●♦♦❞ ❧✉❝❦✦ ❆✉❣✉st ✶✵✱ ✷✵✷✵

Pr♦❣r❛♠♠✐♥❣ ✐s ❛♥♦t❤❡r

❚♦ t❤❡ t❡❛❝❤❡r



❚♦ t❤❡ t❡❛❝❤❡r ❚❤❡ ❜✉❧❦ ♦❢ t❤❡ ♠❛t❡r✐❛❧ ✐♥ t❤❡ ❜♦♦❦ ❝♦♠❡s ❢r♦♠ ♠② ❧❡❝t✉r❡ ♥♦t❡s✳ ❚❤❡r❡ ✐s ❧✐tt❧❡ ❡♠♣❤❛s✐s ♦♥ ❝❧♦s❡❞✲❢♦r♠ ❝♦♠♣✉t❛t✐♦♥s ❛♥❞ ❛❧❣❡❜r❛✐❝ ♠❛♥✐♣✉❧❛t✐♦♥s✳ ■ ❞♦ t❤✐♥❦ t❤❛t ❛ ♣❡rs♦♥ ✇❤♦ ❤❛s ♥❡✈❡r ✐♥t❡❣r❛t❡❞ ❜② ❤❛♥❞ ✭♦r ❞✐✛❡r❡♥t✐❛t❡❞✱ ♦r ❛♣♣❧✐❡❞ t❤❡ q✉❛❞r❛t✐❝ ❢♦r♠✉❧❛✱ ❡t❝✳✮ ❝❛♥♥♦t ♣♦ss✐❜❧② ✉♥❞❡rst❛♥❞ ✐♥t❡❣r❛t✐♦♥ ✭♦r ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ♦r q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥s✱ ❡t❝✳✮✳ ❍♦✇❡✈❡r✱ ❛ ❧❛r❣❡ ♣r♦♣♦rt✐♦♥ ♦❢ t✐♠❡ ❛♥❞ ❡✛♦rt ❝❛♥ ❛♥❞ s❤♦✉❧❞ ❜❡ ❞✐r❡❝t❡❞ t♦✇❛r❞✿



✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❝♦♥❝❡♣ts ❛♥❞



♠♦❞❡❧✐♥❣ ✐♥ r❡❛❧✐st✐❝ s❡tt✐♥❣s✳

❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ t❤✐s ❛♣♣r♦❛❝❤ ✐s t❤❛t ✐t r❡q✉✐r❡s ♠♦r❡ ❛❜str❛❝t✐♦♥ r❛t❤❡r t❤❛♥ ❧❡ss✳ ❱✐s✉❛❧✐③❛t✐♦♥ ✐s t❤❡ ♠❛✐♥ t♦♦❧ ✉s❡❞ t♦ ❞❡❛❧ ✇✐t❤ t❤✐s ❝❤❛❧❧❡♥❣❡✳ ■❧❧✉str❛t✐♦♥s ❛r❡ ♣r♦✈✐❞❡❞ ❢♦r ❡✈❡r② ❝♦♥❝❡♣t✱ ❜✐❣ ♦r s♠❛❧❧✳ ❚❤❡ ♣✐❝t✉r❡s t❤❛t ❝♦♠❡ ♦✉t ❛r❡ s♦♠❡t✐♠❡s ✈❡r② ♣r❡❝✐s❡ ❜✉t s♦♠❡t✐♠❡s s❡r✈❡ ❛s ♠❡r❡ ♠❡t❛♣❤♦rs ❢♦r t❤❡ ❝♦♥❝❡♣ts t❤❡② ✐❧❧✉str❛t❡✳ ❚❤❡ ❤♦♣❡ ✐s t❤❛t t❤❡② ✇✐❧❧ s❡r✈❡ ❛s ✈✐s✉❛❧ ✏❛♥❝❤♦rs✑ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ✇♦r❞s ❛♥❞ ❢♦r♠✉❧❛s✳ ■t ✐s ✉♥❧✐❦❡❧② t❤❛t ❛ ♣❡rs♦♥ ✇❤♦ ❤❛s ♥❡✈❡r ♣❧♦tt❡❞ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❜② ❤❛♥❞ ❝❛♥ ✉♥❞❡rst❛♥❞ ❣r❛♣❤s ♦r ❢✉♥❝t✐♦♥s✳ ❍♦✇❡✈❡r✱ ✇❤❛t ✐❢ ✇❡ ✇❛♥t t♦ ♣❧♦t ♠♦r❡ t❤❛♥ ❥✉st ❛ ❢❡✇ ♣♦✐♥ts ✐♥ ♦r❞❡r t♦ ✈✐s✉❛❧✐③❡ ❝✉r✈❡s✱ s✉r❢❛❝❡s✱ ✈❡❝t♦r ✜❡❧❞s✱ ❡t❝✳❄

❙♣r❡❛❞s❤❡❡ts ✇❡r❡ ❝❤♦s❡♥ ♦✈❡r ❣r❛♣❤✐❝ ❝❛❧❝✉❧❛t♦rs ❢♦r ✈✐s✉❛❧✐③❛t✐♦♥ ♣✉r♣♦s❡s

❜❡❝❛✉s❡ t❤❡② r❡♣r❡s❡♥t t❤❡ s❤♦rt❡st st❡♣ ❛✇❛② ❢r♦♠ ♣❡♥ ❛♥❞ ♣❛♣❡r✳

■♥❞❡❡❞✱ t❤❡ ❞❛t❛ ✐s ♣❧♦tt❡❞ ✐♥ t❤❡

s✐♠♣❧❡st ♠❛♥♥❡r ♣♦ss✐❜❧❡✿ ♦♥❡ ❝❡❧❧ ✲ ♦♥❡ ♥✉♠❜❡r ✲ ♦♥❡ ♣♦✐♥t ♦♥ t❤❡ ❣r❛♣❤✳ ❋♦r ♠♦r❡ ❛❞✈❛♥❝❡❞ t❛s❦s s✉❝❤ ❛s ♠♦❞❡❧✐♥❣✱ s♣r❡❛❞s❤❡❡ts ✇❡r❡ ❝❤♦s❡♥ ♦✈❡r ♦t❤❡r s♦❢t✇❛r❡ ❛♥❞ ♣r♦❣r❛♠♠✐♥❣ ♦♣t✐♦♥s ❢♦r t❤❡✐r ✇✐❞❡ ❛✈❛✐❧❛❜✐❧✐t② ❛♥❞✱ ❛❜♦✈❡ ❛❧❧✱ t❤❡✐r s✐♠♣❧✐❝✐t②✳ ◆✐♥❡ ♦✉t ♦❢ t❡♥✱ t❤❡ s♣r❡❛❞s❤❡❡t s❤♦✇♥ ✇❛s ✐♥✐t✐❛❧❧② ❝r❡❛t❡❞ ❢r♦♠ s❝r❛t❝❤ ✐♥ ❢r♦♥t ♦❢ t❤❡ st✉❞❡♥ts ✇❤♦ ✇❡r❡ ❧❛t❡r ❛❜❧❡ t♦ ❢♦❧❧♦✇ ♠② ❢♦♦tst❡♣s ❛♥❞ ❝r❡❛t❡ t❤❡✐r ♦✇♥✳ ❆❜♦✉t t❤❡ t❡sts✳ ❚❤❡ ❜♦♦❦ ✐s♥✬t ❞❡s✐❣♥❡❞ t♦ ♣r❡♣❛r❡ t❤❡ st✉❞❡♥t ❢♦r s♦♠❡ ♣r❡❡①✐st✐♥❣ ❡①❛♠❀ ♦♥ t❤❡ ❝♦♥tr❛r②✱ ❛ss✐❣♥♠❡♥ts s❤♦✉❧❞ ❜❡ ❜❛s❡❞ ♦♥ ✇❤❛t ❤❛s ❜❡❡♥ ❧❡❛r♥❡❞✳ ❚❤❡ st✉❞❡♥ts✬ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❝♦♥❝❡♣ts ♥❡❡❞s t♦ ❜❡ t❡st❡❞ ❜✉t✱ ♠♦st ♦❢ t❤❡ t✐♠❡✱ t❤✐s ❝❛♥ ❜❡ ❞♦♥❡ ♦♥❧② ✐♥❞✐r❡❝t❧②✳ ❚❤❡r❡❢♦r❡✱ ❛ ❝❡rt❛✐♥ s❤❛r❡ ♦❢ r♦✉t✐♥❡✱ ♠❡❝❤❛♥✐❝❛❧ ♣r♦❜❧❡♠s ✐s ✐♥❡✈✐t❛❜❧❡✳ ◆♦♥❡t❤❡❧❡ss✱ ♥♦ t♦♣✐❝ ❞❡s❡r✈❡s ♠♦r❡ ❛tt❡♥t✐♦♥ ❥✉st ❜❡❝❛✉s❡ ✐t✬s ❧✐❦❡❧② t♦ ❜❡ ♦♥ t❤❡ t❡st✳ ■❢ ❛t ❛❧❧ ♣♦ss✐❜❧❡✱ ❞♦♥✬t ♠❛❦❡ t❤❡ st✉❞❡♥ts ♠❡♠♦r✐③❡ ❢♦r♠✉❧❛s✳ ■♥ t❤❡ ♦r❞❡r ♦❢ t♦♣✐❝s✱ t❤❡ ♠❛✐♥ ❞✐✛❡r❡♥❝❡ ❢r♦♠ ❛ t②♣✐❝❛❧ ❝❛❧❝✉❧✉s t❡①t❜♦♦❦ ✐s t❤❛t s❡q✉❡♥❝❡s ❝♦♠❡ ❜❡❢♦r❡ ❡✈❡r②t❤✐♥❣ ❡❧s❡✳ ❚❤❡ r❡❛s♦♥s ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿



❙❡q✉❡♥❝❡s ❛r❡ t❤❡ s✐♠♣❧❡st ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥s✳



▲✐♠✐ts ♦❢ s❡q✉❡♥❝❡s ❛r❡ s✐♠♣❧❡r t❤❛♥ ❧✐♠✐ts ♦❢ ❣❡♥❡r❛❧ ❢✉♥❝t✐♦♥s ✭✐♥❝❧✉❞✐♥❣ t❤❡ ♦♥❡s ❛t ✐♥✜♥✐t②✮✳



❚❤❡ s✐❣♠❛ ♥♦t❛t✐♦♥✱ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✱ ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ♠❛❦❡ ♠♦r❡ s❡♥s❡ t♦ ❛ st✉❞❡♥t ✇✐t❤



❆ q✉✐❝❦ tr❛♥s✐t✐♦♥ ❢r♦♠ s❡q✉❡♥❝❡s t♦ s❡r✐❡s ♦❢t❡♥ ❧❡❛❞s t♦ ❝♦♥❢✉s✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦✳



❙❡q✉❡♥❝❡s ❛r❡ ♥❡❡❞❡❞ ❢♦r ♠♦❞❡❧✐♥❣✱ ✇❤✐❝❤ s❤♦✉❧❞ st❛rt ❛s ❡❛r❧② ❛s ♣♦ss✐❜❧❡✳

❛ s♦❧✐❞ ❜❛❝❦❣r♦✉♥❞ ✐♥ s❡q✉❡♥❝❡s✳

❋r♦♠ t❤❡ ❞✐s❝r❡t❡ t♦ t❤❡ ❝♦♥t✐♥✉♦✉s



❋r♦♠ t❤❡ ❞✐s❝r❡t❡ t♦ t❤❡ ❝♦♥t✐♥✉♦✉s ■t✬s ♥♦ s❡❝r❡t t❤❛t ❛ ✈❛st ♠❛❥♦r✐t② ♦❢ ❝❛❧❝✉❧✉s st✉❞❡♥ts ✇✐❧❧ ♥❡✈❡r ✉s❡ ✇❤❛t t❤❡② ❤❛✈❡ ❧❡❛r♥❡❞✳ P♦♦r ❝❛r❡❡r ❝❤♦✐❝❡s ❛s✐❞❡✱ ❛ ❢♦r♠❡r ❝❛❧❝✉❧✉s st✉❞❡♥t ✐s ♦❢t❡♥ ✉♥❛❜❧❡ t♦ r❡❝♦❣♥✐③❡ t❤❡ ♠❛t❤❡♠❛t✐❝s t❤❛t ✐s s✉♣♣♦s❡❞ t♦ s✉rr♦✉♥❞ ❤✐♠✳ ❲❤② ❞♦❡s t❤✐s ❤❛♣♣❡♥❄ ❈❛❧❝✉❧✉s ✐s t❤❡ s❝✐❡♥❝❡ ♦❢ ❝❤❛♥❣❡✳ ❋r♦♠ t❤❡ ✈❡r② ❜❡❣✐♥♥✐♥❣✱ ✐ts ♣❡❝✉❧✐❛r ❝❤❛❧❧❡♥❣❡ ❤❛s ❜❡❡♥ t♦ st✉❞② ❛♥❞

❝♦♥t✐♥✉♦✉s ❝❤❛♥❣❡✿ ❝✉r✈❡s ❛♥❞ ♠♦t✐♦♥ ❛❧♦♥❣ ❝✉r✈❡s✳ ❢♦r♠✉❧❛s✳ ❙❦✐❧❧❢✉❧ ♠❛♥✐♣✉❧❛t✐♦♥ ♦❢ t❤♦s❡ ❢♦r♠✉❧❛s ✐s ✇❤❛t

♠❡❛s✉r❡

❚❤❡s❡ ❝✉r✈❡s ❛♥❞ t❤✐s ♠♦t✐♦♥ ❛r❡ r❡♣r❡s❡♥t❡❞

❜②

s♦❧✈❡s ❝❛❧❝✉❧✉s ♣r♦❜❧❡♠s✳ ❋♦r ♦✈❡r ✸✵✵ ②❡❛rs✱

t❤✐s ❛♣♣r♦❛❝❤ ❤❛s ❜❡❡♥ ❡①tr❡♠❡❧② s✉❝❝❡ss❢✉❧ ✐♥ s❝✐❡♥❝❡s ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣✳

❚❤❡ s✉❝❝❡ss❡s ❛r❡ ✇❡❧❧✲❦♥♦✇♥✿

♣r♦❥❡❝t✐❧❡ ♠♦t✐♦♥✱ ♣❧❛♥❡t❛r② ♠♦t✐♦♥✱ ✢♦✇ ♦❢ ❧✐q✉✐❞s✱ ❤❡❛t tr❛♥s❢❡r✱ ✇❛✈❡ ♣r♦♣❛❣❛t✐♦♥✱ ❡t❝✳ ❚❡❛❝❤✐♥❣ ❝❛❧❝✉❧✉s ❢♦❧❧♦✇s t❤✐s ❛♣♣r♦❛❝❤✿ ❆♥ ♦✈❡r✇❤❡❧♠✐♥❣ ♠❛❥♦r✐t② ♦❢ ✇❤❛t t❤❡ st✉❞❡♥t ❞♦❡s ✐s ♠❛♥✐♣✉❧❛t✐♦♥ ♦❢ ❢♦r♠✉❧❛s ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r✳ ❇✉t t❤✐s ♠❡❛♥s t❤❛t ❛❧❧ t❤❡ ♣r♦❜❧❡♠s t❤❡ st✉❞❡♥t ❢❛❝❡s ✇❡r❡ ✭♦r ❝♦✉❧❞ ❤❛✈❡ ❜❡❡♥✮ s♦❧✈❡❞ ✐♥ t❤❡ ✶✽t❤ ♦r ✶✾t❤ ❝❡♥t✉r✐❡s✦ ❚❤✐s ✐s♥✬t ❣♦♦❞ ❡♥♦✉❣❤ ❛♥②♠♦r❡✳ ❲❤❛t ❤❛s ❝❤❛♥❣❡❞ s✐♥❝❡ t❤❡♥❄ ❚❤❡ ❝♦♠♣✉t❡rs ❤❛✈❡ ❛♣♣❡❛r❡❞✱ ♦❢ ❝♦✉rs❡✱ ❛♥❞ ❝♦♠♣✉t❡rs ❞♦♥✬t ♠❛♥✐♣✉❧❛t❡ ❢♦r♠✉❧❛s✳

❚❤❡② ❞♦♥✬t ❤❡❧♣ ✇✐t❤ s♦❧✈✐♥❣ ✕ ✐♥ t❤❡ tr❛❞✐t✐♦♥❛❧ s❡♥s❡ ♦❢

t❤❡ ✇♦r❞ ✕ t❤♦s❡ ♣r♦❜❧❡♠s ❢r♦♠ t❤❡ ♣❛st ❝❡♥t✉r✐❡s✳

✐♥❝r❡♠❡♥t❛❧

■♥st❡❛❞ ♦❢

❝♦♥t✐♥✉♦✉s✱

❝♦♠♣✉t❡rs ❡①❝❡❧ ❛t ❤❛♥❞❧✐♥❣

♣r♦❝❡ss❡s✱ ❛♥❞ ✐♥st❡❛❞ ♦❢ ❢♦r♠✉❧❛s t❤❡② ❛r❡ ❣r❡❛t ❛t ♠❛♥❛❣✐♥❣ ❞✐s❝r❡t❡ ✭❞✐❣✐t❛❧✮ ❞❛t❛✳ ❚♦ ✉t✐❧✐③❡

t❤❡s❡ ❛❞✈❛♥t❛❣❡s✱ s❝✐❡♥t✐sts ✏❞✐s❝r❡t✐③❡✑ t❤❡ r❡s✉❧ts ♦❢ ❝❛❧❝✉❧✉s ❛♥❞ ❝r❡❛t❡ ❛❧❣♦r✐t❤♠s t❤❛t ♠❛♥✐♣✉❧❛t❡ t❤❡ ❞✐❣✐t❛❧ ❞❛t❛✳

❚❤❡ s♦❧✉t✐♦♥s ❛r❡ ❛♣♣r♦①✐♠❛t❡ ❜✉t t❤❡ ❛♣♣❧✐❝❛❜✐❧✐t② ✐s ✉♥❧✐♠✐t❡❞✳

❙✐♥❝❡ t❤❡ ✷✵t❤ ❝❡♥t✉r②✱

t❤✐s ❛♣♣r♦❛❝❤ ❤❛s ❜❡❡♥ ❡①tr❡♠❡❧② s✉❝❝❡ss❢✉❧ ✐♥ s❝✐❡♥❝❡s ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣✿ ❛❡r♦❞②♥❛♠✐❝s ✭❛✐r♣❧❛♥❡ ❛♥❞ ❝❛r ❞❡s✐❣♥✮✱ s♦✉♥❞ ❛♥❞ ✐♠❛❣❡ ♣r♦❝❡ss✐♥❣✱ s♣❛❝❡ ❡①♣❧♦r❛t✐♦♥✱ str✉❝t✉r❡ ♦❢ t❤❡ ❛t♦♠ ❛♥❞ t❤❡ ✉♥✐✈❡rs❡✱ ❡t❝✳ ❚❤❡ ❛♣♣r♦❛❝❤ ✐s ❛❧s♦ ❝✐r❝✉✐t♦✉s✿ ❊✈❡r② ❝♦♥❝❡♣t ✐♥ ❝❛❧❝✉❧✉s

st❛rts

✕ ♦❢t❡♥ ✐♠♣❧✐❝✐t❧② ✕ ❛s ❛ ❞✐s❝r❡t❡ ❛♣♣r♦①✐♠❛t✐♦♥

♦❢ ❛ ❝♦♥t✐♥✉♦✉s ♣❤❡♥♦♠❡♥♦♥✦

❈❛❧❝✉❧✉s ✐s t❤❡ s❝✐❡♥❝❡ ♦❢ ❝❤❛♥❣❡✱

❜♦t❤

✐♥❝r❡♠❡♥t❛❧ ❛♥❞ ❝♦♥t✐♥✉♦✉s✳ ❚❤❡ ❢♦r♠❡r ♣❛rt ✕ t❤❡ s♦✲❝❛❧❧❡❞ ❞✐s❝r❡t❡

❝❛❧❝✉❧✉s ✕ ♠❛② ❜❡ s❡❡♥ ❛s t❤❡ st✉❞② ♦❢ ✐♥❝r❡♠❡♥t❛❧ ♣❤❡♥♦♠❡♥❛ ❛♥❞ t❤❡ q✉❛♥t✐t✐❡s

✐♥❞✐✈✐s✐❜❧❡

❜② t❤❡✐r

✈❡r② ♥❛t✉r❡✿ ♣❡♦♣❧❡✱ ❛♥✐♠❛❧s✱ ❛♥❞ ♦t❤❡r ♦r❣❛♥✐s♠s✱ ♠♦♠❡♥ts ♦❢ t✐♠❡✱ ❧♦❝❛t✐♦♥s ♦❢ s♣❛❝❡✱ ♣❛rt✐❝❧❡s✱ s♦♠❡ ❝♦♠♠♦❞✐t✐❡s✱ ❞✐❣✐t❛❧ ✐♠❛❣❡s ❛♥❞ ♦t❤❡r ♠❛♥✲♠❛❞❡ ❞❛t❛✱ ❡t❝✳ ❲✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡ ❝❛❧❝✉❧✉s ♠❛❝❤✐♥❡r② ❝❛❧❧❡❞ ✏❧✐♠✐ts✑✱ ✇❡ ✐♥✈❛r✐❛❜❧② ❝❤♦♦s❡ t♦ tr❛♥s✐t✐♦♥ t♦ t❤❡ ❝♦♥t✐♥✉♦✉s ♣❛rt ♦❢ ❝❛❧❝✉❧✉s✱ ❡s♣❡❝✐❛❧❧② ✇❤❡♥ ✇❡ ❢❛❝❡ ❝♦♥t✐♥✉♦✉s ♣❤❡♥♦♠❡♥❛ ❛♥❞ t❤❡ q✉❛♥t✐t✐❡s

✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡

❡✐t❤❡r ❜② t❤❡✐r ♥❛t✉r❡ ♦r ❜② ❛ss✉♠♣t✐♦♥✿ t✐♠❡✱

s♣❛❝❡✱ ♠❛ss✱ t❡♠♣❡r❛t✉r❡✱ ♠♦♥❡②✱ s♦♠❡ ❝♦♠♠♦❞✐t✐❡s✱ ❡t❝✳ ❈❛❧❝✉❧✉s ♣r♦❞✉❝❡s ❞❡✜♥✐t✐✈❡ r❡s✉❧ts ❛♥❞ ❛❜s♦❧✉t❡ ❛❝❝✉r❛❝② ✕ ❜✉t ♦♥❧② ❢♦r ♣r♦❜❧❡♠s ❛♠❡♥❛❜❧❡ t♦ ✐ts ♠❡t❤♦❞s✦ ■♥ t❤❡ ❝❧❛ssr♦♦♠✱ t❤❡ ♣r♦❜❧❡♠s ❛r❡ s✐♠♣❧✐✜❡❞ ✉♥t✐❧ t❤❡② ❜❡❝♦♠❡ ♠❛♥❛❣❡❛❜❧❡❀ ♦t❤❡r✇✐s❡✱ ✇❡ ❝✐r❝❧❡ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡ ♠❡t❤♦❞s ✐♥ s❡❛r❝❤ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❲✐t❤✐♥ ❛ t②♣✐❝❛❧ ❝❛❧❝✉❧✉s ❝♦✉rs❡✱ t❤❡ st✉❞❡♥t s✐♠♣❧② ♥❡✈❡r ❣❡ts t♦ ❝♦♠♣❧❡t❡ t❤❡ ✏❝✐r❝❧❡✑✦

▲❛t❡r ♦♥✱ t❤❡

❣r❛❞✉❛t❡ ✐s ❧✐❦❡❧② t♦ t❤✐♥❦ ♦❢ ❝❛❧❝✉❧✉s ♦♥❧② ✇❤❡♥ ❤❡ s❡❡s ❢♦r♠✉❧❛s ❛♥❞ r❛r❡❧② ✇❤❡♥ ❤❡ s❡❡s ♥✉♠❡r✐❝❛❧ ❞❛t❛✳ ■♥ t❤✐s ❜♦♦❦✱ ❡✈❡r② ❝♦♥❝❡♣t ♦❢ ❝❛❧❝✉❧✉s ✐s ✜rst ✐♥tr♦❞✉❝❡❞ ✐♥ ✐ts ❞✐s❝r❡t❡✱ ✏♣r❡✲❧✐♠✐t✑✱ ✐♥❝❛r♥❛t✐♦♥ ✕ ❡❧s❡✇❤❡r❡ t②♣✐❝❛❧❧② ❤✐❞❞❡♥ ✐♥s✐❞❡ ♣r♦♦❢s ✕ ❛♥❞ t❤❡♥ ✉s❡❞ ❢♦r ♠♦❞❡❧✐♥❣ ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ✇❡❧❧ ❜❡❢♦r❡ ✐ts ❝♦♥t✐♥✉♦✉s ❝♦✉♥t❡r♣❛rt ❡♠❡r❣❡s✳ ❚❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❢♦r♠❡r ❛r❡ ❞✐s❝♦✈❡r❡❞ ✜rst ❛♥❞ t❤❡♥ t❤❡ ♠❛t❝❤✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❧❛tt❡r ❛r❡ ❢♦✉♥❞ ❜② ♠❛❦✐♥❣ t❤❡ ✐♥❝r❡♠❡♥t s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r✱ ❛t t❤❡ ❞✐s❝r❡t❡ ❝❛❧❝✉❧✉s

∆x→0

−−−−−−−−−−→

❧✐♠✐t ✿

❝♦♥t✐♥✉♦✉s ❝❛❧❝✉❧✉s

❚❤❡ ✈♦❧✉♠❡ ❛♥❞ ❝❤❛♣t❡r r❡❢❡r❡♥❝❡s ❢♦r ❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞



❚❤❡ ✈♦❧✉♠❡ ❛♥❞ ❝❤❛♣t❡r r❡❢❡r❡♥❝❡s ❢♦r ❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞ ❚❤✐s ❜♦♦❦ ✐s ❛ ♣❛rt ♦❢ t❤❡ s❡r✐❡s ❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❚❤❡ s❡r✐❡s ❝♦✈❡rs t❤❡ st❛♥❞❛r❞ ♠❛t❡r✐❛❧ ♦❢ t❤❡ ✉♥❞❡r✲ ❣r❛❞✉❛t❡ ❝❛❧❝✉❧✉s ✇✐t❤ ❛ s✉❜st❛♥t✐❛❧ r❡✈✐❡✇ ♦❢ ♣r❡❝❛❧❝✉❧✉s ❛♥❞ ❛ ♣r❡✈✐❡✇ ♦❢ ❡❧❡♠❡♥t❛r② ♦r❞✐♥❛r② ❛♥❞ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ❇❡❧♦✇ ✐s t❤❡ ❧✐st ♦❢ t❤❡ ❜♦♦❦s ♦❢ t❤❡ s❡r✐❡s✱ t❤❡✐r ❝❤❛♣t❡rs✱ ❛♥❞ t❤❡ ✇❛② t❤❡ ♣r❡s❡♥t ❜♦♦❦ ✭♣❛r❡♥t❤❡t✐❝❛❧❧②✮ r❡❢❡r❡♥❝❡s t❤❡♠✳ 

✶ P❈✲✶ ✶ P❈✲✷ ✶ P❈✲✸ ✶ P❈✲✹ ✶ P❈✲✺



✷ ❉❈✲✶ ✷ ❉❈✲✷ ✷ ❉❈✲✸ ✷ ❉❈✲✹ ✷ ❉❈✲✺ ✷ ❉❈✲✻

❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❱♦❧✉♠❡ ✹✿ ❈❛❧❝✉❧✉s ✐♥ ❍✐❣❤❡r ❉✐♠❡♥s✐♦♥s

❋✉♥❝t✐♦♥s ✐♥ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡s P❛r❛♠❡tr✐❝ ❝✉r✈❡s ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❚❤❡ ❣r❛❞✐❡♥t ❚❤❡ ✐♥t❡❣r❛❧ ❱❡❝t♦r ✜❡❧❞s 

✺ ❉❊✲✶ ✺ ❉❊✲✷ ✺ ❉❊✲✸ ✺ ❉❊✲✹ ✺ ❉❊✲✺ ✺ ❉❊✲✻

❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❱♦❧✉♠❡ ✸✿ ■♥t❡❣r❛❧ ❈❛❧❝✉❧✉s

❚❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ■♥t❡❣r❛t✐♦♥ ❲❤❛t ✇❡ ❝❛♥ s♦ ✇✐t❤ ✐♥t❡❣r❛❧ ❝❛❧❝✉❧✉s ❙❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❙❡r✐❡s 

✹ ❍❉✲✶ ✹ ❍❉✲✷ ✹ ❍❉✲✸ ✹ ❍❉✲✹ ✹ ❍❉✲✺ ✹ ❍❉✲✻

❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❱♦❧✉♠❡ ✷✿ ❉✐✛❡r❡♥t✐❛❧ ❈❛❧❝✉❧✉s

▲✐♠✐ts ♦❢ s❡q✉❡♥❝❡s ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ❚❤❡ ❞❡r✐✈❛t✐✈❡ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❚❤❡ ♠❛✐♥ t❤❡♦r❡♠s ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s ❲❤❛t ✇❡ ❝❛♥ ❞♦ ✇✐t❤ ❝❛❧❝✉❧✉s 

✸ ■❈✲✶ ✸ ■❈✲✷ ✸ ■❈✲✸ ✸ ■❈✲✹ ✸ ■❈✲✺

❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❱♦❧✉♠❡ ✶✿ Pr❡❝❛❧❝✉❧✉s

❈❛❧❝✉❧✉s ♦❢ s❡q✉❡♥❝❡s ❙❡ts ❛♥❞ ❢✉♥❝t✐♦♥s ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s ❈❧❛ss❡s ♦❢ ❢✉♥❝t✐♦♥s ❆❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr②

❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❱♦❧✉♠❡ ✺✿ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s

❖r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❱❡❝t♦r ✈❛r✐❛❜❧❡s ❱❡❝t♦r ❛♥❞ ❝♦♠♣❧❡① ✈❛r✐❛❜❧❡s ❙②st❡♠s ♦❢ ❖❉❊s ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❖❉❊s P❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s

❊❛❝❤ ✈♦❧✉♠❡ ❝❛♥ ❜❡ r❡❛❞ ✐♥❞❡♣❡♥❞❡♥t❧②✳

❚❤❡ ✈♦❧✉♠❡ ❛♥❞ ❝❤❛♣t❡r r❡❢❡r❡♥❝❡s ❢♦r ❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞



❆ ♣♦ss✐❜❧❡ s❡q✉❡♥❝❡ ♦❢ ❝❤❛♣t❡rs ✐s ♣r❡s❡♥t❡❞ ❜❡❧♦✇✳ ❆♥ ❛rr♦✇ ❢r♦♠ ❆ t♦ ❇ ♠❡❛♥s t❤❛t ❝❤❛♣t❡r ❇ s❤♦✉❧❞♥✬t ❜❡ r❡❛❞ ❜❡❢♦r❡ ❝❤❛♣t❡r ❆✳

❆❜♦✉t t❤❡ ❛✉t❤♦r

❆❜♦✉t t❤❡ ❛✉t❤♦r P❡t❡r ❙❛✈❡❧✐❡✈ ✐s ❛ ♣r♦❢❡ss♦r ♦❢ ♠❛t❤❡♠❛t✐❝s ❛t ▼❛rs❤❛❧❧ ❯♥✐✈❡rs✐t②✱ ❍✉♥t✲ ✐♥❣t♦♥✱ ❲❡st ❱✐r❣✐♥✐❛✱ ❯❙❆✳ ❆❢t❡r ❛ P❤✳❉✳ ❢r♦♠ t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ■❧❧✐♥♦✐s ❛t ❯r❜❛♥❛✲❈❤❛♠♣❛✐❣♥✱ ❤❡ ❞❡✈♦t❡❞ t❤❡ ♥❡①t ✷✵ ②❡❛rs t♦ t❡❛❝❤✐♥❣ ♠❛t❤❡♠❛t✐❝s✳ P❡t❡r ✐s t❤❡ ❛✉t❤♦r ♦❢ ❛ ❣r❛❞✉❛t❡ t❡①t❜♦♦❦ ❚♦♣♦❧♦❣② ■❧❧✉str❛t❡❞ ♣✉❜❧✐s❤❡❞ ✐♥ ✷✵✶✻✳ ❍❡ ❤❛s ❛❧s♦ ❜❡❡♥ ✐♥✈♦❧✈❡❞ ✐♥ r❡s❡❛r❝❤ ✐♥ ❛❧❣❡❜r❛✐❝ t♦♣♦❧♦❣② ❛♥❞ s❡✈❡r❛❧ ♦t❤❡r ✜❡❧❞s✳ ❍✐s ♥♦♥✲❛❝❛❞❡♠✐❝ ♣r♦❥❡❝ts ❤❛✈❡ ❜❡❡♥✿ ❞✐❣✐t❛❧ ✐♠❛❣❡ ❛♥❛❧②s✐s✱ ❛✉t♦♠❛t❡❞ ✜♥❣❡r♣r✐♥t ✐❞❡♥t✐✜❝❛t✐♦♥✱ ❛♥❞ ✐♠❛❣❡ ♠❛t❝❤✐♥❣ ❢♦r ♠✐s✲ s✐❧❡ ♥❛✈✐❣❛t✐♦♥✴❣✉✐❞❛♥❝❡✳



❈♦♥t❡♥ts Pr❡❢❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 

❈❤❛♣t❡r ✶✿ ❖r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✶ ■♥❝r❡♠❡♥t❛❧ ♠♦t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ❉✐s❝r❡t❡ ❢♦r♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ❉✐✛❡r❡♥t✐❛❧ ❢♦r♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺ ❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✻ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✼ ❚❤❡ ♠❡t❤♦❞ ♦❢ ✐♥t❡❣r❛t✐♥❣ ❢❛❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✽ ❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✾ ❊✉❧❡r✬s ♠❡t❤♦❞✿ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✵ ❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄ ✶✳✶✶ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✷ ▲✐♥❡❛r✐③❛t✐♦♥ ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✸ ▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✹ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✺ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



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✶✷ ✶✼ ✷✽ ✸✼ ✹✸ ✻✶ ✻✺ ✻✼ ✼✸ ✼✾ ✽✽ ✾✻ ✶✵✵ ✶✵✻ ✶✶✻

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✶✷✻ ✶✸✸ ✶✹✶ ✶✹✾ ✶✻✷ ✶✻✽ ✶✼✾ ✶✽✹ ✶✾✶

❈❤❛♣t❡r ✸✿ ❱❡❝t♦r ❛♥❞ ❝♦♠♣❧❡① ✈❛r✐❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✽ ✸✳✶ ❆❧❣❡❜r❛ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛♥❞ ♠❛tr✐❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✸ ❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✺ ❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ R2 ✳ ✳ ✸✳✻ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿ C ✐s♥✬t ❥✉st R2 ✸✳✼ ❈♦♠♣❧❡① ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✽ ❈♦♠♣❧❡① ❧✐♥❡❛r ♦♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✾ ▲✐♥❡❛r ♦♣❡r❛t♦rs ✇✐t❤ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s ✳ ✳ ✳ ✳ ✳ ✸✳✶✵ ❈♦♠♣❧❡① ❝❛❧❝✉❧✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✶ ❙❡r✐❡s ❛♥❞ ♣♦✇❡r s❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✷ ❙♦❧✈✐♥❣ ❖❉❊s ✇✐t❤ ♣♦✇❡r s❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



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❈❤❛♣t❡r ✷✿ ❱❡❝t♦r ✈❛r✐❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✻ ✷✳✶ ❲❤❡r❡ ♠❛tr✐❝❡s ❝♦♠❡ ❢r♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ ▲✐♥❡❛r ♦♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹ ❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺ ❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻ ■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✼ ❚❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ ❡✐❣❡♥✈❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✽ ❇❛s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✾ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❡✐❣❡♥✈❛❧✉❡s





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✶✾✽ ✷✵✹ ✷✵✾ ✷✶✻ ✷✶✾ ✷✷✸ ✷✷✽ ✷✸✸ ✷✸✻ ✷✹✷ ✷✹✹ ✷✺✵

❈❤❛♣t❡r ✹✿ ❙②st❡♠s ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺✹

✹✳✶ P❛r❛♠❡tr✐❝ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺✹ ✹✳✷ ❚❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻✸

❈♦♥t❡♥ts

✶✶

✹✳✸ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧ ✹✳✹ ❙♦❧✈✐♥❣ t❤❡ ▲♦t❦❛✕❱♦❧t❡rr❛ ❡q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✺ ❱❡❝t♦r ✜❡❧❞s ❛♥❞ s②st❡♠s ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✻ ❉✐s❝r❡t❡ s②st❡♠s ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✼ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ s②st❡♠s ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✹✳✽ ❚❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥ ❛♥❞ ❧✐♥❡❛r s②st❡♠s ✳ ✳ ✳ ✳ ✳ ✹✳✾ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✶✵ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞ ✳ ✳ ✳



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✷✻✼ ✷✼✵ ✷✼✸ ✷✼✾ ✷✽✸ ✷✽✼ ✷✾✸ ✷✾✽

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✸✵✻

✳ ✳ ✳ ✳ ✳ ✳ ✻✳✶ ❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✷ ❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ ♣❡r♠❡❛❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✸ ❍❡❛t tr❛♥s❢❡r ✐s ❝❛✉s❡❞ ❜② t❡♠♣❡r❛t✉r❡ ❞✐✛❡r❡♥❝❡s ✳ ✳ ✻✳✹ ❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ t❤❡ ❣❡♦♠❡tr② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✺ ❚❤❡ ❤❡❛t P❉❊ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✻ ❈❡❧❧s ❛♥❞ ❢♦r♠s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✼ ❍❡❛t tr❛♥s❢❡r ✐♥ ❞✐♠❡♥s✐♦♥ 2✿ ❛ ♣❧❛t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✽ ❚❤❡ ❤❡❛t P❉❊ ❢♦r ❞✐♠❡♥s✐♦♥ 2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✾ ❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥ 1✿ s♣r✐♥❣s ❛♥❞ str✐♥❣s ✻✳✶✵ ❚❤❡ ✇❛✈❡ P❉❊ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✶✶ ❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥ 2✿ ❛ ♠❡♠❜r❛♥❡ ✳ ✳ ✳

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✸✹✸

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✹✶✵

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✷✷

❈❤❛♣t❡r ✺✿ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❖❉❊s ✺✳✶ ✺✳✷ ✺✳✸ ✺✳✹ ✺✳✺ ✺✳✻ ✺✳✼ ✺✳✽ ✺✳✾



✳ ✳ ❱❡❝t♦r✲✈❛❧✉❡❞ ❢♦r♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚❤❡ ♣✉rs✉✐t ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❛s s②st❡♠s ✳ ✳ ✳ ❱❡❝t♦r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r✿ ❛ ❞♦✉❜❧❡ ❆ ♣❡♥❞✉❧✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ P❧❛♥❡t❛r② ♠♦t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚❤❡ t✇♦✲ ❛♥❞ t❤r❡❡✲❜♦❞② ♣r♦❜❧❡♠s ✳ ✳ ❆ ❝❛♥♥♦♥ ✐s ✜r❡❞✳✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ s♣r✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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❈❤❛♣t❡r ✻✿ P❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s

❊①❡r❝✐s❡s ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ❊①❡r❝✐s❡s✿ ❇❛s✐❝s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❊①❡r❝✐s❡s✿ ❆♥❛❧②t✐❝❛❧ ♠❡t❤♦❞s ✳ ✳ ✳ ✳ ✳ ✳ ✸ ❊①❡r❝✐s❡s✿ ❊✉❧❡r✬s ♠❡t❤♦❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ❊①❡r❝✐s❡s✿ ●❡♥❡r❛❧✐t✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ❊①❡r❝✐s❡s✿ ▼♦❞❡❧s ❛♥❞ s❡tt✐♥❣ ✉♣ ❖❉❊s ✻ ❊①❡r❝✐s❡s✿ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ✳ ✳ ✳ ✳ ✳ ✳ ✼ ❊①❡r❝✐s❡s✿ ❙②st❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ❊①❡r❝✐s❡s✿ ❙❡❝♦♥❞ ♦r❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ❊①❡r❝✐s❡s✿ ❆❞✈❛♥❝❡❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ❊①❡r❝✐s❡s✿ P❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ❊①❡r❝✐s❡s✿ ❈♦♠♣✉t✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

■♥❞❡①

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✵✻ ✸✵✼ ✸✶✷ ✸✶✺ ✸✷✶ ✸✷✹ ✸✸✵ ✸✸✻ ✸✸✾

✸✹✸ ✸✺✸ ✸✻✵ ✸✻✺ ✸✼✵ ✸✼✺ ✸✽✸ ✸✾✶ ✸✾✸ ✹✵✵ ✹✵✺ ✹✶✵ ✹✶✷ ✹✶✸ ✹✶✹ ✹✶✺ ✹✶✻ ✹✶✼ ✹✶✽ ✹✶✾ ✹✷✵ ✹✷✶

❈❤❛♣t❡r ✶✿ ❖r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s

❈♦♥t❡♥ts

✶✳✶ ■♥❝r❡♠❡♥t❛❧ ♠♦t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ❉✐s❝r❡t❡ ❢♦r♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ❉✐✛❡r❡♥t✐❛❧ ❢♦r♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺ ❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✻ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✼ ❚❤❡ ♠❡t❤♦❞ ♦❢ ✐♥t❡❣r❛t✐♥❣ ❢❛❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✽ ❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✾ ❊✉❧❡r✬s ♠❡t❤♦❞✿ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✵ ❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄ ✶✳✶✶ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✷ ▲✐♥❡❛r✐③❛t✐♦♥ ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✸ ▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✹ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✺ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✶✳ ■♥❝r❡♠❡♥t❛❧ ♠♦t✐♦♥

❲❡ ❝❛♥ ❡❛s✐❧② ❞❡r✐✈❡ t❤❡ s♣❡❡❞ ❢r♦♠ t❤❡ ❞✐st❛♥❝❡ t❤❛t ✇❡ ❤❛✈❡ ❝♦✈❡r❡❞✿

❖♥ t❤❡ ✢✐♣ s✐❞❡✱ t❤❡ ❞❡r✐✈❡ t❤❡ ❞✐st❛♥❝❡ ✇❡ ❤❛✈❡ ❝♦✈❡r❡❞ ❢r♦♠ t❤❡ ❦♥♦✇♥ ✈❡❧♦❝✐t②✿

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✶✷ ✳ ✶✼ ✳ ✷✽ ✳ ✸✼ ✳ ✹✸ ✳ ✻✶ ✳ ✻✺ ✳ ✻✼ ✳ ✼✸ ✳ ✼✾ ✳ ✽✽ ✳ ✾✻ ✳ ✶✵✵ ✳ ✶✵✻ ✳ ✶✶✻

✶✳✶✳

■♥❝r❡♠❡♥t❛❧ ♠♦t✐♦♥

✶✸

❚❤❡ t✇♦ ♣r♦❜❧❡♠s ❛r❡ s♦❧✈❡❞✱ r❡s♣❡❝t✐✈❡❧②✱ ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡s❡ t✇♦ ✈❡rs✐♦♥s ♦❢ t❤❡ s❛♠❡ ❡❧❡♠❡♥t❛r② s❝❤♦♦❧ ❢♦r♠✉❧❛✿ s♣❡❡❞

=

❞✐st❛♥❝❡

/

t✐♠❡

❛♥❞

❞✐st❛♥❝❡

=

❲❤❛t t❛❦❡s t❤✐s ✐❞❡❛ ❜❡②♦♥❞ ❡❧❡♠❡♥t❛r② s❝❤♦♦❧ ✐s t❤❡ ♣♦ss✐❜✐❧✐t② t❤❛t

s♣❡❡❞

×

t✐♠❡

✈❡❧♦❝✐t② ✈❛r✐❡s✳

. ❚❤❡ s✐♠♣❧❡st ❝❛s❡ ✐s

✇❤❡♥ ✐t ✈❛r✐❡s ✐♥❝r❡♠❡♥t❛❧❧②✳ ❲❡ ♥❡①t ❝♦♥s✐❞❡r ♠♦r❡ ❝♦♠♣❧❡① ❡①❛♠♣❧❡s ♦❢ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❧♦❝❛t✐♦♥ ❛♥❞ ✈❡❧♦❝✐t②✳ ❋✐rst✱

t♦ ✈❡❧♦❝✐t②✳✳✳

❙✉♣♣♦s❡ t❤❛t t❤✐s t✐♠❡ ✇❡ ❤❛✈❡ ❛

s❡q✉❡♥❝❡

♦❢ ♠♦r❡ t❤❛♥

❢r♦♠ ❧♦❝❛t✐♦♥

30 ❞❛t❛ ♣♦✐♥ts ✭♠♦r❡ ✐s ✐♥❞✐❝❛t❡❞ ❜② ✏✳✳✳✑✮❀

t❤❡② ❛r❡

t❤❡ ❧♦❝❛t✐♦♥s ♦❢ ❛ ♠♦✈✐♥❣ ♦❜❥❡❝t r❡❝♦r❞❡❞ ❡✈❡r② ♠✐♥✉t❡✿ t✐♠❡

♠✐♥✉t❡s

❧♦❝❛t✐♦♥

♠✐❧❡s

0 1 2 3 4 5 6 7 8 9 10 ... 0.00 0.10 0.20 0.30 0.39 0.48 0.56 0.64 0.72 0.78 0.84 ...

❚❤✐s ❞❛t❛ ✐s ❛❧s♦ s❡❡♥ ✐♥ t❤❡ ✜rst t✇♦ ❝♦❧✉♠♥s ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t ✭❧❡❢t✮✿

❊✈❡r② ♣❛✐r ♦❢ ♥✉♠❜❡rs ✐♥ t❤❡ t❛❜❧❡ ✐s t❤❡♥ ♣❧♦tt❡❞ ✭r✐❣❤t✮✳ ♦♥❡✲♠✐♥✉t❡ ✐♥t❡r✈❛❧s✱ ✇❡ ❝♦♠♣✉t❡ t❤❡

❞✐✛❡r❡♥❝❡s

❚♦ ✉♥❞❡rst❛♥❞ ❤♦✇ ❢❛st ✇❡ ♠♦✈❡ ♦✈❡r t❤❡s❡

♦❢ ❧♦❝❛t✐♦♥s ❢♦r ❡❛❝❤ ♣❛✐r ♦❢ ❝♦♥s❡❝✉t✐✈❡ ❧♦❝❛t✐♦♥s✳

❋✐rst✱ t❤❡ t❛❜❧❡✳ ❲❡ ✉s❡ t❤❡ ❞❛t❛ ❢r♦♠ t❤❡ r♦✇ ♦❢ ❧♦❝❛t✐♦♥s✳ ❚❤✐s ✐s ❤♦✇ t❤❡ ✜rst ♦♥❡ ✐s ❝♦♠♣✉t❡❞✿ t✐♠❡

♠✐♥

❧♦❝❛t✐♦♥

♠✐❧❡s

❞✐✛❡r❡♥❝❡ ✈❡❧♦❝✐t②

♠✐❧❡s✴♠✐♥

0 0.00 ց

1 0.10 ↓ 0.10 − 0.00 || 0.10

... ... ... ...

❲❡ ❝♦♠♣✉t❡ t❤✐s ❞✐✛❡r❡♥❝❡ ❢♦r ❡❛❝❤ ♣❛✐r ♦❢ ❝♦♥s❡❝✉t✐✈❡ ❧♦❝❛t✐♦♥s ❛♥❞ t❤❡♥ ♣❧❛❝❡ ✐t ✐♥ ❛ r♦✇ ❢♦r t❤❡ ✈❡❧♦❝✐t✐❡s

✶✳✶✳ ■♥❝r❡♠❡♥t❛❧ ♠♦t✐♦♥

✶✹

t❤❛t ✇❡ ❝r❡❛t❡❞ ❛t t❤❡ ❜♦tt♦♠ ♦❢ ♦✉r t❛❜❧❡✿ t✐♠❡

♠✐♥

❧♦❝❛t✐♦♥

♠✐❧❡s

✈❡❧♦❝✐t②

♠✐❧❡s✴♠✐♥

0 0.00

1 2 3 4 5 6 7 8 9 0.10 0.20 0.30 0.39 0.48 0.56 0.64 0.72 0.78 ց ↓ ց ↓ ց ↓ ց ↓ ց ↓ ց ↓ ց ↓ ց ↓ ց ↓ 0.10 0.10 0.10 0.09 0.09 0.09 0.08 0.07 0.07

... ... ... ...

Pr❛❝t✐❝❛❧❧②✱ ✇❡✬❞ r❛t❤❡r ✉s❡ t❤❡ ❝♦♠♣✉t✐♥❣ ❝❛♣❛❜✐❧✐t✐❡s ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t✳ ❊①❛♠♣❧❡ ✶✳✶✳✶✿ s♣r❡❛❞s❤❡❡t ❢♦r♠✉❧❛s

❲❡ ✉s❡ ❢♦r♠✉❧❛s t♦ ♣✉❧❧ ❞❛t❛ ❢r♦♠ ♦t❤❡r ❝❡❧❧s✳ ❚❤❡r❡ ❛r❡ t✇♦ ✇❛②s✳ ❋✐rst✱ t❤❡ ✏❛❜s♦❧✉t❡✧ r❡❢❡r❡♥❝❡✿

❂❘✷❈✸✂✷ ❆♥② ❝❡❧❧ ✇✐t❤ t❤✐s ❢♦r♠✉❧❛ ✇✐❧❧ t❛❦❡ t❤❡ ✈❛❧✉❡ ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ ❝❡❧❧ ❧♦❝❛t❡❞ ❛t r♦✇

2

❛♥❞ ❝♦❧✉♠♥

3

❛♥❞

sq✉❛r❡ ✐t✿

❙❡❝♦♥❞✱ t❤❡ ✏r❡❧❛t✐✈❡✧ r❡❢❡r❡♥❝❡✿

❂❘❬✷❪❈❬✸❪✂✷ ❆♥② ❝❡❧❧ ✇✐t❤ t❤✐s ❢♦r♠✉❧❛ ✇✐❧❧ t❛❦❡ t❤❡ ✈❛❧✉❡ ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ ❝❡❧❧ ❧♦❝❛t❡❞

2 r♦✇s ❞♦✇♥ ❛♥❞ 3 ❝♦❧✉♠♥s

r✐❣❤t ❢r♦♠ ✐t ❛♥❞ sq✉❛r❡ ✐t✿

❲❡ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡s ❜② ♣✉❧❧✐♥❣ ❞❛t❛ ❢r♦♠ t❤❡ ❝♦❧✉♠♥ ♦❢ ❧♦❝❛t✐♦♥s ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛✿

❂❘❈❬✲✶❪✲❘❬✲✶❪❈❬✲✶❪ ❍❡r❡✱ t❤❡ t✇♦ ✈❛❧✉❡s ❝♦♠❡ ❢r♦♠ t❤❡ ❧❛st ❝♦❧✉♠♥✱

❈❬✲✶❪ ✱

s❛♠❡ r♦✇✱

❘✱

❛♥❞ ❧❛st r♦✇✱

❘❬✲✶❪ ✳

❇❡❧♦✇✱ ②♦✉

❝❛♥ s❡❡ t❤❡ t✇♦ r❡❢❡r❡♥❝❡s ✐♥ t❤❡ ❢♦r♠✉❧❛s ♠❛r❦❡❞ ✇✐t❤ r❡❞ ❛♥❞ ❜❧✉❡ ✭❧❡❢t✮ ❛♥❞ t❤❡ ❞❡♣❡♥❞❡♥❝❡ s❤♦✇♥ ✇✐t❤ t❤❡ ❛rr♦✇s ✭r✐❣❤t✮✿

✶✳✶✳ ■♥❝r❡♠❡♥t❛❧ ♠♦t✐♦♥

✶✺

❲❡ ♣❧❛❝❡ t❤❡ r❡s✉❧t ✐♥ ❛ ♥❡✇ ❝♦❧✉♠♥ ✇❡ ❝r❡❛t❡❞ ❢♦r t❤❡ ✈❡❧♦❝✐t✐❡s✿

❚❤✐s ♥❡✇ ❞❛t❛ ✐s ✐❧❧✉str❛t❡❞ ✇✐t❤ t❤❡ s❡❝♦♥❞ s❝❛tt❡r ♣❧♦t✳ ❚♦ ❡♠♣❤❛s✐③❡ t❤❡ ❢❛❝t t❤❛t t❤❡ ✈❡❧♦❝✐t② ❞❛t❛✱ ✉♥❧✐❦❡ t❤❡ ❧♦❝❛t✐♦♥✱ ✐s r❡❢❡rr✐♥❣ t♦ t✐♠❡ ✐♥t❡r✈❛❧s r❛t❤❡r t❤❛♥ t✐♠❡ ✐♥st❛♥❝❡s✱ ✇❡ ♣❧♦t ✐t ✇✐t❤ ❤♦r✐③♦♥t❛❧ s❡❣♠❡♥ts✳ ■♥ ❢❛❝t✱ t❤❡ ❞❛t❛ t❛❜❧❡ ❝❛♥ ❜❡ r❡❛rr❛♥❣❡❞ ❛s ❢♦❧❧♦✇s t♦ ♠❛❦❡ t❤✐s ♣♦✐♥t ❝❧❡❛r❡r✿ t✐♠❡ ❧♦❝❛t✐♦♥ ✈❡❧♦❝✐t②

0 1 2 3 4 ... 0.00 − 0.10 − 0.20 − 0.30 − .39 − ... · 0.10 · 0.10 · 0.10 · 0.09 · 0.09 ...

■♥ ❛ ♠♦r❡ ❣❡♥❡r❛❧ s❡tt✐♥❣✱ t❤❡ t✐♠❡ ♣r♦❣r❡ss❡s ✐♥ ✐♥❝r❡♠❡♥ts✳ ▲❡t✬s ❝❛❧❧ ✐t h✳ ❲❡ ❝❛♥ t❤✐♥❦ ♦❢ ✐t ❛s ❞❡♣❡♥❞❡♥t ♦♥ t✐♠❡ ✐❢ ♥❡❡❞❡❞✳ ■❢ ❛ ❢✉♥❝t✐♦♥ f r❡♣r❡s❡♥ts t❤❡ ❧♦❝❛t✐♦♥s ❛t t❤❡ t✐♠❡ ♠♦♠❡♥ts a, a + h, a + 2h, ...✱ t❤❡♥ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ♦✈❡r ❡❛❝❤ ♣❡r✐♦❞ ♦❢ t✐♠❡ [t, t + h] ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣r❡ss✐♦♥✿ ∆f = f (t + h) − f (t)

❚❤✐s ✐s ❛ ❢✉♥❝t✐♦♥ ✐s ❝❛❧❧❡❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ f ✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ✈❡❧♦❝✐t② ♦✈❡r ❡❛❝❤ ♣❡r✐♦❞ ♦❢ t✐♠❡ [t, t + h] ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣r❡ss✐♦♥✿ f (t + h) − f (t) ∆f = ∆t h

❚❤✐s ✐s ❛ ❢✉♥❝t✐♦♥ ✐s ❝❛❧❧❡❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ✳ ❆ ♠♦r❡ ❝❤❛❧❧❡♥❣✐♥❣ ❛♥❞ ♠♦r❡ ✐♠♣♦rt❛♥t tr❛♥s✐t✐♦♥ ✐s ❢r♦♠ ✈❡❧♦❝✐t② t♦ ❧♦❝❛t✐♦♥✳✳✳ ❆❣❛✐♥✱ ✇❡ ❝♦♥s✐❞❡r 30 ❞❛t❛ ♣♦✐♥ts✳ ❚❤❡s❡ ♥✉♠❜❡rs ❛r❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ✈❡❧♦❝✐t② ♦❢ ❛♥ ♦❜❥❡❝t r❡❝♦r❞❡❞ ❡✈❡r② ♠✐♥✉t❡✿ t✐♠❡ ✈❡❧♦❝✐t②

♠✐♥✉t❡s ♠✐❧❡s✴❤♦✉r

0 1 2 3 4 5 6 7 8 9 10 ... 0.10 0.20 0.30 0.39 0.48 0.56 0.64 0.72 0.78 0.84 ...

❚❤✐s ❞❛t❛ ✐s ❛❧s♦ s❡❡♥ ✐♥ t❤❡ ✜rst t✇♦ ❝♦❧✉♠♥s ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t ♣❧♦tt❡❞ ♦♥❡ ❜❛r ❛t ❛ t✐♠❡✿

✶✳✶✳

✶✻

■♥❝r❡♠❡♥t❛❧ ♠♦t✐♦♥

❚❤✐s ♣❛rt ❞❛t❛ ✐s ❢✉rt❤❡r♠♦r❡ ✐❧❧✉str❛t❡❞ ❛s ❛ s❝❛tt❡r ♣❧♦t ♦♥ t❤❡ r✐❣❤t✳ ❆❣❛✐♥✱ ✇❡ ❡♠♣❤❛s✐③❡ t❤❡ ❢❛❝t t❤❛t t❤❡ ✈❡❧♦❝✐t② ❞❛t❛ ✐s r❡❢❡rr✐♥❣ t♦ t✐♠❡ ✐♥t❡r✈❛❧s ❜② ♣❧♦tt✐♥❣ ✐ts ✈❛❧✉❡s ✇✐t❤ ❤♦r✐③♦♥t❛❧ ❜❛rs✳ ❚❤✐s ♣❛rt✐❝✉❧❛r s❡t ♦❢ ❞❛t❛ ♠❛② ❜❡ ❞❡s❝r✐❜✐♥❣ t❤❡ ❤♦r✐③♦♥t❛❧ s♣❡❡❞ ♦❢ ❛ ❜❛❧❧ r♦❧❧✐♥❣ t❤r♦✉❣❤ ❛ tr♦✉❣❤✿

❚♦ ✜♥❞ ♦✉t ✇❤❡r❡ ✇❡ ❛r❡ ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ ♦❢ t❤❡s❡ ♦♥❡✲♠✐♥✉t❡ ✐♥t❡r✈❛❧s✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ s✉♠ ♦❢ t❤❡ ❝♦♥s❡❝✉t✐✈❡ ✈❡❧♦❝✐t✐❡s ❛s t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ❢♦r ❡❛❝❤ ✐♥t❡r✈❛❧ ❜② ♣✉❧❧✐♥❣ t❤❡ ❞❛t❛ ❢r♦♠ t❤❡ r♦✇ ♦❢ ✈❡❧♦❝✐t✐❡s ✇✐t❤ t❤❡ ♣r❡✈✐♦✉s r❡s✉❧t✳ ❚❤✐s ✐s ❤♦✇ t❤❡ ✜rst ♦♥❡ ✐s ❝♦♠♣✉t❡❞✱ ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ ✐s 0✿ t✐♠❡ ♠✐♥ 0 1 ... ✈❡❧♦❝✐t② ♠✐❧❡s 0.10 ... s✉♠

↓ 0.00+ 0.10 ... ↑ || 0.00 0.10 ...

❧♦❝❛t✐♦♥ ♠✐❧❡s✴♠✐♥ ❲❡ ♣❧❛❝❡ t❤✐s ❞❛t❛ ✐♥ ❛ ♥❡✇ r♦✇ ❛❞❞❡❞ t♦ t❤❡ ❜♦tt♦♠ ♦❢ ♦✉r t❛❜❧❡✿ t✐♠❡ ♠✐♥ 0 1 2 3 4 ✈❡❧♦❝✐t② ♠✐❧❡s 0.10 0.20 0.30 0.39 ❧♦❝❛t✐♦♥

♠✐❧❡s✴♠✐♥

5 0.48 ↓ ↓ ↓ ↓ ↓ 0.00 → 0.10 → 0.30 → 0.59 → 0.98 → 1.46 →

6 0.56 ↓ 2.03 →

7 0.64 ↓ 2.67 →

8 0.72 ↓ 3.39 →

Pr❛❝t✐❝❛❧❧②✱ ✇❡ ✉s❡ t❤❡ s♣r❡❛❞s❤❡❡t✳ ❲❡ ❝♦♠♣✉t❡ t❤❡ s✉♠s ❜② ♣✉❧❧✐♥❣ t❤❡ ❞❛t❛ ❢r♦♠ t❤❡ ❝♦❧✉♠♥ ♦❢ ✈❡❧♦❝✐t✐❡s ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛✿ ❂❘❬✲✶❪❈✰❘❈❬✲✶❪

❍❡r❡✱ t❤❡ t✇♦ ✈❛❧✉❡s ❝♦♠❡ ❢r♦♠ t❤❡ s❛♠❡✱ ❈ ✱ ♦r ❧❛st✱ ❈❬✲✶❪ ✱ ❝♦❧✉♠♥ ❛♥❞ t❤❡ s❛♠❡✱ ❘ ✱ ❛♥❞ ❧❛st✱ ❘❬✲✶❪ ✱ r♦✇✱ ❛s ❢♦❧❧♦✇s✿

✶✳✷✳

✶✼

❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s

❲❡ ♣❧❛❝❡ t❤❡ r❡s✉❧t ✐♥ ❛ ♥❡✇ ❝♦❧✉♠♥ ❢♦r t❤❡ ❧♦❝❛t✐♦♥s✿

❚❤❡ ❞❛t❛ ✐s ❛❧s♦ ✐❧❧✉str❛t❡❞ ❛s t❤❡ s❡❝♦♥❞ s❝❛tt❡r ♣❧♦t ♦♥ t❤❡ r✐❣❤t✳ ❲❡✱ ❛❣❛✐♥✱ r❡❛rr❛♥❣❡ t❤❡ ❞❛t❛ t❛❜❧❡ t♦ ♠❛❦❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ t②♣❡s ♦❢ ❞❛t❛ ❝❧❡❛r❡r✿ t✐♠❡ ✈❡❧♦❝✐t② ❧♦❝❛t✐♦♥

0 1 2 3 4 ... · 0.00 · 0.10 · 0.20 · 0.30 · 0.39 ... 0.00 − 0.10 − 0.30 − 0.59 − .98 − ...

■♥ ❛ ♠♦r❡ ❣❡♥❡r❛❧ s❡tt✐♥❣✱ t❤❡ t✐♠❡ ♣r♦❣r❡ss❡s ✐♥ ✐♥❝r❡♠❡♥ts✱ h✳ ■❢ ❛ ❢✉♥❝t✐♦♥ g r❡♣r❡s❡♥ts t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ♦✈❡r ❡❛❝❤ t✐♠❡ ✐♥t❡r✈❛❧ [t, t + h]✱ t❤❡♥ t❤❡ ♦✈❡r t❤❡ ♣❡r✐♦❞ ♦❢ t✐♠❡ [a, a + nh] ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣r❡ss✐♦♥✿ n X

t♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t

g(a + kh) .

k=1

❚❤✐s ✐s ❛ ❢✉♥❝t✐♦♥ ✐s ❝❛❧❧❡❞ t❤❡ s✉♠ ♦❢ g ❛♥❞ ✐t ✐s ❝♦♠♣✉t❡❞ r❡❝✉rs✐✈❡❧② ❛s ❡①♣❧❛✐♥❡❞ ❛❜♦✈❡✳ ❋✉rt❤❡r♠♦r❡✱ ✐❢ ❛ ❢✉♥❝t✐♦♥ v r❡♣r❡s❡♥ts t❤❡ ✈❡❧♦❝✐t✐❡s ♦✈❡r ❡❛❝❤ ✐♥t❡r✈❛❧ [t, t + h]✱ t❤❡♥ t❤❡ t♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t ♦✈❡r t❤❡ ♣❡r✐♦❞ ♦❢ t✐♠❡ [a, a + nh] ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣r❡ss✐♦♥✿ n X

v(a + kh)h .

k=1

❚❤✐s ✐s ❛ ❢✉♥❝t✐♦♥ ✐s ❝❛❧❧❡❞ t❤❡ ❘✐❡♠❛♥♥

s✉♠

♦❢ v ✳

■♥ t❤❡ ❡①❛♠♣❧❡s ❛❜♦✈❡✱ t❤❡ ✈❛❧✉❡s ♦❢ g ❛♥❞ v ❛r❡ ♣❧❛❝❡❞ ❛t t❤❡ ❣❡♥❡r❛❧ s❡t✉♣ ♣r❡s❡♥t❡❞ ❜❡❧♦✇✱ ✇❡ tr❛♥s❝❡♥❞ t❤✐s ✐ss✉❡✳

❡♥❞s

♦❢ t❤❡ ✐♥t❡r✈❛❧s✳ ❲❤❡r❡ ❡❧s❡❄ ■♥ t❤❡

✶✳✷✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s

❇❡❧♦✇✱ ✇❡ ♣r♦❞✉❝❡ ❞✐s❝r❡t❡ ♠♦❞❡❧s ❛♥❞ t❤❡♥ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❢r♦♠ ✈❡r❜❛❧ ❞❡s❝r✐♣t✐♦♥s✳ ❲❡ ❤❛✈❡ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b] ♦r [a, +∞]✳ ■t ✐s✱ ✜rst✱ ❛ s❡q✉❡♥❝❡ ♦❢ n ♥♦❞❡s✱ ti ✿ a = t0 < t1 < t2 < ...

✶✳✷✳

❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s

❚❤❡

✐♥❝r❡♠❡♥ts

♦❢

t

✶✽

❛r❡ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ✐♥t❡r✈❛❧s✿

∆ti = ti − ti−1 , i = 1, 2, ...

■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ♥♦❞❡s✱ t❤❡r❡ ❛r❡

❡❞❣❡s ✿

c1 = [t0 , t1 ], c2 = [t1 , t2 ], ... ❲❡ t❤❡♥ s♣❡❛❦ ♦❢ ❛

❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥

♦❢ t❤❡ ✐♥t❡r✈❛❧✳

■♥ ❛❧❧ ❡①❛♠♣❧❡s ✐♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ✐♥❝r❡♠❡♥ts ❛r❡ ❡q✉❛❧✿

∆ti = ti − ti−1 = h, i = 1, 2, ... ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

a = t0 , t1 = a + h, t2 = a + 2h, ... ◆❡①t✱ t❤❡

❞✐✛❡r❡♥❝❡

♦❢ ❛ ❢✉♥❝t✐♦♥

♣❛rt✐t✐♦♥✳ ❖♥ t❤❡ ❡❞❣❡

[tk−1 , tk ]✱

y

❞❡✜♥❡❞ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ✐s ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t t❤❡ ❡❞❣❡s ♦❢ t❤❡

✇❡ ❤❛✈❡✿

∆y = y(tk ) − y(tk−1 ), k = 1, 2, ... ❙✉♣♣♦s❡ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡

y(a)

✐s s❡t ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢

y

✐s ❣✐✈❡♥ ♦♥❡ st❡♣ ❛t ❛ t✐♠❡✳ ❚❤❡♥ ✇❡ ✜♥❞

y

❜② ❛

r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿

y(tk+1 ) = y(tk ) + ∆y , k = 0, 1, 2, ... ▼❡❛♥✇❤✐❧❡✱ t❤✐s ✐s t❤❡ ❢❛♠✐❧✐❛r

❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✿ ∆y =f. ∆t

❊①❛♠♣❧❡ ✶✳✷✳✶✿ ✉♥✐❢♦r♠ ♠♦t✐♦♥

❯♥✐❢♦r♠ ♠♦t✐♦♥ ♠❡❛♥s t❤❛t ✇❡ ❝♦✈❡r t❤❡ s❛♠❡ ❞✐st❛♥❝❡ ♦✈❡r ❡q✉❛❧ ✐♥t❡r✈❛❧s ♦❢ t✐♠❡✳ ■❢ t❤✐s tr❛♥s❧❛t❡s ✐♥t♦✿ ❚❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ❧♦❝❛t✐♦♥

∆y

y ✐s ♦✉r ❧♦❝❛t✐♦♥✱ ∆t✳

✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t✐♠❡

❚❤❡ ❛❧❣❡❜r❛ ✐s ❛s ❢♦❧❧♦✇s✿

❢♦r s♦♠❡ ❝♦♥st❛♥t

∆y = k · ∆t ,

k✳

■❢ ✇❡ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t ❛ ❢❛♠✐❧✐❛r ✐❞❡❛ ❢r♦♠ ♣❤②s✐❝s✱ ✇❡ ❤❛✈❡ ❛



❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❞②♥❛♠✐❝s ✿

✏❚❤❡ ✈❡❧♦❝✐t② ✐s ❝♦♥st❛♥t✑✳

❚❤❡ ❛❧❣❡❜r❛ ✐s ❛s ❢♦❧❧♦✇s✿

∆y = k. ∆t

❙✉♣♣♦s❡ ✇❡ ❛❧s♦ ❤❛✈❡ ❛♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✿

y(t0 ) = y0 . ❚❤❡♥ ♦✉r ❡q✉❛t✐♦♥ ❣✐✈❡s ✉s ❛ s❡q✉❡♥❝❡ ✈✐❛ t❤✐s

r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ✿

y(tn+1 ) = y(tn ) + k · ∆t . ❆s ❛♥ ✐❧❧✉str❛t✐♦♥✱ ❝♦♥s✐❞❡r ❛ ✏❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✑✿

❲❡ ❝❤♦♦s❡ t❤❡ t✐♠❡ ✐♥❝r❡♠❡♥ts t♦ ❜❡

∆y = 2. ∆t ❝♦♥st❛♥t ∆t = h = 0.2 ✳

❚❤❡♥✿

y(tn+1 ) = y(tn ) + 2h . ❍❡r❡ ❛r❡ ❛ ❢❡✇ s♦❧✉t✐♦♥s ❢♦r ✈❛r✐♦✉s ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✭♦♥ t❤❡

y ✲❛①✐s✮✿

✶✳✷✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s

✶✾

❚❤❡② ❛r❡ ❥✉st ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥s✿ st❡♣ r✐❣❤t ❜②

h

❛♥❞ t❤❡♥ ✉♣ ❜②

2h✳

●❡♥❡r❛❧❧②✱ t❤❡② ❞♦♥✬t ❤❛✈❡

t♦ ❜❡ ✕ ✇❤❡♥ t❤❡ ♣❛rt✐t✐♦♥ ✐s ✉♥❡✈❡♥✿

❇✉t t❤❡ ♣♦✐♥ts st✐❧❧ ❧✐❡ ♦♥ t❤❡ s❛♠❡ str❛✐❣❤t ❧✐♥❡s✦

❊①❛♠♣❧❡ ✶✳✷✳✷✿ ❧♦❝❛t✐♦♥ ❢r♦♠ ✈❡❧♦❝✐t②

❚❤❡ ❢♦r♠✉❧❛ ❤❛s ❜❡❡♥ ✉s❡❞ ✐♥ ❱♦❧✉♠❡s ✷✱ ✸✱ ❛♥❞ ✹ t♦ ♠♦❞❡❧ ✉♥✐❢♦r♠ ♠♦t✐♦♥ ❛♥❞ ❛❝q✉✐r❡ ❧♦❝❛t✐♦♥ ❢r♦♠ t❤✐s ✈❡❧♦❝✐t②✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✇❡ ❤❛✈❡✿



✏❚❤❡ ✈❡❧♦❝✐t② ❞❡♣❡♥❞s ♦♥ t✐♠❡✑✳

❲❡ t❛❦❡ ❛♥② ❢✉♥❝t✐♦♥

z = f (t)

❞❡✜♥❡❞ ❛t t❤❡ ❡❞❣❡s ♦❢ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❛♥❞ t❤✐♥❦ ♦❢

✈❛❧✉❡ ♦❢ t❤❡ ✈❡❧♦❝✐t② ❞✉r✐♥❣ t❤❡ t✐♠❡ ✐♥t❡r✈❛❧

[ti−1 , ti ]✳

f (ci )

❛s t❤❡

❲❡ s❡t ✉♣ ❛ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✿

∆y = f (cn ) . ∆t ❘❡❛rr❛♥❣❡❞ ✐t ♣r♦❞✉❝❡s ✐ts ♦✇♥ s♦❧✉t✐♦♥ ❛s ❛ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿

y(tn+1 ) = y(tn ) + f (cn ) · ∆t , ✇❤❡r❡

tn+1 = tn + ∆t . f ❄ ■t ✐s ❢✉♥❝t✐♦♥ ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡ z = f (t) ✭♣❧♦tt❡❞ ❜❡❧♦✇ ❧❡❢t✮✳ ❍♦✇❡✈❡r✱ t❤❡ yn = y(tn ) ✐s t♦ ❜❡ ♣❧♦tt❡❞ ♦♥ t❤❡ ty ✲♣❧❛♥❡✳ ■t ✐s✱ t❤❡r❡❢♦r❡✱ ❜❡♥❡✜❝✐❛❧ t♦ t❤✐♥❦ ♦❢ f ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s z = f (t, y) t❤❛t ❥✉st ❤❛♣♣❡♥s t♦ ❤❛✈❡ t❤❡ s❛♠❡ ✈❛❧✉❡ ❢♦r ❡❛❝❤ t r❡❣❛r❞❧❡ss ♦❢ y ✳ ❲❡ s❛♠♣❧❡ f ♦✈❡r ✐ts ❞♦♠❛✐♥✱ ❛ r❡❝t❛♥❣❧❡ ♦❢ t✬s ❛♥❞ y ✬s✱ ❛♥❞ ✇r✐t❡ t❤❡ ♦✉t♣✉ts ✭♠✐❞❞❧❡✮✿

❲❤❛t ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥ ✐s ❣r❛♣❤ ♦❢ ❛ s♦❧✉t✐♦♥

✶✳✷✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s

✷✵

❲❡ ❡♥❤❛♥❝❡ t❤❡ ✈✐s✉❛❧✐③❛t✐♦♥ ❜② s❤♦✇✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢ f ✐♥ t❡r♠s ♦❢ ❝♦❧♦r ✕ ❜❧✉❡ ❢♦r ♥❡❣❛t✐✈❡ ❛♥❞ r❡❞ ❢♦r ♣♦s✐t✐✈❡✳ ❲❡ ❛❧s♦ s❤♦✇ f ❛s ❛ s✉r❢❛❝❡ ✭r✐❣❤t✮✳ ❚❤❡ ♠✐❞❞❧❡ ✐s t❤❡ ♠♦st ❝♦♥✈❡♥✐❡♥t ♦❢ t❤❡s❡ ❛s ✐t ✐s s❡❡♥ ❛s ❛ ✜❡❧❞ ♦❢ ✈❡❧♦❝✐t✐❡s✳ ❙♦✱ ❛s y ✐s ♣❧♦tt❡❞ ♣♦✐♥t ❜② ♣♦✐♥t✱ ✇❡ ❦♥♦✇ t❤❡ ♣r♦❝❡❞✉r❡ ❢♦r ❝♦♥str✉❝t✐♥❣ ❛ s♦❧✉t✐♦♥✿ • ❲❡ ❣♦ ✉♣ ✇❤❡♥ t❤❡ ❛r❡❛ ✐s r❡❞✳ • ❲❡ ❣♦ ❞♦✇♥ ✇❤❡♥ t❤❡ ❛r❡❛ ✐s ❜❧✉❡✳ ❲❡ ❝❛♥ ❞r❛✇ ❛ ❝✉r✈❡ ✐♥ t❤✐s ♠❛♥♥❡r✿

▼♦r❡ ♣r❡❝✐s❡❧②✱ ✇❡ r❡❛❞ t❤❡ ❞❛t❛ ❢r♦♠ t❤❡ t❛❜❧❡ ❛s ✇❡ ♣r♦❣r❡ss ♦♥❡ ♣♦✐♥t ❛t ❛ t✐♠❡✿ ✈❡❧♦❝✐t② t✐♠❡ ✐♥t❡r✈❛❧ h ❥✉♠♣ ∆y 0.2 · 0.1 = 0.02 0.7 · 0.1 = 0.07 1.2 · 0.1 = 0.12 .. .. .. ❖r ✐t ✐s ❞♦♥❡ ❜② t❤❡ ❝♦♠♣✉t❡r✿

■♥ t❤❡ ♠❡❛♥t✐♠❡✱ ✇❡ ❣♦ ❜❛❝❦ t♦ ♦✉r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✿

∆y =f. ∆t ❋✉rt❤❡r♠♦r❡✱ ✇❤❛t ✐❢ f ✐s♥✬t ❥✉st ❛ str❡❛♠ ♦❢ ♥✉♠❜❡rs ❜✉t ❛ s❛♠♣❧❡❞ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥❄ ❆♥❞ ✇❤❛t ✐❢ y ✐s ❛ s❛♠♣❧❡❞ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥❄ ❲❡ ❝❛♥ t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❧❛tt❡r ❡q✉❛t✐♦♥✱

✶✳✷✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s

∆t → 0✳

✷✶

❚❤❡ r❡s✉❧t ✐s ❛ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✿

dy =f, dt ♦r

y ′ = f (t) . ❊①❡r❝✐s❡ ✶✳✷✳✸

P❧♦t ♠♦r❡ s♦❧✉t✐♦♥s st❛rt✐♥❣ ❛t ♦t❤❡r ♣♦✐♥ts✳ ❲❤❛t ✐s ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡♠❄

❊①❛♠♣❧❡ ✶✳✷✳✹✿ ❞❛t❛ str❡❛♠

❚❤❡ q✉❛♥t✐t✐❡s ✉s❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ ♥❡①t ❧♦❝❛t✐♦♥ ✭♦r ❛ st❛t❡✮ ✕ ❝✉rr❡♥t t✐♠❡ ❛♥❞ ✈❡❧♦❝✐t② ✕ ❞♦❡s♥✬t ❤❛✈❡ t♦ ❝♦♠❡ ❢r♦♠ ❛ ❢♦r♠✉❧❛✦ ❚❤✐s ✐s ♥♦t❤✐♥❣ ❜✉t ❞❛t❛ ❛♥❞ ✐t ♠❛② ❝♦♠❡ ❛s str✐♥❣s ♦❢ ♥✉♠❜❡rs ✭♦r ✐t ❝❛♥ ❜❡ ❢❡❞ ✐♥t♦ t❤❡ ❝♦♠♣✉t❡r ❝♦♥t✐♥✉♦✉s❧②✮✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ✐♥✐t✐❛❧ t✐♠❡ t0 ❛♥❞ t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥

y0

❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ✜rst r♦✇ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t

❛♥❞✱ ❛s ✇❡ ♣r♦❣r❡ss ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡✱ ♥❡✇ ♥✉♠❜❡rs ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ♥❡①t r♦✇ ♦❢ ♦✉r s♣r❡❛❞s❤❡❡t✿

tn , vn , yn , n = 1, 2, 3, ... ❚❤❡ s❛♠❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ✐s ✉s❡❞ t♦ ✜♥❞ t❤❡ ♥❡①t ❧♦❝❛t✐♦♥✿

yn+1 = yn + vn+1 · ∆t . ❚❤❡ r❡s✉❧t ✐s ❛ ❣r♦✇✐♥❣ t❛❜❧❡ ♦❢ ✈❛❧✉❡s✿ ✐t❡r❛t✐♦♥

n

t✐♠❡

0 1 ... 1000 ...

✐♥✐t✐❛❧✿

tn

✈❡❧♦❝✐t②

3.5 3.6 ... 103.5 ...

vn

❧♦❝❛t✐♦♥

−− 33 ... 4 ...

yn

22 25.3 ... 336 ...

❆❧❧ ❜✉t t❤❡ ❧❛st ❝♦❧✉♠♥ ♠❛② ❝♦♠❡ ❛s ❛ ❞❛t❛ ✜❧❡✳

❊①❛♠♣❧❡ ✶✳✷✳✺✿ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤

❚❤❡ ❞②♥❛♠✐❝s ✐s ✈❡r② ❞✐✛❡r❡♥t ✐♥ t❤❡ ♥❡①t ❞❡s❝r✐♣t✐♦♥✿



✏❚❤❡ r❛t❡ ♦❢ ❣r♦✇t❤✴❞❡❝❛② ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❝✉rr❡♥t ♣♦♣✉❧❛t✐♦♥✳✑

❆♥ ❡①❛♠♣❧❡ ✐s ❜❛❝t❡r✐❛ ♦r r❛❜❜✐ts ❣r♦✇✐♥❣ ✇✐t❤ ❛♥ ✉♥r❡str✐❝t❡❞ ❛♠♦✉♥t ♦❢ ❢♦♦❞ ❜❡❝❛✉s❡ t❤❡ ♥✉♠❜❡r ♦❢ t❤❡ ♥❡✇❜♦r♥ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ♥✉♠❜❡r ♦❢ ❛❞✉❧ts✳ ❲❡ r❡✇r✐t❡ t❤❡ ❞❡s❝r✐♣t✐♦♥✿

❢♦r s♦♠❡ ❝♦♥st❛♥t ❛s

k✳

∆y = ky , ∆t

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥

∆y

✐s ♣r♦♣♦rt✐♦♥❛❧ t♦

y

❛s ✇❡❧❧

∆t✿ ∆y = k · y · ∆t .

❚❤❡ ✏r✐❣❤t✲❤❛♥❞ s✐❞❡✑ ❢✉♥❝t✐♦♥ ✐s ✈❡r② s✐♠♣❧❡✿

f (y) = ky . yn = y(tn ) ✐s ♣❧♦tt❡❞ ♦♥ t❤❡ ty ✲♣❧❛♥❡ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✐t ✈❛r✐❛❜❧❡s z = f (t, y) t❤❛t ❥✉st ❤❛♣♣❡♥s t♦ ❤❛✈❡ t❤❡ s❛♠❡

❖♥❝❡ ❛❣❛✐♥✱ ❤♦✇❡✈❡r✱ t❤❡ ❣r❛♣❤ ♦❢ ❛ s♦❧✉t✐♦♥ ✐s ❜❡tt❡r t♦ t❤✐♥❦ ♦❢ ✈❛❧✉❡ ❢♦r ❡❛❝❤

y

f

❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦

r❡❣❛r❞❧❡ss ♦❢

t✿

✶✳✷✳

✷✷

❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s

❚❤❡♥ t❤❡ ❡q✉❛t✐♦♥ ❣✐✈❡s ✉s ❛ s♦❧✉t✐♦♥ ✈✐❛ t❤✐s r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿ y(tn+1 ) = y(tn ) + ky(tn ) · ∆t .

❲❡ ❛ss✉♠❡ ❤❡r❡ t❤❛t ✇❡ ❛❧s♦ ❤❛✈❡ ❛♥ ✐♥✐t✐❛❧

❝♦♥❞✐t✐♦♥



y(t0 ) = y0 .

❚❤❡ ❢♦r♠✉❧❛ ❤❛s ❜❡❡♥ ✉s❡❞ ✐♥ ❈❤❛♣t❡r ✷❉❈✲✻ t♦ st✉❞② ❡①♣♦♥❡♥t✐❛❧ ♠♦❞❡❧s ❛♥❞ ♥♦✇ ✇❡ ✇✐❧❧ ❛♣♣❧② ✐t t♦ ♠♦❞❡❧s ✇✐t❤ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ ❛ q✉❛♥t✐t② ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ q✉❛♥t✐t②✬s ❝✉rr❡♥t ✈❛❧✉❡✿

❲❡ s❡❡ ❤♦✇ ✇❡ ♣r♦❣r❡ss ✐♥ t❤❡ ✉♣✇❛r❞ ❞✐r❡❝t✐♦♥ ✇❤✐❝❤ ♠❛❦❡s t❤❡ ✈❛❧✉❡s ♦❢ f ❤✐❣❤❡r ❛♥❞✱ ❛s ❛ r❡s✉❧t✱ t❤❡ ❣r♦✇t❤ ♦❢ y ❢❛st❡r✳ ❲❡ ❝❛♥ ♣❧♦t ❛ ❧♦t ♠♦r❡ s♦❧✉t✐♦♥s ✐♥ t❤✐s ♠❛♥♥❡r✿

■♥ t❤❡ ♠❡❛♥t✐♠❡✱ ✐❢ ✇❡ t❤✐♥❦ ♦❢ y ❛♥❞ f ❛s s❛♠♣❧❡❞ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ❢♦r ❛❧❧ r❡❛❧ t ❛♥❞ y ✱ t❤❡ ❡q✉❛t✐♦♥ ✐s ❝♦♥✈❡rt❡❞ t♦ ❛ ♥❡✇ ❡q✉❛t✐♦♥✿ ∆y dy = ky =⇒ = ky ♦r y ′ = ky . ∆t→0 ∆t dt lim

❊①❡r❝✐s❡ ✶✳✷✳✻

P❧♦t ♠♦r❡ s♦❧✉t✐♦♥s✳ ❲❤❛t ✐s ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡♠❄ ❈♦♠♣❛r❡ t♦ t❤❡ ❦♥♦✇♥ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s t❤❛t s❛t✐s❢② y ′ = y ✳

✶✳✷✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s

✷✸

❊①❡r❝✐s❡ ✶✳✷✳✼

P❧♦t s♦❧✉t✐♦♥s ❢♦r

k < 0✳

❲❤❛t ❞♦❡s t❤❡ s②st❡♠ ♠♦❞❡❧❄

❊①❛♠♣❧❡ ✶✳✷✳✽✿ ❧♦❣✐st✐❝ ❣r♦✇t❤

❆ ❞✐✛❡r❡♥t✱ ❛♥❞ ♠♦r❡ ❝♦♠♣❧❡①✱ ♣♦♣✉❧❛t✐♦♥ ❞②♥❛♠✐❝s ✐s ❣✐✈❡ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡s❝r✐♣t✐♦♥✿



✏❚❤❡ r❛t❡ ♦❢ ❣r♦✇t❤ ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❝✉rr❡♥t ♣♦♣✉❧❛t✐♦♥ ❛♥❞ t♦

t❤❡ r❡♠❛✐♥✐♥❣ ♣♦♣✉❧❛t✐♦♥ ♦❢ t❤❡ t♦t❛❧ t❤❛t ❝❛♥ ❜❡ s✉st❛✐♥❡❞✑✳ ❆♥ ❡①❛♠♣❧❡ ✐s ❜❛❝t❡r✐❛ ❣r♦✇✐♥❣ ✐♥ ❛ ❥❛r✳ ❲❡ r❡✇r✐t❡ t❤❡ ❞❡s❝r✐♣t✐♦♥✿

❢♦r s♦♠❡ ❝♦♥st❛♥t

T − y✱

✇❤❡r❡

T

k✳

∆y = ky(T − y) , ∆t

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥

✐s t❤❡ t♦t❛❧ ♣♦ss✐❜❧❡ ♣♦♣✉❧❛t✐♦♥ ✭❛s ✇❡❧❧ ❛s

∆t

∆y

✐s ♣r♦♣♦rt✐♦♥❛❧ t♦

y

✳✳✳ ❛♥❞

❛s ❜❡❢♦r❡✮✿

∆y = k · y · (T − y) · ∆t . ❚❤❡ ✏r✐❣❤t✲❤❛♥❞ s✐❞❡✑ ❢✉♥❝t✐♦♥ ✐s s❧✐❣❤t❧② ♠♦r❡ ❝♦♠♣❧❡①✿

f (t, y) = ky(T − y) . ❚❤✐s ✐s ♦✉r ❢✉♥❝t✐♦♥ ✭k

= 5, T = 1✮✿

❲❡ s❡❡ ❧♦✇ ❧❡✈❡❧ ♦❢ ♣♦t❡♥t✐❛❧ ❣r♦✇t❤ ✐♥ t❤❡ ❛r❡❛s ✇❤❡r❡ t❤❡ ♣♦♣✉❧❛t✐♦♥ ✐s ❧♦✇ ♦r ✇❤❡r❡ ✐t ✐s t♦♦ ❝❧♦s❡ t♦ ❜❡ ❡①❤❛✉st✐♥❣ t❤❡ r❡s♦✉r❝❡s✳ ❲✐t❤ ❛♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✿

y(t0 ) = y0 , ✇❡ ♣r♦❞✉❝❡ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧ ✈✐❛ t❤✐s r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿

y(tn+1 ) = y(tn ) + ky(tn )(T − y(tn )) · ∆t, k > 0 . ❚❤❡ ♠♦❞❡❧ ❞❡s❝r✐❜❡s ❛ r❡str✐❝t❡❞ ❣r♦✇t❤ ✭k

= 5, ∆t = h = .05✮✿

✶✳✷✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s

✷✹

❲❡ s❡❡ t❤❡ ❢❛st❡st ❣r♦✇t❤ ✇❤❡♥ t❤❡ ♣♦♣✉❧❛t✐♦♥ ✐s ❛❜♦✉t ❤❛❧❢ ♦❢ ✇❤❛t ❝❛♥ ❜❡ s✉st❛✐♥❡❞✳ ❆❧❧ s♦❧✉t✐♦♥s ❡①❤✐❜✐t t❤✐s ❜❡❤❛✈✐♦r✿

❈❤♦♦s✐♥❣

∆t

s♠❛❧❧ r❡❧❛t✐✈❡ t♦

k

❝♦✉❧❞ ❜r❡❛❦ ♦✉r ♠♦❞❡❧ ✭k

= 25, ∆t = .05✮✿

❚❤❡ ♣♦♣✉❧❛t✐♦♥ ✐s s♦♠❡t✐♠❡s ❧❛r❣❡r t❤❛♥ t❤❡ ❝❛♣❛❝✐t② ♦❢ t❤❡ ❥❛r✦ ■❢

y

✐s ❛ s❛♠♣❧❡❞ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥✱ t❤❡♥ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥

∆y = y(T − y) . ∆t ♣r♦❞✉❝❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✿

y ′ = ky(T − y) . ❊①❡r❝✐s❡ ✶✳✷✳✾

❲❤❛t ❞♦❡s t❤❡ ❝❛s❡ ♦❢

k 0✳

❚❤❡ ✏r✐❣❤t✲❤❛♥❞ s✐❞❡✑ ❢✉♥❝t✐♦♥ ✐s s❧✐❣❤t❧②

s✐♠♣❧❡r t❤❛♥ ❧❛st✿

f (t, y) = k · (r − y) . ■t ✐s ♣❧♦tt❡❞ ❜❡❧♦✇ ✭k

= 10, ∆t = .05✮✿

✶✳✷✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s

✷✺

❲❡ ❝❛♥ s❡❡ t❤❛t ✇❤❡♥ t❤❡ t❡♠♣❡r❛t✉r❡ ✐s ❤✐❣❤❡r t❤❛♥ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡✱ ✐t ❞❡❝r❡❛s❡s ❛♥❞ ✇❤❡♥ t❤❡ t❡♠♣❡r❛t✉r❡ ✐s ❧♦✇❡r t❤❛♥ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡✱ ✐t ✐♥❝r❡❛s❡s✳ ❚❤✐s ✐s ✇❤② t❤❡ ♦❜❥❡❝t✬s t❡♠♣❡r❛t✉r❡ ✐s♥✬t ❡①♣❡❝t❡❞ t♦ ✏♦✈❡rs❤♦♦t✑ t❤❛t ♦❢ t❤❡ r♦♦♠✳

❚❤❡ ❡q✉❛t✐♦♥ ❣✐✈❡s ✉s ❛ s♦❧✉t✐♦♥ ✈✐❛ t❤✐s r❡❝✉rs✐✈❡

❢♦r♠✉❧❛✿

y(tn+1 ) = y(tn ) + k(r − y(tn )) · ∆t .

❚❤❡ ❢♦r♠✉❧❛ ✇❛s ✉s❡❞ ✐♥ ❈❤❛♣t❡r ✷❉❈✲✻ t♦ ♠♦❞❡❧ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ❈♦♦❧✐♥❣✿

■♥ t❤❡ ♠❡❛♥t✐♠❡✱ t❤❡ ❞❡r✐✈❛t✐✈❡✱ ✐❢ ❛♥②✱ ✇♦✉❧❞ s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣✿

y ′ = k(r − y) . ❊①❛♠♣❧❡ ✶✳✷✳✶✶✿ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ❈♦♦❧✐♥❣ ❝♦♥t✐♥✉❡❞

❈❤♦♦s✐♥❣

∆t

s♠❛❧❧ r❡❧❛t✐✈❡ t♦

k

❝♦✉❧❞ ❜r❡❛❦ ♦✉r ♠♦❞❡❧ ✭k

= 25, h = ∆t = .05✮✿

✶✳✷✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s

✷✻

❚❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ❝♦✛❡❡ ✐s s♦♠❡t✐♠❡s ❝♦❧❞❡r t❤❛♥ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡✦ ❈❛♥ ✇❡ ❣✉❛r❛♥t❡❡ t❤❛t t❤✐s ✇♦♥✬t ❤❛♣♣❡♥❄ ❙✉♣♣♦s❡ ❢♦r ❛❧❧

u0 > r ✱

✇❤❛t ❝♦♥❞✐t✐♦♥ ✇♦✉❧❞ ❡♥s✉r❡ t❤❛t

un > r

n❄

❇② ✐♥❞✉❝t✐♦♥✱ ✇❡ ❛ss✉♠❡ t❤❛t

un > r un+1 ⇐= ⇐= ⇐= ⇐=

❚❤❛t✬s t❤❡ ❝♦♥❞✐t✐♦♥✦

❛♥❞ ♣r♦✈❡ t❤❛t

un+1 > r✿

= un + k(r − un )h un − kun h un (1 − kh) 1 − kh kh

>r > r − krh > r(1 − kh) >0 0 s♠❛❧❧ ❡♥♦✉❣❤✳ ❚❛❦✐♥❣ t❤❡ ❧✐♠✐t ♦✈❡r ∆t → 0 ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣✿ y ′ (t) = f (y(t)) ,

♣r♦✈✐❞❡❞ y = y(t) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t ❛♥❞ z = f (y) ✐s ❝♦♥t✐♥✉♦✉s ❛t y(t)✳ ❙✉❝❤ ❛♥ ❡q✉❛t✐♦♥ ♠❛② ❜❡ ♣♦ss✐❜❧❡ t♦ s♦❧✈❡✳ ▲❡t✬s r❡✈✐❡✇ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇❡ ❤❛✈❡ s❡❡♥✿ ❡q✉❛t✐♦♥

❛ s♦❧✉t✐♦♥

1. f ′ (x)

= x2

−→ f (x) = x3 /3

2. f ′ (x)

= f (x) −→ f (x) = ex

❲❤❡♥ x r❡♣r❡s❡♥ts t✐♠❡ ❛♥❞ y = f (x) r❡♣r❡s❡♥ts ❧♦❝❛t✐♦♥✱ t❤❡ ❡q✉❛t✐♦♥s ❤❛✈❡ s✐♠♣❧❡ ✐♥t❡r♣r❡t❛t✐♦♥s✿ ✶✳ ❚❤❡ ✈❡❧♦❝✐t② ✐s ❦♥♦✇♥ ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡✿ ♠✐ss✐❧❡✳ ✷✳ ❚❤❡ ✈❡❧♦❝✐t② ✐s ❦♥♦✇♥ ❛t ❡✈❡r② ❧♦❝❛t✐♦♥✿ ❧✐q✉✐❞ ✢♦✇✳ ❚❤✐s ✐♥t❡r♣r❡t❛t✐♦♥ ❥✉st✐✜❡s ✉s✐♥❣ t ❢♦r t❤❡ ♥❛♠❡ ♦❢ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✳ ■♥ t❤❡ ✜❡❧❞ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ✐t ✐s ❛❧s♦ ❝♦♠♠♦♥ ♥♦t t♦ ♥❛♠❡ t❤❡ ♥❛♠❡s ♦❢ t❤❡ ✈❛r✐❛❜❧❡s ✉♥❞❡r t❤✐s ♠♦r❡ ❝♦♠♣❛❝t ♥♦t❛t✐♦♥✿

t❤❡ ❢✉♥❝t✐♦♥s ❛♥②♠♦r❡ ❜✉t ✉s❡ ✐♥st❡❛❞

✶✳✺✳

❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s

✹✹

❖❉❊s

1. y ′ = t2 −→ y = t3 /3 2. y ′ = y −→ y = et

✐s ❛ s♦❧✉t✐♦♥✳ ✐s ❛ s♦❧✉t✐♦♥✳

❚❤❡② ❛r❡ ❣❡♥❡r❛❧✐③❡❞ ❛s ❢♦❧❧♦✇s✳ ❋✐rst✱ ✇❡ ❤❛✈❡

g

✐♥❞❡♣❡♥❞❡♥t ♦❢

y✳

❲❡ ❛r❡ ❛❢t❡r ❢✉♥❝t✐♦♥s

y = y(t)

t❤❛t s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥✿

y ′ (t) = g(t) , ❢♦r s♦♠❡ ❢✉♥❝t✐♦♥

(t, y(t))✳

g✳

■t ✐s

❧♦❝❛t✐♦♥✲✐♥❞❡♣❡♥❞❡♥t✳

❚❤✐s s❧♦♣❡ ✐s t❤❡ s❛♠❡ ❛❧♦♥❣ ❛♥② ❣✐✈❡♥

❲❤❡♥ t❤❡ ❢✉♥❝t✐♦♥

g

❚❤❡② ❛r❡ r❡♣r❡s❡♥t❡❞ ❜②✿

C

✈❡rt✐❝❛❧

❤❛♣♣❡♥s t♦ ❜❡ ❝♦♥st❛♥t✱ s❛②

♣❛r❛❜♦❧❛s✿

✇❤❡r❡

❍❡r❡✱ ❢♦r ❡✈❡r②

−1✱

t✱

✇❡ ❦♥♦✇ t❤❡ s❧♦♣❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ❛t

❧✐♥❡✿

✐ts s♦❧✉t✐♦♥ s❡t ✐s t❤❡ ❢❛♠✐❧② ♦❢ t❤❡s❡ ✈❡rt✐❝❛❧❧② s❤✐❢t❡❞

1 y = − t2 + C , 2

✐s ❛♥② r❡❛❧ ♥✉♠❜❡r✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡ ❛♥ ❡①♣❧✐❝✐t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ s♦❧✉t✐♦♥ s❡t✿



1 y = − t2 + C : C ∈ R 2



.

❆♥ ❡①❛♠✐♥❛t✐♦♥ ♦❢ t❤❡ ♣✐❝t✉r❡ r❡✈❡❛❧s t❤❛t t❤✐s ❢❛♠✐❧② ♦❢ ❝✉r✈❡s s❛t✐s❢② t❤❡s❡ t✇♦ ♣r♦♣❡rt✐❡s✿ ✶✳ ❚❤❡ ♣❧❛♥❡ ✐s ❢✉❧❧② ❝♦✈❡r❡❞ ❜② t❤❡ ❝✉r✈❡s✳ ✷✳ ❚❤❡ ❝✉r✈❡s ❞♦♥✬t ✐♥t❡rs❡❝t✳

❊①❛♠♣❧❡ ✶✳✺✳✶✿ q✉❛❞r❛t✐❝ ❝❛s❡ ▲❡t✬s ♣r♦✈❡ t❤❡s❡ st❛t❡♠❡♥ts✳

✶✳✺✳ ❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s

✹✺

❋✐rst✱ ✏❚❤❡ ♣❧❛♥❡ ✐s ❢✉❧❧② ❝♦✈❡r❡❞ ❜② t❤❡ ❝✉r✈❡s✑ ✐s tr❛♥s❧❛t❡❞ ❛s ✏❚❤❡ ♣❧❛♥❡ ✐s t❤❡ ✉♥✐♦♥ t❤❡ ❝✉r✈❡s✑ ❛♥❞✱ ❢✉rt❤❡r♠♦r❡✱ ✏❊✈❡r② ♣♦✐♥t ♦♥ t❤❡ ♣❧❛♥❡ ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❝✉r✈❡s✑✳ ❙✉♣♣♦s❡ ♣♦✐♥t✳ ▲❡t✬s ✜♥❞ ❛ ❝✉r✈❡ ❢♦r ✐t✳ ❚❤✐s ♠❡❛♥s t❤❛t ✇❡ ♥❡❡❞ t♦ ✜♥❞ ❛ s♣❡❝✐✜❝

(t0 , y0 )

❜❡❧♦♥❣s t♦ t❤❡ ❝✉r✈❡

1 y = − t2 + C 2

C✱

(t0 , y0 ) ✐s ♦♥❡ s✉❝❤

t❤❛t✬s ❛❧❧✳ ❚❤❡ ♣♦✐♥t

♠❡❛♥s t❤❛t t❤❡ ❡q✉❛t✐♦♥ ✐s s❛t✐s✜❡❞ ❢♦r ✐t✿

1 y0 = − t20 + C . 2 ❙♦❧✈❡ ❢♦r

C✿

1 C = y0 + t20 . 2

❉♦♥❡✦ ❙❡❝♦♥❞✱ ✏❚❤❡ ❝✉r✈❡s ❞♦♥✬t ✐♥t❡rs❡❝t✑ ✐s tr❛♥s❧❛t❡❞ ❛s ✏❚✇♦ ❝✉r✈❡s t❤❛t ✐♥t❡rs❡❝t ❛r❡ t❤❡ s❛♠❡ ❝✉r✈❡✑✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ s✉❝❤ ❝✉r✈❡s✿

1 y = − t2 + C , 2

❛♥❞

1 y = − t2 + K , 2

❢♦r s♦♠❡ r❡❛❧ ♥✉♠❜❡rs

C

❛♥❞

K✳

❙✉♣♣♦s❡ t❤❡② ✐♥t❡rs❡❝t✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡r❡ ✐s ❛ ♣♦✐♥t

(t0 , y0 ) t❤❛t

❜❡❧♦♥❣s t♦ ❜♦t❤✳ ❚❤❡r❡❢♦r❡✱ ❜♦t❤ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ❛r❡ s❛t✐s✜❡❞ ❢♦r t❤✐s ♣♦✐♥t✿

1 y0 = − t20 + C , 2 ❛♥❞

1 y0 = − t20 + K . 2

❲❡ s♦❧✈❡ t❤✐s s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s t♦ ❝♦♥❝❧✉❞❡✿

C =K. ❚❤❛t✬s t❤❡ s❛♠❡ ❝✉r✈❡✦

❙❡❝♦♥❞✱ ✇❡ ❤❛✈❡

g

✐♥❞❡♣❡♥❞❡♥t ♦❢

t✳

❲❡ ❛r❡ ❛❢t❡r ❢✉♥❝t✐♦♥s

y = y(x)

t❤❛t s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥✿

y ′ (t) = h(y(t)) , ❢♦r s♦♠❡

h✳ ❲❛r♥✐♥❣✦ ❨♦✉ ❝❛♥✬t ✐♥t❡❣r❛t❡

■t ✐s t✐♠❡✲✐♥❞❡♣❡♥❞❡♥t✳ ❍❡r❡✱ ❢♦r ❡✈❡r② s❛♠❡ ❛❧♦♥❣ ❛♥② ❣✐✈❡♥ ❤♦r✐③♦♥t❛❧ ❧✐♥❡✿

y✱

y′ = y✳

✇❡ ❦♥♦✇ t❤❡ s❧♦♣❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ❛t

(t, y)✳

❚❤✐s s❧♦♣❡ ✐s t❤❡

✶✳✺✳

❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s

❲❤❡♥ t❤❡ ❢✉♥❝t✐♦♥

✹✻

h(x) = x

❤❛♣♣❡♥s t♦ ❜❡ t❤❡ ✐❞❡♥t✐t②✱ ✐ts s♦❧✉t✐♦♥ ✐s t❤❡ ❢❛♠✐❧② ♦❢ ✈❡rt✐❝❛❧❧② str❡t❝❤❡❞

❝✉r✈❡s✿

❚❤❡② ❛r❡ r❡♣r❡s❡♥t❡❞ ❜②✿

y = Cet , ✇❤❡r❡

C

✐s ❛♥② r❡❛❧ ♥✉♠❜❡r✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡ ❛♥ ❡①♣❧✐❝✐t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ s♦❧✉t✐♦♥ s❡t✿

{y = Cet : C ∈ R} . ❆♥ ❡①❛♠✐♥❛t✐♦♥ ♦❢ t❤❡ ♣✐❝t✉r❡ r❡✈❡❛❧s t❤❛t t❤✐s ❢❛♠✐❧② ♦❢ ❝✉r✈❡s s❛t✐s❢② t❤❡s❡ t✇♦ ♣r♦♣❡rt✐❡s✿ ✶✳ ❚❤❡ ♣❧❛♥❡ ✐s ❢✉❧❧② ❝♦✈❡r❡❞ ❜② t❤❡ ❝✉r✈❡s✳ ✷✳ ❚❤❡ ❝✉r✈❡s ❞♦♥✬t ✐♥t❡rs❡❝t✳ ❊①❛♠♣❧❡ ✶✳✺✳✷✿ ❡①♣♦♥❡♥t✐❛❧ ❝❛s❡

▲❡t✬s ♣r♦✈❡ t❤❡s❡ st❛t❡♠❡♥ts✳ ❋✐rst✱ ✏❚❤❡ ♣❧❛♥❡ ✐s ❢✉❧❧② ❝♦✈❡r❡❞ ❜② t❤❡ ❝✉r✈❡s✑ ✐s ✉♥❞❡rst♦♦❞ ❛s ✏❊✈❡r② ♣♦✐♥t ♦♥ t❤❡ ♣❧❛♥❡ ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❝✉r✈❡s✑✳ ❙✉♣♣♦s❡ ✇❡ ♥❡❡❞ t♦ ✜♥❞ ❛ s♣❡❝✐✜❝

C✳

(t0 , y0 )

✐s ♦♥❡ s✉❝❤ ♣♦✐♥t✳ ▲❡t✬s ✜♥❞ ❛ ❝✉r✈❡ ❢♦r ✐t✳ ❚❤✐s ♠❡❛♥s t❤❛t

❚❤❡ ♣♦✐♥t

(t0 , y0 )

❜❡❧♦♥❣s t♦ t❤❡ ❝✉r✈❡

❡q✉❛t✐♦♥ ✐s s❛t✐s✜❡❞ ❢♦r ✐t✿

1 y = − t2 + C 2

♠❡❛♥s t❤❛t t❤❡

y0 = Cet0 . ❙♦❧✈❡ ❢♦r

C✿ C = y0 /et0 .

❉♦♥❡✦ ❙❡❝♦♥❞✱ ✏❚❤❡ ❝✉r✈❡s ❞♦♥✬t ✐♥t❡rs❡❝t✑ ✐s ✉♥❞❡rst♦♦❞ ❛s ✏❚✇♦ ❝✉r✈❡s t❤❛t ✐♥t❡rs❡❝t ❛r❡ t❤❡ s❛♠❡ ❝✉r✈❡✑✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ s✉❝❤ ❝✉r✈❡s✿

y = Cet , ❛♥❞

y = Ket , ❢♦r s♦♠❡ r❡❛❧ ♥✉♠❜❡rs

C

❛♥❞

K✳

❙✉♣♣♦s❡ t❤❡② ✐♥t❡rs❡❝t✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡r❡ ✐s ❛ ♣♦✐♥t

❜❡❧♦♥❣s t♦ ❜♦t❤✳ ❚❤❡r❡❢♦r❡✱ ❜♦t❤ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ❛r❡ s❛t✐s✜❡❞ ❢♦r t❤✐s ♣♦✐♥t✿

y0 = Cet0 , ❛♥❞

y0 = Ket0 .

❲❡ s♦❧✈❡ t❤✐s

s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s t♦ ❝♦♥❝❧✉❞❡✿

C =K. ❚❤❛t✬s t❤❡ s❛♠❡ ❝✉r✈❡✦

(t0 , y0 ) t❤❛t

✶✳✺✳

❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s

✹✼

❲❡ ❝♦♥s✐❞❡r❡❞ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ✭❞✐s❝r❡t❡ ❖❉❊✮ ♦❢ ✜rst ♦r❞❡r ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

∆y = f (t, y) . ∆t ❲❡ t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✳

❉❡✜♥✐t✐♦♥ ✶✳✺✳✸✿ ❖❉❊s ❙✉♣♣♦s❡

f

✐s s♦♠❡ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳ ❆♥

♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥

♦❢ ✜rst ♦r❞❡r✱ ♦r s✐♠♣❧② ❛♥ ❖❉❊✱ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ dy = f (t, y) dt

♦r

y ′ = f (t, y)

❍❡r❡✱ ✏♦r❞✐♥❛r②✑ r❡❢❡rs t♦ t❤❡ ❢❛❝t t❤❛t t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❛♥❞ ♦♥❧② ♦♥❡ ❞❡r✐✈❛t✐✈❡✳ ❚❤❡ ✏♦r❞❡r✑ r❡❢❡rs t♦ t❤❡ ♦r❞❡r ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✳ ❲❡ ✇✐❧❧ r❡❢❡r t♦

f

❛♥❞

h

❛s t❤❡ ✏r✐❣❤t✲❤❛♥❞ s✐❞❡✑✳

●❡♥❡r❛❧❧②✱ t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ ❛♥ ❖❉❊ ❤❛s ♥❡✐t❤❡r ♦❢ t❤❡ ❛❜♦✈❡ ♣❛tt❡r♥s✿

❈♦♥s✐❞❡r t❤❡ t✇♦ ❡①❛♠♣❧❡s ♦❢ ❖❉❊s ❢r♦♠ ❜❡❢♦r❡✿

y ′ = −gt + v0 =⇒ y(t) = −gt2 /2 + v0 t + C y′ = y =⇒ y(t) = Cet ❚❤❡r❡ ✐s ♦♥❡ s♦❧✉t✐♦♥ ❢♦r ❡❛❝❤

C✱

❛s ✇❡ ❦♥♦✇✳ ❊❛❝❤ ♦❢ t❤❡♠ ❤❛s ❞♦♠❛✐♥

(−∞, ∞)✳

❯♥❧✐❦❡ t❤❡s❡ t✇♦ ❖❉❊s✱ s♦♠❡ ♦t❤❡rs ❤❛✈❡ s♦❧✉t✐♦♥s t❤❛t ❝❛♥♥♦t ❜❡ ❞❡✜♥❡❞ t♦ t❤❡ ✇❤♦❧❡ r❡❛❧ ❧✐♥❡✳

❊①❛♠♣❧❡ ✶✳✺✳✹✿ ❛s②♠♣t♦t❡s ❙♦♠❡ ♦❢ t❤❡♠ s✐♠♣❧② r✉♥ ❛✇❛②✳ ❋♦r ❡①❛♠♣❧❡✱ ❤❡r❡ ✐s ❛ s✐♠♣❧❡ ❡①❛♠♣❧❡ ♦❢ s✉❝❤ ❛♥ ❖❉❊✿

y ′ = 1/x . ❚❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❢✉♥❝t✐♦♥✱

f (x, y) = 1/x , ✐s ✉♥❞❡✜♥❡❞ ❛t

x = 0✳

❋♦r ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✱ ✐ts ❞♦♠❛✐♥ ❝♦♥s✐sts ♥♦✇ ♦❢ t✇♦ ♦♣❡♥ ❤❛❧❢✲♣❧❛♥❡s✿

x 0. x = 0✳

✶✳✺✳

❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s

✹✽

❲❡ ❤❛✈❡✱ ✐♥ ❢❛❝t✱ t✇♦ ❢❛♠✐❧✐❡s ♦❢ s♦❧✉t✐♦♥s✿ ❢♦r x > 0; ❢♦r x < 0.

y = ln(−x) + C y = ln(x) + K

❚❤❡② ❡①✐st s❡♣❛r❛t❡❧② ❢r♦♠ ❡❛❝❤ ♦t❤❡r ❛♥❞ s❤♦✉❧❞ ♥♦t ❜❡ ❝♦♠❜✐♥❡❞ ❛r❜✐tr❛r✐❧②✳ ❚❤✐s ✐s t❤❡ r❡❛s♦♥ ✇❤② ✇❡ t❛❦❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦♥ s♦❧✉t✐♦♥s✳

❉❡✜♥✐t✐♦♥ ✶✳✺✳✺✿ s♦❧✉t✐♦♥ ♦❢ ❖❉❊ ❆ s♦❧✉t✐♦♥ ♦❢ ❖❉❊ y ′ = f (t, y) ✐s ❛ ❢✉♥❝t✐♦♥ y = y(t) ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I s✉❝❤ t❤❛t ❢♦r ❡✈❡r② t ✐♥ I ✇❡ ❤❛✈❡✿ y ′ (t) = f (t, y(t))

■t ♠❛② ❛❧s♦ ❜❡ ❝❛❧❧❡❞ ❛ str♦♥❣

s♦❧✉t✐♦♥✳

◆♦t❡ t❤❛t ✇❡ r❡q✉✐r❡ t❤❡ ❞♦♠❛✐♥ t♦ ❜❡ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ s♦ t❤❛t t❤❡ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ✐s ♣r♦♣❡r❧② ❞❡✜♥❡❞✳ ▲❡t✬s ✐♥t❡❣r❛t❡ t❤❡ ❡q✉❛t✐♦♥ ♦✈❡r s♦♠❡ ✐♥t❡r✈❛❧ [a, x] ✇✐t❤ ✈❛r✐❛❜❧❡ r✐❣❤t ❡♥❞✿ Z

x ′

y (t) dt = a

Z

x

f (t, y(t)) dt . a

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ s✐♠♣❧✐✜❡s✿ y(x) − y(a) =

❲❡ ❤❛✈❡ ❛♥ ✐♥t❡❣r❛❧

Z

x

f (t, y(t)) dt a

❡q✉❛t✐♦♥✳ ❉❡✜♥✐t✐♦♥ ✶✳✺✳✻✿ ✇❡❛❦ s♦❧✉t✐♦♥ ♦❢ ❖❉❊ ❆ ✇❡❛❦ s♦❧✉t✐♦♥ ♦❢ ❖❉❊ y ′ = f (t, y) ✐s ❛ ❢✉♥❝t✐♦♥ y ❝♦♥t✐♥✉♦✉s ♦♥ ❛♥ ❝❧♦s❡❞ ✐♥t❡r✈❛❧ [a, b] s✉❝❤ t❤❛t ❢♦r ❡✈❡r② x ✐♥ (a, b]✱ t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ✐s s❛t✐s✜❡❞✳

◆♦t❡ t❤❛t ✇❡ r❡q✉✐r❡ t❤❡ ❞♦♠❛✐♥ t♦ ❜❡ ❛♥ ❝❧♦s❡❞ ✐♥t❡r✈❛❧ s♦ t❤❛t ✇❡ ❝❛♥ ✐♥t❡❣r❛t❡✳

❊①❛♠♣❧❡ ✶✳✺✳✼✿ st❡♣✲❢✉♥❝t✐♦♥ ■♥ t❤❡ ❧❛t❡r ❝❛s❡ t❤❡ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ❛ st❡♣✲❢✉♥❝t✐♦♥✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ❤♦✇ ❡❛s② ✐t ✐s t♦ ♣r♦❞✉❝❡ t❤❡ s♦❧✉t✐♦♥s✿

✶✳✺✳

❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s

✹✾

❙✉❝❤ ❢✉♥❝t✐♦♥s ❛r❡ ❝❛❧❧❡❞ ✏♣✐❡❝❡✲✇✐s❡ ❧✐♥❡❛r✑✳

❊①❡r❝✐s❡ ✶✳✺✳✽

❉♦ ❛♥② ♦❢ t❤❡ s♦❧✉t✐♦♥s ✐♥t❡rs❡❝t❄

❊①❡r❝✐s❡ ✶✳✺✳✾

P❧♦t ❛ ❢❡✇ s♦❧✉t✐♦♥s ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❢✉♥❝t✐♦♥✿

1 0 2 −1

1 −1 0 1 −1 −1 1 −2 0 1 0 2

❆❜♦✈❡✱ ✇❡ s❤♦✇❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡♦r❡♠ ✶✳✺✳✶✵✿ ✇❡❛❦ s♦❧✉t✐♦♥s

❊✈❡r② ✭str♦♥❣✮ s♦❧✉t✐♦♥ ✐s ❛ ✇❡❛❦ s♦❧✉t✐♦♥✳ ❊①❛♠♣❧❡ ✶✳✺✳✶✶✿ ❝♦♥✈❡rs❡

❚❤❡ ❝♦♥✈❡rs❡ ✐s ❢❛❧s❡✳ ▲❡t✬s ❝❤♦♦s❡ t❤❡ r✐❣❤t✲❤❛♥❞ ❢✉♥❝t✐♦♥

f

t♦ ❜❡ t❤❡ s✐❣♥ ❢✉♥❝t✐♦♥✿

❚❤❡♥ ❛♥ ✏♦❜✈✐♦✉s✑ s♦❧✉t✐♦♥ ✐s t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡✿

❚❤❡ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❢❛✐❧s ❛t

t=0

❤♦✇❡✈❡r✿

y ′ = sign(t), ❈♦♥t✐♥✉♦✉s ✈s✳ ❞✐✛❡r❡♥t✐❛❜❧❡ ✈s✳ ✐♥t❡❣r❛❜❧❡✿

y = |t| .

✶✳✺✳

❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s

✺✵

❊①❛♠♣❧❡ ✶✳✺✳✶✷✿ ✐♥❝r❡♠❡♥t❛❧ ❝❤❛♥❣❡

❲❡❛❦ s♦❧✉t✐♦♥ ♠❛② ❡♠❡r❣❡ ❢r♦♠ ♣r♦❝❡ss❡s ❣✉✐❞❡❞ ❜② ❢✉♥❝t✐♦♥s t❤❛t ❝❤❛♥❣❡ ✐♥❝r❡♠❡♥t❛❧❧②✳ ❖♥❡ ❡①❛♠♣❧❡ ✐s ❤♦✇ t❤❡ ✐♥❝♦♠❡ t❛① ❡♠❡r❣❡s ❢r♦♠ t❤❡ ✐♥❝♦♠❡ t❛① r❛t❡✿

❲❡ ❤❛✈❡ t❤❡ ❖❉❊ ♦♥ t❤❡ ❧❡❢t ❛♥❞ ❛ s♦❧✉t✐♦♥ ♦♥ t❤❡ r✐❣❤t✳ ■t ✐s s♦❧✈❡❞ ❜② ❛ s✐♠♣❧❡ ✐♥t❡❣r❛t✐♦♥✳ ❲❡ ✐❣♥♦r❡ t❤❡ ♥♦♥✲❞✐✛❡r❡♥t✐❛❜✐❧✐t② ❛t t❤❡ ❡♥❞s ♦❢ t❤❡ ❜r❛❝❦❡ts✱ ❜✉t t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ✇❡❛❦ s♦❧✉t✐♦♥ ✐s st✐❧❧ r❡q✉✐r❡❞✦ ❆♥♦t❤❡r s♦✉r❝❡ ✐s s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣✳ ❲❡❛❦ s♦❧✉t✐♦♥s ♠❛② ❛❧s♦ ❝♦♠❡ ❢r♦♠ s❛♠♣❧✐♥❣ ♦❢ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ❢✉♥❝t✐♦♥ ♦❢ ❛♥ ❖❉❊ ❛s ❛ ✇❛② t♦ s♦❧✈❡ ✐t ♥✉♠❡r✐❝❛❧❧②✿

❚❤❡ ❢✉♥❝t✐♦♥ ✐s r❡♣❧❛❝❡❞ ✇✐t❤ ❛ st❡♣✲❢✉♥❝t✐♦♥✳ ❚❤❡ ✈❡r② ✜rst ♦❜s❡r✈❛t✐♦♥ ✇❡ s❤♦✉❧❞ ♠❛❦❡ ✐s t❤❛t✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡✜♥✐t✐♦♥✱ t❤❡r❡ ✐s ❛ s❡♣❛r❛t❡ s♦❧✉t✐♦♥ ❢♦r ❡❛❝❤ ✐♥t❡r✈❛❧ ✐♥s✐❞❡ I ✱ ✐✳❡✳✱ ✐❢ y ✐s ❛ s♦❧✉t✐♦♥ ♦♥ ✐♥t❡r✈❛❧ I t❤❡♥ s♦ ✐s ✐ts r❡str✐❝t✐♦♥ z = y t♦ ❛♥② ✭♦♣❡♥ J ♦r ❝❧♦s❡❞ r❡s♣❡❝t✐✈❡❧②✮ ✐♥t❡r✈❛❧ J ✐♥ I ✳

✶✳✺✳

❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s

✺✶

❚❤✐s ♥❡✇ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ✈❡r② s✐♠♣❧②✿

z(t) = y(t)

❢♦r ❡❛❝❤

t

✐♥

J.

❋♦r t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡✱ ❡❛❝❤ s❡❣♠❡♥t ♦❢ ❛ ❝✉r✈❡ s❡❡♥ ❜❡❧♦✇ ✐s ❛ s♦❧✉t✐♦♥✿

❲❛r♥✐♥❣✦

❚❤❡ ✐♥t❡r✈❛❧ I ♠❛② ❜❡ ✐♥✜♥✐t❡✿ I = (−∞, +∞), (−∞, b), (a, +∞) . ❘❡❝❛❧❧ t❤❛t s♦❧✈✐♥❣ ❛♥ ❖❉❊ ♠❡❛♥s t♦ ❢❛❝❡ ❛ ✏✜❡❧❞ ♦❢ s❧♦♣❡s✑ ❛♥❞ t❤❡♥ s❡❛r❝❤ ❢♦r ❝✉r✈❡s ✇✐t❤ t❤❡s❡ t❛♥❣❡♥ts✿

❙♦❧✈✐♥❣ ♣r♦❜❧❡♠s ❛❜♦✉t t❤❡ ❢❛❧❧ ♦❢ ❛ ❜❛❧❧ t❤r♦✇♥ ❢r♦♠ ❛ ♣❛rt✐❝✉❧❛r ❧♦❝❛t✐♦♥ ❛t ❛ ♣❛rt✐❝✉❧❛r t✐♠❡ r❡q✉✐r❡❞ ♣✐❝❦✐♥❣ ♦✉t ♦♥❡ ♦❢ t❤❡♠✿

❚❤❡r❡ ✐s ❛ r❡❧❛t❡❞ ❝♦♥❝❡♣t ✐♥ t❤❡ t❤❡♦r② ♦❢ ❖❉❊s✳

✶✳✺✳

❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s

✺✷

❉❡✜♥✐t✐♦♥ ✶✳✺✳✶✸✿ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ ❋♦r ❛ ❣✐✈❡♥ ❖❉❊ y ′ = f (t, y) ❛♥❞ ❛ ❣✐✈❡♥ ♣❛✐r ♦❢ ✈❛❧✉❡s (t0 , y0 )✱ t❤❡ ✐♥✐t✐❛❧ ♣r♦❜❧❡♠✱ ♦r ❛♥ ■❱P✱ ✐s y ′ = f (t, y),

❛♥❞ ✐ts

s♦❧✉t✐♦♥

y(t0 ) = y0 ✳

✈❛❧✉❡

y(t0 ) = y0

✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❖❉❊ t❤❛t s❛t✐s✜❡s t❤❡

✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✱

❲❛r♥✐♥❣✦ ❆ ❞✐✛❡r❡♥t ✐♥✐t✐❛❧ ✈❛❧✉❡ ❝♦♥❞✐t✐♦♥ ♠❛② ♣r♦❞✉❝❡ t❤❡ s❛♠❡ s♦❧✉t✐♦♥✳

■♥ t❤❡ ❡①❛♠♣❧❡ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❛❜❧❡ ❢♦r ❡❛❝❤ (t0 , y0 ) ✐♥ t❤❡ ❞♦♠❛✐♥✳ ❚❤❛t ✐s ✇❤② t❤❡ ❝✉r✈❡s ✜❧❧ t❤❡ ♣❧❛♥❡✳ ■♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ t❤❡② ✜❧❧ ❛❧❧ ♣❧❛♥❡ ❡①❝❡♣t t❤❡ ❧✐♥❡ t = 0✳ ❚❤❡ ✜rst ♣r♦♣❡rt② ♦❢ t❤❡ s♦❧✉t✐♦♥ s❡t ✐s✿ ◮ ❚❤❡ ❝✉r✈❡s ❝♦✈❡r t❤❡ ♣❧❛♥❡✳

❉❡✜♥✐t✐♦♥ ✶✳✺✳✶✹✿ ❡①✐st❡♥❝❡

❲❡ s❛② t❤❛t ❛♥ ❖❉❊ s❛t✐s✜❡s t❤❡ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ❛t ❛ ♣♦✐♥t (t0 , y0 ) ✇❤❡♥ t❤❡ ■❱P✿ y ′ = f (t, y),

y(t0 ) = y0 ,

❤❛s ❛ s♦❧✉t✐♦♥✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❡✈❡r② ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ t❤❡r❡ ❡①✐sts ❛t ❧❡❛st ♦♥❡ s♦❧✉t✐♦♥ t❤❛t st❛rts t❤❡r❡✳ ■t ❞♦❡s♥✬t ♠❛tt❡r ❤♦✇ s♠❛❧❧ ✐s t❤❡ ❞♦♠❛✐♥ ♦❢ t❤✐s s♦❧✉t✐♦♥✳ ❚❤❡r❡ ❛r❡ ♥♦ ✭str♦♥❣ ♦r ✇❡❛❦✮ s♦❧✉t✐♦♥s ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡✿

■❢ ②♦✉r ♠♦❞❡❧ ♦❢ ❛ r❡❛❧✲❧✐❢❡ ♣r♦❝❡ss ❞♦❡s♥✬t s❛t✐s❢② t❤✐s ♣r♦♣❡rt②✱ ✐t ♠❛② ❜❡ ✐♥❛❞❡q✉❛t❡✳ ■t ✐s ❛s ✐❢ t❤❡ ♣r♦❝❡ss st❛rts ❜✉t ♥❡✈❡r ❝♦♥t✐♥✉❡s✳ ❋r♦♠ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t ❛♥t✐❞❡r✐✈❛t✐✈❡s✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳

❚❤❡♦r❡♠ ✶✳✺✳✶✺✿ ❊①✐st❡♥❝❡ ❢♦r ▲♦❝❛t✐♦♥✲■♥❞❡♣❡♥❞❡♥t ❖❉❊ ❆♥ ❖❉❊ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ✇❤✐❝❤ ✐s ❛ ❢✉♥❝t✐♦♥ ✐♥❞❡♣❡♥❞❡♥t ♦❢ y ❛♥❞ ✐♥t❡✲ ❣r❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ t ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I ✿ y ′ = f (t) ,

s❛t✐s✜❡s t❤❡ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ❢♦r ❡✈❡r② ♣♦✐♥t (t0 , y0 ) ✇✐t❤ t0 ✐♥ I ✳

✶✳✺✳

❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s

✺✸

Pr♦♦❢✳ ■♥❞❡❡❞✱ t❤❡ s♦❧✉t✐♦♥ ✐s ❥✉st t❤❡ ❛♣♣r♦♣r✐❛t❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ f ✳ ❚❤❡ s❡❝♦♥❞ ♣r♦♣❡rt② ♦❢ t❤❡ s♦❧✉t✐♦♥ s❡t ✐s✿ ◮ ❚❤❡ ❝✉r✈❡s ❞♦♥✬t ✐♥t❡rs❡❝t✳

❉❡✜♥✐t✐♦♥ ✶✳✺✳✶✻✿ ✉♥✐q✉❡♥❡ss ❲❡ s❛② t❤❛t ❛♥ ❖❉❊ s❛t✐s✜❡s t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt② ❛t ❛ ♣♦✐♥t (t0 , y0 ) ✐❢ ❡✈❡r② ♣❛✐r ♦❢ s♦❧✉t✐♦♥s✱ y1 , y2 ✱ ♦❢ t❤❡ ■❱P✿ y ′ = f (t, y),

y(t0 ) = y0

❛r❡ ❡q✉❛❧✱ y1 (t) = y2 (t)

❢♦r ❡✈❡r② t ✐♥ s♦♠❡ ♦♣❡♥ ✐♥t❡r✈❛❧ t❤❛t ❝♦♥t❛✐♥s t0 ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❡✈❡r② ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ t❤❡r❡ ✐s ❛t ♠♦st ♦♥❡ s♦❧✉t✐♦♥✱ ✐✳❡✳✱ t❤❡r❡✳

✉♥✐q✉❡

s♦❧✉t✐♦♥✱ t❤❛t st❛rts

❚♦ ✈✐♦❧❛t❡ ✉♥✐q✉❡♥❡ss✱ t❤❡ t✇♦ str♦♥❣ s♦❧✉t✐♦♥s ✇✐❧❧ ❤❛✈❡ t♦ ❤❛✈❡ t❤❡ s❛♠❡ s❧♦♣❡ ✭✐✳❡✳✱ t❤❡② ❛r❡ t❛♥❣❡♥t t♦ ❡❛❝❤ ♦t❤❡r✮ ❛t t❤❡ ❢♦r❦✿

❊①❛♠♣❧❡ ✶✳✺✳✶✼✿ ♥♦♥✲✉♥✐q✉❡♥❡ss ❲❡❛❦ s♦❧✉t✐♦♥s ❝❛♥ ❡❛s✐❧② ♣❛rt ✇❛②s✿

❚❤❡ ❞❡✜♥✐t✐♦♥ s❛②s t❤❛t t❤❡✐r r❡str✐❝t✐♦♥s t♦ s♦♠❡ ✐♥t❡r✈❛❧ J ❛r❡ ❡q✉❛❧✿ y1 = y2 . J

J

■❢ ②♦✉r ♠♦❞❡❧ ♦❢ ❛ r❡❛❧✲❧✐❢❡ ♣r♦❝❡ss ❞♦❡s♥✬t s❛t✐s❢② t❤✐s ♣r♦♣❡rt②✱ ✐t ♠❛② ❜❡ ✐♥❛❞❡q✉❛t❡✳ ■t✬s ❛s ✐❢ ②♦✉ ❤❛✈❡ ❛❧❧ t❤❡ ❞❛t❛ ❜✉t ❝❛♥✬t ♣r❡❞✐❝t ❡✈❡♥ t❤❡ ♥❡❛r❡st ❢✉t✉r❡✳ ❋r♦♠ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t ❛♥t✐❞❡r✐✈❛t✐✈❡s✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳

✶✳✺✳

❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s

✺✹

❚❤❡♦r❡♠ ✶✳✺✳✶✽✿ ❯♥✐q✉❡♥❡ss ❢♦r ▲♦❝❛t✐♦♥✲■♥❞❡♣❡♥❞❡♥t ❖❉❊

❆♥ ❖❉❊ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ✇❤✐❝❤ ✐s ❛ ❢✉♥❝t✐♦♥ ✐♥❞❡♣❡♥❞❡♥t ♦❢ y ❛♥❞ ✐♥t❡✲ ❣r❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ t ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I ✿ y ′ = f (t) ,

s❛t✐s✜❡s t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt② ❢♦r ❡✈❡r② ♣♦✐♥t (t0 , y0 ) ✇✐t❤ t0 ✐♥ I ✳ Pr♦♦❢✳

■♥❞❡❡❞✱ t❤❡ s♦❧✉t✐♦♥ ✐s ❛❧✇❛②s ♦♥❡ ♦❢ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ f ✳ ❲❡ ❛❧s♦ ❦♥♦✇ t❤❛t t❤❡s❡ ❛♥t✐❞❡r✐✈❛t✐✈❡s ❛❧✇❛②s ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡✐r ❣r❛♣❤s ♥❡✈❡r ✐♥t❡rs❡❝t✳ ❲❡ ✇✐❧❧ s❡❡ ❖❉❊s ✇✐t❤♦✉t t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt② ❧❛t❡r✳ ❊①❛♠♣❧❡ ✶✳✺✳✶✾✿ ❧♦❣✐st✐❝ ❡q✉❛t✐♦♥

❋♦r t❤❡ ❧♦❣✐st✐❝ ❡q✉❛t✐♦♥✱ t❤❡ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ∆y = ky(1 − y)∆t ♠❛② ♣r♦❞✉❝❡ str❛♥❣❡ r❡s✉❧ts ✭❧❡❢t✮ ❜✉t ♥♦t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❖❉❊ y ′ = ky(1 − y) ✭r✐❣❤t✮✿

❖♥❡ ❝❛♥ s❤♦✇ t❤❛t t❤✐s ✐s tr✉❡ ❜② ❛♣♣❡❛❧✐♥❣ t♦ t❤❡ ✉♥✐q✉❡♥❡ss ♦❢ t❤❡ s♦❧✉t✐♦♥s✿ t❤❡② ❝❛♥✬t ❝r♦ss t❤❡ tr✐✈✐❛❧ s♦❧✉t✐♦♥s y = 0 ❛♥❞ y = 1✳ ❙♦✱ ❢♦r ❛ ♠♦❞❡❧ t♦ ❜❡ ✈❛❧✐❞✱ ✐t ❤❛s t♦ s❛t✐s❢②✿ ✶✳ ❊①✐st❡♥❝❡✿ ❚❤❡ ♣❧❛♥❡ ✐s ❢✉❧❧② ❝♦✈❡r❡❞ ❜② t❤❡ s♦❧✉t✐♦♥ ❝✉r✈❡s✳ ✷✳ ❯♥✐q✉❡♥❡ss✿ ❚❤❡ s♦❧✉t✐♦♥ ❝✉r✈❡s ❞♦♥✬t ✐♥t❡rs❡❝t✳ ❚❤❡r❡ ✐s ❛♥♦t❤❡r ✐♠♣♦rt❛♥t ❝♦♥❞✐t✐♦♥✦ ❊①❛♠♣❧❡ ✶✳✺✳✷✵✿ ✇❡❛t❤❡r ❢♦r❡❝❛st

■t ✐s ✇❡❧❧ ❦♥♦✇♥ t❤❛t ❛ ✇❡❛t❤❡r ❢♦r❡❝❛st ❜❡②♦♥❞ 3 ❞❛②s ✐s ✈❡r② ✉♥r❡❧✐❛❜❧❡✳ ❆ s✐t✉❛t✐♦♥ ❧✐❦❡ t❤❛t ♦❝❝✉rs ✇❤❡♥ s♠❛❧❧ ❝❤❛♥❣❡ ✐♥ t❤❡ ✐♥♣✉t ♣r♦❞✉❝❡s ❛ ❧❛r❣❡ ❝❤❛♥❣❡ ✐♥ t❤❡ ♦✉t♣✉t✿

✶✳✺✳

❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s

✺✺

❙✉♣♣♦s❡ t❤❡ ❢♦r❡❝❛st ❞❡♣❡♥❞s ♦♥ t❤❡ ♠❡❛s✉r✐♥❣ t❤❡ ❝✉rr❡♥t t❡♠♣❡r❛t✉r❡✳ ■♥ ❛♥ ❛tt❡♠♣t t♦ ✐♠♣r♦✈❡ t❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❢♦r❡❝❛st✱ ✇❡ ♠❛❦❡ t❤✐s ♠❡❛s✉r❡♠❡♥t ♠♦r❡ ❛♥❞ ♠♦r❡ ♣r❡❝✐s❡✳ ❍♦✇❡✈❡r✱ t❤❡ r❡s✉❧ts ✇✐❧❧ ❡✈❡♥t✉❛❧❧② r✉♥ ❛✇❛② ❢r♦♠ t❤❡ ♥♦♠✐♥❛❧✱ y = 0✳ ❲❤❛t ✇❡ ❛r❡ ❛❢t❡r ✐s ♠♦❞❡❧s t❤❛t ♣r♦❞✉❝❡ s♦❧✉t✐♦♥s t❤❛t ❛♣♣r♦❛❝❤ t❤❡ ♥♦♠✐♥❛❧ ❛s t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❛♣♣r♦❛❝❤ t❤❡ ♥♦♠✐♥❛❧✿

❚❤✐s ✐s ✇❤❛t ✇❡ ♠❛② r❡q✉✐r❡ ✐♥ ♦r❞❡r t♦ ❛✈♦✐❞ t❤✐s ❦✐♥❞ ♦❢ ✐♥st❛❜✐❧✐t②✳

❉❡✜♥✐t✐♦♥ ✶✳✺✳✷✶✿ ❝♦♥t✐♥✉✐t② ♦❢ ■❱P ❲❡ s❛② t❤❛t ❛♥ ❖❉❊ s❛t✐s✜❡s t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ■❱P ♣r♦♣❡rt② ❛t (t0 , y0 ) ✐❢ ❢♦r ❡❛❝❤ s♦❧✉t✐♦♥ y ♦❢ t❤❡ ■❱P✱ ❛♥② t1 > t0 ✇✐t❤✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ y ✱ ❛♥❞ ❛♥② s❡q✉❡♥❝❡ ♦❢ s♦❧✉t✐♦♥s✱ yn ❞❡✜♥❡❞ ♦♥ [t0 , t1 ]✱ ♦❢ t❤❡ ❖❉❊ ✇❡ ❤❛✈❡✱ ❛s n → ∞✱ yn (t0 ) → y(t0 ) =⇒ yn (t1 ) → y(t1 ) .

❙✉❝❤ ❛♥ ❖❉❊ ❝❛♥ ❛❧s♦ ❜❡ ❝❛❧❧❡❞ st❛❜❧❡✳

❊①❛♠♣❧❡ ✶✳✺✳✷✷✿ ♣❛r❛❜♦❧❛s ▲❡t✬s t❛❦❡ t❤✐s ❢❛♠✐❧② ♦❢ ♣❛r❛❜♦❧❛s ❛❣❛✐♥✿ y = −t2 /2 + C .

❆ s❡q✉❡♥❝❡ ♦❢ s♦❧✉t✐♦♥s ✇✐❧❧ ❧♦♦❦ ❧✐❦❡ t❤✐s✿ yn (t) = −t2 /2 + Cn , n = 1, 2, 3...

❚❤❡♥✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❤❛✈❡ ❛t ♦♥❡ ❡♥❞✱ t = t0 ✿ yn (t0 ) = −t20 /2 + Cn , n = 1, 2, 3...

■❢ t❤✐s s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs ❝♦♥✈❡r❣❡s✱ yn (t0 ) = −t20 /2 + Cn → y(t0 ) = −t20 /2 + C ,

t❤❡♥ Cn → C ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ❝♦♥✈❡r❣❡♥❝❡ ♦♥ t❤❡ ♦t❤❡r ❡♥❞✱ t = t0 ✿ yn (t1 ) = −t21 /2 + Cn → y(t1 ) = −t21 /2 + C .

❊①❡r❝✐s❡ ✶✳✺✳✷✸ Pr♦✈❡ t❤❡ ♣r♦♣❡rt② ❢♦r t❤❡ ❢❛♠✐❧② ♦❢ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s y = Cet ✳

✶✳✺✳

❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s

✺✻

❊①❡r❝✐s❡ ✶✳✺✳✷✹ ❲❤❛t ❛❜♦✉t ❝♦♥✈❡r❣❡♥❝❡ ♦❢

❢✉♥❝t✐♦♥s yn → y❄ ❲❤❛t ❛❜♦✉t t❤❡ ✉♥✐❢♦r♠ ❝♦♥✈❡r❣❡♥❝❡❄

❊①❛♠♣❧❡ ✶✳✺✳✷✺✿ ❝♦♥t✐♥✉✐t②❄ ■♥ ♦r❞❡r t♦ ❡①♣❧❛✐♥ t❤❡ t❡r♠ ✏❝♦♥t✐♥✉✐t②✑✱ ❧❡t✬s ❞❡✜♥❡ ❛ ❢✉♥❝t✐♦♥

Q(z) = y(t1 )

✇❤❡♥

Q

♦❢ t❤❡

y ✲❛①✐s

t♦ ✐ts❡❧❢ ❜②✿

y(t0 ) = z .

■t✬s ❛s ✐❢ t❤❡ ✢♦✇ r❡❛rr❛♥❣❡s t❤❡ ♠♦❧❡❝✉❧❡s✳ ❲❡ t❤❡♥ ❜✉✐❧❞ t❤❡ ❣r❛♣❤ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✿

■t ✐s ❝♦♥t✐♥✉♦✉s✳

❊①❡r❝✐s❡ ✶✳✺✳✷✻ Pr♦✈❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s✳

■❢ ②♦✉r ♠♦❞❡❧ ♦❢ ❛ r❡❛❧✲❧✐❢❡ ♣r♦❝❡ss ❞♦❡s♥✬t s❛t✐s❢② t❤✐s ♣r♦♣❡rt②✱ ✐t ♠❛② ❜❡ ✐♥❛❞❡q✉❛t❡✳ ■t✬s ❛s ✐❢ ❛♥② ❡rr♦r ✐♥ t❤❡ ✐♥✐t✐❛❧ ❞❛t❛ ✕ ♥♦ ♠❛tt❡r ❤♦✇ s♠❛❧❧ ✕ ♠❛② ❝❛✉s❡ ❛ s✐❣♥✐✜❝❛♥t ❡rr♦r ✐♥ ②♦✉r ♣r❡❞✐❝t✐♦♥✳

❊①❛♠♣❧❡ ✶✳✺✳✷✼✿ ❞✐s❝♦♥t✐♥✉✐t② ♦❢ ■❱P ❲❡❛❦ s♦❧✉t✐♦♥ ❝❛♥ ❡❛s✐❧② ✈✐♦❧❛t❡ t❤✐s ♣r♦♣❡rt②✿

❋r♦♠ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t ❛♥t✐❞❡r✐✈❛t✐✈❡s✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳

❚❤❡♦r❡♠ ✶✳✺✳✷✽✿ ❈♦♥t✐♥✉✐t② ♦❢ ■❱P ❢♦r ▲♦❝❛t✐♦♥✲■♥❞❡♣❡♥❞❡♥t ❖❉❊

❆♥ ❖❉❊ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ✇❤✐❝❤ ✐s ❛ ❢✉♥❝t✐♦♥ ✐♥❞❡♣❡♥❞❡♥t ♦❢ y ❛♥❞ ✐♥t❡✲ ❣r❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ t ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I ✿ y ′ = f (t) ,

s❛t✐s✜❡s t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ■❱P ♣r♦♣❡rt② ❢♦r ❡✈❡r② ♣♦✐♥t (t0 , y0 ) ✇✐t❤ t0 ✐♥ I ✳

✶✳✺✳

❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s

✺✼

Pr♦♦❢✳

■♥❞❡❡❞✱ t❤❡ s♦❧✉t✐♦♥ ✐s ❛❧✇❛②s ♦♥❡ ♦❢ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ f ✳ ❲❡ ❛❧s♦ ❦♥♦✇ t❤❛t t❤❡s❡ ❛♥t✐❞❡r✐✈❛t✐✈❡s ❛❧✇❛②s ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❝♦♥✈❡r❣❡ t♦ ❡❛❝❤ ♦t❤❡r ✇❤❡♥ t❤✐s ❝♦♥st❛♥t ❣♦❡s t♦ ③❡r♦✳ ◆❡①t✱ ❖❉❊s ♣r♦❞✉❝❡ ❢❛♠✐❧✐❡s ♦❢ ❝✉r✈❡s ❛s t❤❡ s❡ts ♦❢ t❤❡✐r s♦❧✉t✐♦♥s✳✳✳ ❛♥❞ ✈✐❝❡ ✈❡rs❛✿ ■❢ ❛ ❢❛♠✐❧② ♦❢ ❝✉r✈❡s ✐s ❣✐✈❡♥ ❜② ❛♥ ❡q✉❛t✐♦♥ ✇✐t❤ ❛ s✐♥❣❧❡ ♣❛r❛♠❡t❡r C ✱ t❤❡♥ ❞✐✛❡r❡♥t✐❛t✐♥❣ t❤❡ ❡q✉❛t✐♦♥ ✐♥ ❛ ♣❛rt✐❝✉❧❛r ✇❛② ✇✐❧❧ ♣r♦❞✉❝❡ ❛♥ ❖❉❊✳ ❊①❛♠♣❧❡ ✶✳✺✳✷✾✿ ✈❡rt✐❝❛❧❧② s❤✐❢t❡❞

❋✐rst✱ t❤❡ ❢❛♠✐❧② ♦❢ ✈❡rt✐❝❛❧❧② s❤✐❢t❡❞ ❣r❛♣❤s✱ y = g(t) + C ,

❝r❡❛t❡s✱ ✇❤❡♥ ❞✐✛❡r❡♥t✐❛t❡❞✱ ❛♥ ❖❉❊✿

y ′ = g ′ (t) .

❙❡❝♦♥❞✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❢❛♠✐❧② ♦❢ ✈❡rt✐❝❛❧❧② str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ❣r❛♣❤s✱ y = Cet .

❲❤❡♥ ❞✐✛❡r❡♥t✐❛t❡❞✱ t❤❡ r❡s✉❧t✐♥❣ ❡q✉❛t✐♦♥ st✐❧❧ ❤❛s C ❤♦✇❡✈❡r✦ ❚♦ ❝r❡❛t❡s ❛♥ ❖❉❊ ✇✐t❤♦✉t C ✱ ❧❡t✬s ✜rst s♦❧✈❡ ❢♦r C ❛♥❞ t❤❡♥ ❞✐✛❡r❡♥t✐❛t❡✿ C = ye−t =⇒ 0 = y ′ e−t + y e−t

❙✐♥❝❡ e−t > 0✱ ✇❡ ❤❛✈❡✿

′

= y ′ e−t − ye−t = (y ′ − y)e−t .

y′ − y = 0 .

❊①❛♠♣❧❡ ✶✳✺✳✸✵✿ ❝✐r❝❧❡s

❲❤❛t ❛❜♦✉t t❤❡s❡ ❝♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s❄

❚❤❡② ❛r❡ ❣✐✈❡♥ ❜②

x2 + y 2 = C > 0 .

❲❡ ❞✐✛❡r❡♥t✐❛t❡ ✭✐♠♣❧✐❝✐t❧②✮ ✇✐t❤ r❡s♣❡❝t t♦ x✿ 2x + 2yy ′ = 0 ,

♦r✱

x y′ = − . y

❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s y 6= 0✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ❝❛♥♥♦t ❜❡ s❛t✐s✜❡❞ ♦♥ t❤❡ x✲❛①✐s✳ ❋✉rt❤❡r♠♦r❡✱ ❡✈❡♥ ✐❢ t❤✐s ❢✉♥❝t✐♦♥ ✇❛s ❞❡✜♥❡❞ ♦♥ t❤❡ x✲❛①✐s✱ t❤❡ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ✇♦✉❧❞ st✐❧❧ ❜r❡❛❦ ❞♦✇♥✿

✶✳✺✳

❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s

✺✽

❆s ②♦✉ ❝❛♥ s❡❡✱ ✇❡ ❝❛♥♥♦t ❡①t❡♥❞ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❧❡❢t ♦❢ t❤✐s ♣♦✐♥t✳ ❙✐♠✐❧❛r❧②✱ ✇❡ ❝❛♥♥♦t ❡①t❡♥❞ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ r✐❣❤t ♦❢ t❤❡ ♣♦✐♥t ✐❢ ✐t ✐s ♦♥ t❤❡ ♦t❤❡r s✐❞❡ ♦❢ 0✳ ❊①❡r❝✐s❡ ✶✳✺✳✸✶

❲❤❛t ❛❜♦✉t ✉♥✐q✉❡♥❡ss❄ ❊①❛♠♣❧❡ ✶✳✺✳✸✷✿ ❤②♣❡r❜♦❧❛s

❚❤❡s❡ ❤②♣❡r❜♦❧❛s ❛r❡ ❣✐✈❡♥ ❜② t❤❡s❡ ❡q✉❛t✐♦♥s✿ xy = C .

✭■♥ t❤❡ ❝❛s❡ ✇❤❡♥ C = 0✱ ✇❡ ❤❛✈❡ t❤❡ t✇♦ ❛①❡s✳✮ ❚❤❡♥✱ ✇❡ ❤❛✈❡ ❛♥ ❖❉❊✿ y + xy ′ = 0 ,

♦r✱ y′ =

y . x

❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s x 6= 0✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ❝❛♥♥♦t ❜❡ s❛t✐s✜❡❞ ♦♥ t❤❡ y ✲❛①✐s✳ ❋✉rt❤❡r♠♦r❡✱ ❡✈❡♥ ✐❢ t❤✐s ❢✉♥❝t✐♦♥ ✇❛s ❞❡✜♥❡❞ ♦♥ t❤❡ y ✲❛①✐s✱ t❤❡ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ✇♦✉❧❞ st✐❧❧ ❜r❡❛❦ ❞♦✇♥✿

❲❡ ❝❛♥♥♦t ❤❛✈❡ ❛ s♦❧✉t✐♦♥ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ y ✲❛①✐s✳

✶✳✺✳

❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s

✺✾

■t ❛♣♣❡❛rs t❤❛t t❤❡ ♣r❡s❡♥❝❡ ♦r t❤❡ ❛❜s❡♥❝❡ ♦❢ t❤❡ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ❞❡♣❡♥❞s ♦♥ t❤❡ ❝♦♥t✐♥✉✐t② ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❖❉❊✳ ❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t t❤❡♦r❡♠ ❧✐❡s ♦✉ts✐❞❡ t❤❡ s❝♦♣❡ ♦❢ t❤✐s ❜♦♦❦✳

❚❤❡♦r❡♠ ✶✳✺✳✸✸✿ ❊①✐st❡♥❝❡

❙✉♣♣♦s❡ U ✐s ❛♥ ♦♣❡♥ s❡t ♦♥ t❤❡ ♣❧❛♥❡ (t, y) t❤❛t ❝♦♥t❛✐♥s (t0 , y0 ) ❛♥❞ s✉♣♣♦s❡ t❤❛t ❛ ❢✉♥❝t✐♦♥ z = f (t, y) ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❞❡✜♥❡❞ ♦♥ U ✐s • ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ t✱ ❛♥❞ • ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ y ✳ ❚❤❡♥ t❤❡ ❖❉❊ y ′ = f (t, y) s❛t✐s✜❡s t❤❡ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ❛t (t0 , y0 )✳ ❆t ✐ts s✐♠♣❧❡st✱ t❤✐s ♦♣❡♥ s❡t

U

✐s ❛ r❡❝t❛♥❣❧❡✱ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ✐♥t❡r✈❛❧s

j

❛♥❞

G✿

❲❛r♥✐♥❣✦ ❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ s♦❧✉t✐♦♥ ❞♦❡s♥✬t ❤❛✈❡ t♦ ❜❡ t❤❡ ✇❤♦❧❡

J

❜❡❝❛✉s❡ ✐t ♠❛② ❡①✐t t❤❡ r❡❝t❛♥❣❧❡ t❤r♦✉❣❤

✐ts t♦♣ ♦r ✐ts ❜♦tt♦♠✿

❊①❡r❝✐s❡ ✶✳✺✳✸✹ ❲❤❛t ✐❢

G = (−∞, ∞)❄

❊①❛♠♣❧❡ ✶✳✺✳✸✺✿ ♥♦♥✲✉♥✐q✉❡♥❡ss ❈♦♥s✐❞❡r t❤✐s ❖❉❊✿

◆♦t✐❝❡ t❤❛t t❤❡r❡ ✐s ♦♥❡ ♠♦r❡ s♦❧✉t✐♦♥

dy = y 2/3 . dx ♣❛ss✐♥❣ t❤r♦✉❣❤ (0, 0)

■t ✐s

y=

(

0 1 3 x 27

✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ tr✐✈✐❛❧ s♦❧✉t✐♦♥

❢♦r

x < 0,

❢♦r

x ≥ 0.

y = 0✿

✶✳✺✳

❙♦❧✉t✐♦♥ s❡ts ♦❢ ❖❉❊s

✻✵

❲❤❛t ♠❛❦❡s t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ y 2/3 s♣❡❝✐❛❧ ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡s❄ ■ts ❞❡r✐✈❛t✐✈❡ ✐s ✐♥✜♥✐t❡ ❛t y = 0✳ ■t ❛♣♣❡❛rs t❤❛t t❤❡ ♣r❡s❡♥❝❡ ♦r t❤❡ ❛❜s❡♥❝❡ ♦❢ t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt② ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❖❉❊✳ ❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t t❤❡♦r❡♠ ❧✐❡s ♦✉ts✐❞❡ t❤❡ s❝♦♣❡ ♦❢ t❤✐s ❜♦♦❦✳ ❚❤❡ s❡t✲✉♣ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❛t ♦❢ t❤❡ ❧❛st t❤❡♦r❡♠ ❡①❝❡♣t✿ ◮ ❝♦♥t✐♥✉✐t② ❢♦r y ✐s r❡♣❧❛❝❡❞ ✇✐t❤ ❞✐✛❡r❡♥t✐❛❜✐❧✐t②✳

❚❤❡♦r❡♠ ✶✳✺✳✸✻✿ ❯♥✐q✉❡♥❡ss ✭✰❊①✐st❡♥❝❡✮ ❙✉♣♣♦s❡ U ✐s ❛♥ ♦♣❡♥ s❡t ♦♥ t❤❡ ♣❧❛♥❡ (t, y) t❤❛t ❝♦♥t❛✐♥s (t0 , y0 ) ❛♥❞ s✉♣♣♦s❡ t❤❛t ❛ ❢✉♥❝t✐♦♥ z = f (t, y) ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❞❡✜♥❡❞ ♦♥ U ✐s • ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ t✱ ❛♥❞ • ❞✐✛❡r❡♥t✐❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ y ✳ ❚❤❡♥ t❤❡ ❖❉❊ y ′ = f (t, y) s❛t✐s✜❡s t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt② ❛t (t0 , y0 )✳

❲❛r♥✐♥❣✦ ❚❤✐s ✐s

♥♦t

✇❤❛t ♥♦♥✲✉♥✐q✉❡♥❡ss ❧♦♦❦s ❧✐❦❡✿

■t ❛♣♣❡❛rs t❤❛t t❤❡ ♣r❡s❡♥❝❡ ♦r t❤❡ ❛❜s❡♥❝❡ ♦❢ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ■❱P ♣r♦♣❡rt② ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❖❉❊✳ ❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t t❤❡♦r❡♠ ❧✐❡s ♦✉ts✐❞❡ t❤❡ s❝♦♣❡ ♦❢ t❤✐s ❜♦♦❦✳ ❚❤❡ s❡t✲✉♣ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❛t ♦❢ t❤❡ ❧❛st t❤❡♦r❡♠✳

❚❤❡♦r❡♠ ✶✳✺✳✸✼✿ ❈♦♥t✐♥✉✐t② ♦❢ ■❱P ✭✰❯♥✐q✉❡♥❡ss✰❊①✐st❡♥❝❡✮ ❙✉♣♣♦s❡ U ✐s ❛♥ ♦♣❡♥ s❡t ♦♥ t❤❡ ♣❧❛♥❡ (t, y) t❤❛t ❝♦♥t❛✐♥s (t0 , y0 ) ❛♥❞ s✉♣♣♦s❡ t❤❛t ❛ ❢✉♥❝t✐♦♥ z = f (t, y) ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❞❡✜♥❡❞ ♦♥ U ✐s • ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ t✱ ❛♥❞ • ❞✐✛❡r❡♥t✐❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ y ✳ ❚❤❡♥ t❤❡ ❖❉❊ y ′ = f (t, y) s❛t✐s✜❡s t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ■❱P ♣r♦♣❡rt② ❛t (t0 , y0 )✳ ◆❡①t✱ ♦♥❝❡ ✇❡ ❤❛✈❡ ❛ s♦❧✉t✐♦♥ y = y(t) ♦♥ ✐♥t❡r✈❛❧ I ✱ ❛♥② r❡str✐❝t✐♦♥ z = y ♦❢ ❢✉♥❝t✐♦♥ y t♦ ❛♥② ✐♥t❡r✈❛❧ J J ✐♥s✐❞❡ I ✐s ❛❧s♦ ❛ s♦❧✉t✐♦♥✳ ❈♦♥✈❡rs❡❧②✱ ♦♥❝❡ ✇❡ ❤❛✈❡ ❛ s♦❧✉t✐♦♥ z ♦♥ ✐♥t❡r✈❛❧ J ✱ t❤❡r❡ ♠❛② ❜❡ ❛♥ ❡①t❡♥s✐♦♥ y ♦❢ z ✱ ✐✳❡✳✱ y = y(t) ✐s ❛ s♦❧✉t✐♦♥ ❛♥❞ z = y ✳ ❙♦♠❡t✐♠❡s ✐t ✐s ✐♠♣♦ss✐❜❧❡ t♦ ❢✉rt❤❡r ❡①t❡♥❞ ♦✉r s♦❧✉t✐♦♥✳ J

❉❡✜♥✐t✐♦♥ ✶✳✺✳✸✽✿ ♠❛①✐♠❛❧ s♦❧✉t✐♦♥ ❆ s♦❧✉t✐♦♥ y = y(t) ✐s ❝❛❧❧❡❞ ❛ ♠❛①✐♠❛❧

s♦❧✉t✐♦♥ ✐❢ ✐t ❞♦❡s♥✬t ❤❛✈❡ ❡①t❡♥s✐♦♥s✳

❆s ✇❡ ❤❛✈❡ s❡❡♥ ❛❜♦✈❡✱ ❛ ♠❛①✐♠❛❧ s♦❧✉t✐♦♥ ✐s ♦❢t❡♥ ❞❡✜♥❡❞ ♦♥ (−∞, +∞) ❜✉t ❛❧s♦ ♠❛② ❜❡ ❞❡✜♥❡❞ ♦♥ ♦t❤❡r ♦♣❡♥ ✐♥t❡r✈❛❧s✳ ❇❡❧♦✇✱ ✇❡ ✇✐❧❧ r❡❢❡r t♦ ♠❛①✐♠❛❧ s♦❧✉t✐♦♥s ❛s s✐♠♣❧② s♦❧✉t✐♦♥s✳

✶✳✻✳

❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s

✻✶

✶✳✻✳ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s

❙♦♠❡ t②♣❡s ♦❢ ❖❉❊s ❝❛♥ ❜❡ s♦❧✈❡❞ ❡①♣❧✐❝✐t❧②✳ ❉✐✛❡r❡♥t✐❛❧ ❢♦r♠s ❛❧❧♦✇ ✉s s♦♠❡t✐♠❡s t♦ s❡♣❛r❛t❡ x✱ ❛♥❞ dx✱ ❢r♦♠ y ✱ ❛♥❞ dy ✳ ❊①❛♠♣❧❡ ✶✳✻✳✶✿ ✐♥t❡❣r❛t✐♦♥

❲❡ r❡✲✇r✐t❡ t❤❡ ❖❉❊ ❛s ❛♥ ❡q✉❛t✐♦♥ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠s ❛♥❞ t❤❡♥ ✐♥t❡❣r❛t❡ ❜♦t❤ s✐❞❡s✿ dy = x2 =⇒ dy = x2 dx =⇒ dx

Z

dy =

Z

x2 dx =⇒ y + C = x3 /3 + K .

❚❤❡ t✇♦ ✐♥❞❡✜♥✐t❡ ✐♥t❡❣r❛t✐♦♥ ♣r♦❞✉❝❡ t✇♦ ❝♦♥st❛♥ts✳ ❙✐♥❝❡ ❜♦t❤ ❛r❡ ❛r❜✐tr❛r②✱ ❥✉st ♦♥❡ ✐s s✉✣❝✐❡♥t ✭Q = K − C ✮✿ y = x3 /3 + Q .

❚❤❡ ❖❉❊ ✐s s♦❧✈❡❞✳ ❊①❛♠♣❧❡ ✶✳✻✳✷✿ ♣♦♣✉❧❛t✐♦♥ ♠♦❞❡❧

❲❡ ❝♦♥s✐❞❡r t❤❡ ❢❛♠✐❧✐❛r ❖❉❊✿

dy = y. dx

❘❡❝❛❧❧ t❤❛t ♣❧♦tt✐♥❣ ✐ts ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ✭❞✐s❝r❡t❡ ❖❉❊✮ ♣r♦❞✉❝❡s t❤✐s✿

❋♦❧❧♦✇✐♥❣ t❤❡ s❛♠❡ ♣r♦❝❡❞✉r❡✱ ✇❡ ✐♥t❡❣r❛t❡ ❜♦t❤ s✐❞❡s ♦❢ ❛♥ ❡q✉❛t✐♦♥ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠s ❦❡❡♣✐♥❣ ♦♥❧② ♦♥❡ ❝♦♥st❛♥t✿ dy dy = y =⇒ = dx =⇒ ❲❤❡♥ y > 0 : dx y

Z

dy = y

Z

dx =⇒ ln y = x + Q .

❚❤✐s ✐s ❛ ♣❛rt ♦❢ t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ t❤❡ ❖❉❊ ❣✐✈❡♥ ✐♠♣❧✐❝✐t❧②✳ ■t ✐s ❛ ❢❛♠✐❧② ♦❢ r❡❧❛t✐♦♥s✳ ❙✐♠✐❧❛r❧②✱ ✇❡ ♦❜t❛✐♥✿ ❲❤❡♥ y < 0 : ln(−y) = x + P . ❲❤❛t✬s ❧❡❢t ✐s t❤❡ ❝❛s❡ ♦❢ y = 0✳ ❚❤❛t✬s ❛ ✇❤♦❧❡ ✭❝♦♥st❛♥t✮ s♦❧✉t✐♦♥✿ ❲❤❡♥ y = 0 :

dy = 0 =⇒ y = ❝♦♥st❛♥t =⇒ y = 0 . dx

❲❡ ✈❡r✐❢② t❤✐s ❢❛❝t ❜② s✉❜st✐t✉t✐♦♥✳ ◆♦✇ ❡①♣❧✐❝✐t t❤❡ s♦❧✉t✐♦♥s✳ ❲❡ ❛♣♣❧② t❤❡ ❧♦❣❛r✐t❤♠ t♦ ❜♦t❤ s✐❞❡s ♦❢ t❤❡ ✐♠♣❧✐❝✐t ❡q✉❛t✐♦♥s✳ ❋✐rst✿ ❲❤❡♥ y > 0 : ln y = x + Q =⇒ y = ex+Q = eQ ex , Q r❡❛❧.

✶✳✻✳

❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s

✻✷

❚❤❡s❡ ❛r❡ ❛❧❧ ♣♦s✐t✐✈❡ ♠✉❧t✐♣❧❡s ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✳ ❙❡❝♦♥❞✿ ❲❤❡♥ y > 0 : ln(−y) = x + P =⇒ y = −ex+P = −eP ex , P r❡❛❧. ❚❤❡s❡ ❛r❡ ❛❧❧ ♥❡❣❛t✐✈❡ ♠✉❧t✐♣❧❡s ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✳ ❖❢ ❝♦✉rs❡✱ y = 0 ✐s t❤❡ ③❡r♦ ♠✉❧t✐♣❧❡✳ ❲❡ ❤❛✈❡ t❤❡♠ ❛❧❧ ✐♥ ♦♥❡ ❢♦r♠✉❧❛✿ y = Cex , C r❡❛❧. ❚❤❡ s♦❧✉t✐♦♥ s❡t t❤✐s r❡s✉❧t ♣r♦❞✉❝❡s ✐s ❢❛♠✐❧✐❛r✿

■♥ ❣❡♥❡r❛❧✱ t❤❡ ♠❡t❤♦❞ ❛♣♣❧✐❡s ✇❤❡♥❡✈❡r ♦✉r r✐❣❤t✲❤❛♥❞ s✐❞❡ ❢✉♥❝t✐♦♥ s♣❧✐ts ✐♥t♦ ❛ ❢❛❝t♦r ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✇✐t❤ ♦♥❡ ❞❡♣❡♥❞✐♥❣ ♦♥❧② ♦♥ x ❛♥❞ t❤❡ ♦t❤❡r ♦♥❧② ♦♥ y ✿ dy dy = p(x)q(y) =⇒ = p(x) dx . dx q(y)

❙✉❝❤ ❖❉❊s ❛r❡ ❝❛❧❧❡❞ s❡♣❛r❛❜❧❡✳ ❊①❡r❝✐s❡ ✶✳✻✳✸

●✐✈❡ ❡①❛♠♣❧❡s ♦❢ s❡♣❛r❛❜❧❡ ❛♥❞ ♥♦♥✲s❡♣❛r❛❜❧❡ ❖❉❊s✳ ❚❤❡♦r❡♠ ✶✳✻✳✹✿ ❙♦❧✉t✐♦♥s ♦❢ ❙❡♣❛r❛❜❧❡ ❖❉❊s

❊✈❡r② s♦❧✉t✐♦♥ y = y(x) ♦❢ ❛ s❡♣❛r❛❜❧❡ ❖❉❊ y ′ = p(x)q(y) s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥✿ Z

dy = q(y)

Z

p(x) dx

❲❛r♥✐♥❣✦ ❚❤❡r❡ ✐s ♥♦ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ❤❡r❡ ❛s t❤❡s❡ ❛r❡ t❤❡ s❛♠❡ ✈❛r✐❛❜❧❡s✳

❚❤❡ q✉❡st✐♦♥ t❤❡♥ r❡♠❛✐♥s ✇❤❡t❤❡r t❤❡ t✇♦ ✐♥t❡❣r❛❧s ❝❛♥ ❜❡ ❡✈❛❧✉❛t❡❞ ✐♥ ❛♥ ❡❧❡♠❡♥t❛r② ❢❛s❤✐♦♥✳ ❊①❛♠♣❧❡ ✶✳✻✳✺✿ ❛❧❣❡❜r❛

❈♦♥s✐❞❡r t❤✐s ❖❉❊✿

dy = y sin x . dx

❲❡ ❦♥♦✇ t❤❛t t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ t❤❡ ✉♥✐q✉❡♥❡ss ❛r❡ s❛t✐s✜❡❞✳

✶✳✻✳

❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s

✻✸

❲❡ ❝❛♥ ❛❧s♦ t❛❦❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t❤❡ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✿ ∆y = y sin x . ∆x

❛♥❞ ♣❧♦t t❤❡ s♦❧✉t✐♦♥s✿

◆♦✇ t❤❡ ❛❧❣❡❜r❛✳ ❚❤❡ ❖❉❊ ✐s s❡♣❛r❛❜❧❡✿ dy dx dy =⇒ Zy =⇒

= ysin x = sin x dx Z dy = sin x dx y

=⇒ ln y

= − cos x + Q ❢♦r y > 0, ln(−y) = − cos x + P ❢♦r y < 0

=⇒ y

= e− cos x+Q ❢♦r y > 0,

y = −e− cos x+P ❢♦r y < 0

=⇒ y

= e− cos x eQ ❢♦r y > 0,

y = −e− cos x eP ❢♦r y < 0

=⇒ y

= Ce− cos x

❊✈❡♥ t❤♦✉❣❤ s♦♠❡ ♦❢ t❤❡ s♦❧✉t✐♦♥s ❛♣♣❡❛r t♦ t♦✉❝❤ t❤❡ x✲❛①✐s✱ t❤❡ ✉♥✐q✉❡♥❡ss ✐s s❛t✐s✜❡❞✳ ■♥❞❡❡❞✱ ❡✈❡r② s♦❧✉t✐♦♥ ✐s ❥✉st ❛ ✈❡rt✐❝❛❧❧② str❡t❝❤❡❞ ♦❢ t❤❡ s✐♠♣❧❡st ♦♥❡✿ y = e− cos x . ❊①❡r❝✐s❡ ✶✳✻✳✻

Pr♦✈❡ t❤❡ ❧❛st st❛t❡♠❡♥t✳ ▲❡t✬s ❛♣♣❧② t❤❡ ♠❡t❤♦❞ t♦ ❛♥ ❖❉❊ t❤❛t ✇❡ ✇✐❧❧ s❡❡ ❧❛t❡r✳ ❚❤❡♦r❡♠ ✶✳✻✳✼✿ ▲✐♥❡❛r ❖❉❊

❚❤❡ ❖❉❊

y ′ = a(x)y ,

✇✐t❤ ❛♥ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥ a✱ s❛t✐s✜❡s t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss✳ ❚❤❡ s♦❧✉✲ t✐♦♥s ♦❢ t❤❡ ❖❉❊ ❛r❡ ❣✐✈❡♥ ❜②✿ y = CeA(x)

✶✳✻✳

❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s ✇❤❡r❡

A

✻✹

✐s ❛♥② ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

a✿

A(x) =

Z

a(x) dx

❚❤✉s t❤❡ s♦❧✉t✐♦♥ s❡t s♣❧✐ts ✐♥t♦ t❤r❡❡ ♣❛rts✿ (1) y = KeA(x) ✇✐t❤ K < 0, (2) y = 0, (3) y = CeA(x) ✇✐t❤ C > 0 .

❆❧❧ ❛r❡ ♠✉❧t✐♣❧❡s ♦❢ y = eA(x) ✳ ❊①❡r❝✐s❡ ✶✳✻✳✽

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❊①❛♠♣❧❡ ✶✳✻✳✾✿ ❣❡♥❡r✐❝

❈♦♥s✐❞❡r t❤✐s ❖❉❊✿

dy = y2x . dx

❲❡ ❦♥♦✇ t❤❛t t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ t❤❡ ✉♥✐q✉❡♥❡ss ❛r❡ s❛t✐s✜❡❞✳ ❲❡ ❝❛♥ ❛❧s♦ t❛❦❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t❤❡ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✿ ∆y = y sin x . ∆x

❛♥❞ ♣❧♦t t❤❡ s♦❧✉t✐♦♥s✿

◆♦✇ t❤❡ ❛❧❣❡❜r❛✳ ❚❤❡ ❖❉❊ ✐s s❡♣❛r❛❜❧❡✿ dy dx dy =⇒ 2 Zy

= y2x

= x dx Z dy = x dx =⇒ y2 1 1 =⇒ − = x2 + Q y 2 1 ♦♥ ❡❛❝❤ ♦♣❡♥ ✐♥t❡r✈❛❧ ♦❢ x . =⇒ y = −1 2 x +Q 2

✶✳✼✳

❚❤❡ ♠❡t❤♦❞ ♦❢ ✐♥t❡❣r❛t✐♥❣ ❢❛❝t♦rs

✻✺

❊①❛♠♣❧❡ ✶✳✻✳✶✵✿ ❧♦❣✐st✐❝ ♠♦❞❡❧ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❢❛♠✐❧✐❛r ❖❉❊✿

dy = y(1 − y) . dx

❘❡❝❛❧❧ t❤❛t ♣❧♦tt✐♥❣ ✐ts ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ♣r♦❞✉❝❡s t❤✐s✿

❋♦❧❧♦✇✐♥❣ t❤❡ s❛♠❡ ♣r♦❝❡❞✉r❡✱ ✇❡ ✐♥t❡❣r❛t❡ ❜♦t❤ s✐❞❡s ♦❢ ❛♥ ❡q✉❛t✐♦♥ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠s ❦❡❡♣✐♥❣ ♦♥❧② ♦♥❡ ❝♦♥st❛♥t✿

dy dy = y(1 − y) =⇒ = dx =⇒ dx y(1 − y)

Z

dy = y(1 − y)

Z

dx =⇒

y =x+Q y−1

❢♦r

0 < y < 1.

❚❤✐s ✐s ❛ ♣❛rt ♦❢ t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ t❤❡ ❖❉❊ ❣✐✈❡♥ ✐♠♣❧✐❝✐t❧② ❛s ❛ ❢❛♠✐❧② ♦❢ r❡❧❛t✐♦♥s✳

❊①❡r❝✐s❡ ✶✳✻✳✶✶ ❋✐♥❞ t❤❡ ❡①♣❧✐❝✐t s♦❧✉t✐♦♥s✳

✶✳✼✳ ❚❤❡ ♠❡t❤♦❞ ♦❢ ✐♥t❡❣r❛t✐♥❣ ❢❛❝t♦rs

❊①❛♠♣❧❡ ✶✳✼✳✶✿ ♥♦♥✲s❡♣❛r❛❜❧❡ ❡q✉❛t✐♦♥ ▲❡t✬s t❛❦❡ ❛ ❝❛r❡❢✉❧ ❧♦♦❦ ❛t t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✭♥♦♥✲s❡♣❛r❛❜❧❡✮ ❡q✉❛t✐♦♥✿ 2

y ′ · sin x + y · cos x = xex . ❲❡ s❡❡ t✇♦ ♣❛✐rs ♦❢ ❢✉♥❝t✐♦♥s ♣r❡s❡♥t✿ • y ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ y ′ ✱ ❛♥❞

• sin x

❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡

cos x✳

❚❤❡② ❛r❡ ✏❝r♦ss✲♠✉❧t✐♣❧✐❡❞✑✳ ❚❤❡ ❡①♣r❡ss✐♦♥ ✐s ❢❛♠✐❧✐❛r❀ ✐t✬s t❤❡ ♦✉t❝♦♠❡ ♦❢ t❤❡ Pr♦❞✉❝t ❘✉❧❡✿

(y · sin x)′ = y ′ · sin x + y · cos x . ❚❤❡r❡❢♦r❡✱ ♦✉r ❖❉❊ t✉r♥s ✐♥t♦✿

❲❡ ✐♥t❡❣r❛t❡✿

(y · sin x)′ = xex . 2

Z



(y · sin x) dx =

Z

2

xex dx ,

✶✳✼✳

❚❤❡ ♠❡t❤♦❞ ♦❢ ✐♥t❡❣r❛t✐♥❣ ❢❛❝t♦rs

✻✻

❛♥❞ ❛♣♣❧② t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✿

1 2 y · sin x = ex + C . 2 ❲❡ ✇❡r❡ ❧✉❝❦②✳

❊①❛♠♣❧❡ ✶✳✼✳✷✿ ❢✉rt❤❡r st❡♣s ❈♦♥s✐❞❡r t❤❡ ❖❉❊✿

y′ + y = x . ❚❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ✐s ❛♥❞

y



♥♦t

t❤❡ ♦✉t❝♦♠❡ ♦❢ t❤❡ Pr♦❞✉❝t ❘✉❧❡✳✳✳ ❈❛♥ ✇❡ ♠❛❦❡ ✐t s♦❄ ❆❢t❡r ❛❧❧✱ t❤❡ ♣❛✐r

y

✐s ❛❧r❡❛❞② t❤❡r❡✳ ❯♥❢♦rt✉♥❛t❡❧②✱ ✇❤❛t✬s ❧❡❢t ❞♦❡s♥✬t ✇♦r❦✿

y′ · 1 + y · 1 = x . ❲❡ ❝❛♥✬t ❥✉st r❡♣❧❛❝❡ t❤❡ t❤❡

s❛♠❡

1✬s

✇✐t❤ s♦♠❡ ❢✉♥❝t✐♦♥s

F

❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡

F ′✱

❝❛♥ ✇❡❄ ❲❡ ❝❛♥ ✐❢ t❤✐s ✐s

❢✉♥❝t✐♦♥✿

(ex )′ = ex .

❲❡ ♠✉❧t✐♣❧② ❜♦t❤ s✐❞❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥ ❜② t❤✐s ❢❛❝t♦r

F (x) = ex ✿

y ′ · ex + y · ex = xex , ♦r

y ′ · ex + y · (ex )′ = xex . ❇② t❤❡ Pr♦❞✉❝t ❘✉❧❡ ✇❡ ❤❛✈❡ ✐♥st❡❛❞✿

(y · ex )′ = xex . ❆❢t❡r ✐♥t❡❣r❛t✐♦♥✱ ✇❡ ❤❛✈❡

y · ex = −ex + xex + C , ♦r ❡✈❡♥ s✐♠♣❧❡r✿

y = −1 + x + Ce−x . ❚❤❡♥✱ ✐t ✐s ♣r♦♠✐s✐♥❣ ❢♦r t❤✐s ❛♣♣r♦❛❝❤ t♦ ❤❛✈❡ ❜♦t❤ ❡q✉❛t✐♦♥ s❤♦✉❧❞ ❜❡

❧✐♥❡❛r

✇✐t❤ r❡s♣❡❝t t♦

y

❛♥❞

y

❛♥❞

y′✱

❜✉t ♥♦t

y2

♦r

sin y ′ ✳

■♥ ♦t❤❡r ✇♦r❞s✱ ❛♥



y✳

❊①❛♠♣❧❡ ✶✳✼✳✸✿ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s ❲❡ ♠❛❦❡ ❛ s♠❛❧❧ ❛❞❥✉st♠❡♥t t♦ t❤❡ ❖❉❊✿

y ′ + 2y = x . ❚❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ✐s ♥♦t t❤❡ ♦✉t❝♦♠❡ ♦❢ t❤❡ Pr♦❞✉❝t ❘✉❧❡✳✳✳ ❡✈❡♥ ✐❢ ✇❡ ♠✉❧t✐♣❧② ❜②

y ′ · ex + 2y · ex = xex . ❲❤❛t ❝❤♦✐❝❡ ♦❢ ❛ ❢❛❝t♦r

F

✇♦✉❧❞ ✇♦r❦✿

y ′ · F (x) + 2y · F (x) = xF (x)? ❲❡✬❞ ♥❡❡❞ t❤✐s✿

F ′ = 2F . ❇✉t t❤❛t✬s ❥✉st ❛♥♦t❤❡r ❖❉❊✦ ❆♥❞ ❛ ❢❛♠✐❧✐❛r ♦♥❡ t♦♦✿

F (x) = e2x .

ex ✿

✶✳✽✳

❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s

✻✼

❚❤✉s ✇❡ ❤❛✈❡✿

y ′ · e2x + 2y · e2x = xe2x ,

♦r

y ′ · e2x + y · e2x

❚❤❡♥✱

y · e2x

❆❢t❡r ✐♥t❡❣r❛t✐♦♥✱ ✇❡ ❤❛✈❡

′

′

= xe2x .

= xe2x .

y · e2x = −e2x /2 + xe2x + C .

■♥ ❣❡♥❡r❛❧✱ ✇❤❡♥ t❤❡ ❧✐♥❡❛r ❡q✉❛t✐♦♥

y ′ + a(x)y = b(x)

✐s ♠✉❧t✐♣❧✐❡❞ ❜② s♦♠❡ ❢❛❝t♦r F ✱ ✇❡ ❤❛✈❡ y ′ F (x) + a(x)yF (x) = b(x)F (x) .

❚❤❡♥✱ t❤❡ ❛♣♣r♦❛❝❤ ✈✐❛ t❤❡ Pr♦❞✉❝t ❘✉❧❡ ❛♣♣❧✐❡s ✇❤❡♥ t❤❛t ❡q✉❛t✐♦♥ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤✐s ❡q✉❛t✐♦♥✿ y ′ F (x) + yF ′ (x) = b(x)F (x) ,

✐✳❡✳✱ ✇❤❡♥ F s❛t✐s✜❡s✿

F ′ = a(x)F .

❚❤✐s ❖❉❊ ❛❧✇❛②s ❤❛s ❛ s♦❧✉t✐♦♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛st t❤❡♦r❡♠✿ F =e

R

a(x) dx

,

❛s ❧♦♥❣ ❛s ❢✉♥❝t✐♦♥ a ✐s ✐♥t❡❣r❛❜❧❡✳ ❋✉♥❝t✐♦♥ F ❞❡✜♥❡❞ t❤✐s ✇❛② ✐s ❝❛❧❧❡❞ t❤❡ ✐♥t❡❣r❛t✐♥❣ ❡q✉❛t✐♦♥✳

❢❛❝t♦r ♦❢ t❤❡ ❧✐♥❡❛r

❚❤❡♦r❡♠ ✶✳✼✳✹✿ ◆♦♥✲❤♦♠♦❣❡♥❡♦✉s ▲✐♥❡❛r ❖❉❊ ❊✈❡r② s♦❧✉t✐♦♥ ♦❢ t❤❡ ❖❉❊

y ′ + a(x)y = b(x) ✐s ❣✐✈❡♥ ❜②✿

y=e ✇❤❡r❡

A

✐s ❛♥② ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

−A(x)

Z

b(x)eA(x) dx

a✿

A(x) =

Z

a(x) dx

✶✳✽✳ ❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s

❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐s ❛ ❝♦♠♠♦♥ t♦♦❧ ✐♥ s❝✐❡♥❝❡s ❛♥❞ ♠❛t❤❡♠❛t✐❝s✳ ❚❤❡ ✐❞❡❛ ♦❢ ✜♥❞✐♥❣ ❛ ♥❡✇ ✈❛r✐❛❜❧❡ t❤❛t ♠✐❣❤t ♠❛❦❡ t❤❡ ♣r♦❜❧❡♠ s✐♠♣❧❡r ✐s ❡s♣❡❝✐❛❧❧② ♣❡r✈❛s✐✈❡ ✐♥ ♠❛t❤❡♠❛t✐❝s✳ ❙♦♠❡ s✐♠♣❧❡ ❡①❛♠♣❧❡s ❛r❡✿ • ❝❤❛♥❣❡ ♦❢ ✉♥✐ts ✭s❡❝♦♥❞s t♦ ♠✐♥✉t❡s✱ ♠❡t❡rs t♦ ❢❡❡t✱ ❈❡❧s✐✉s t♦ ❋❛❤r❡♥❤❡✐t ❡t❝✳✮

✶✳✽✳

❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s

✻✽

• ❝❤❛♥❣❡ ♦❢ ❢r❛♠❡ ♦❢ r❡❢❡r❡♥❝❡ ✭❢r♦♠ t❤❡ ❊❛rt❤ t♦ t❤❡ ❙✉♥ ❛s t❤❡ ❝❡♥t❡r ✐♥ ♦r❞❡r t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ♠♦t✐♦♥

♦❢ t❤❡ ♣❧❛♥❡t✱ t❤❡♥ ❜❛❝❦ t♦ t❤❡ ❊❛rt❤ ❛s t❤❡ ❝❡♥t❡r t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ♠♦t✐♦♥ ♦❢ ❛ s♣❛❝❡❝r❛❢t ❤❡❛❞✐♥❣ ❢♦r t❤❡ ▼♦♦♥✱ ❡t❝✳✮

• r❡♣r❡s❡♥t✐♥❣ ❛ ❢✉♥❝t✐♦♥ ❛s ❛ ❝♦♠♣♦s✐t✐♦♥

❚❤❡ ✜rst ❛♣♣r♦❛❝❤ ✐s t♦ tr② t♦ ❝❤❛♥❣❡ t❤❡ ♦❢ t❤❡ ♣❧❛♥❡ ❛♥❞ t❤❡ ❣r❛♣❤s ♦♥ ✐t✿

✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✳ ❚❤❡ r❡s✉❧t ✐s ❛ ❤♦r✐③♦♥t❛❧ tr❛♥s❢♦r♠❛t✐♦♥

❊①❛♠♣❧❡ ✶✳✽✳✶✿ ✐♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥

▲❡t✬s r❡✈✐❡✇ ❛ s✐♠♣❧❡ ❡①❛♠♣❧❡ ♦❢ ✐♥t❡❣r❛t✐♦♥

❘❡❝♦❣♥✐③✐♥❣ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ❝♦♠♣♦s✐t✐♦♥ ✿

❜② s✉❜st✐t✉t✐♦♥ ✿

Z

2

xex dx =?

2

y = ex ,

✇❡ s♣❧✐t ✐t✳ ❲❡ ✐♥tr♦❞✉❝❡ ✭✐♥s❡rt✦✮ ❛♥ ✐♥t❡r♠❡❞✐❛t❡ ✈❛r✐❛❜❧❡✱ u✿ x 7→ u 7→ y ,

✇✐t❤

u = x2 .

❚❤❡♥✱ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ♦❢ u ✐s t❤✐s ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠✿ du = 2xdx .

❲❡ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦ ✐♥t♦ t❤❡ ✐♥t❡❣r❛❧✿ Z

❆t t❤✐s ♣♦✐♥t✱

xe

x2

dx =

Z

xe x2

x2 =u

• ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐s ❝♦♠♣❧❡t❡✱ ❛♥❞ • t❤❡ ♥❡✇ ✐♥t❡❣r❛❧ ✐s s✐♠♣❧❡r✦

dx

= dx= du 2x

Z

u du

1 xe = 2x 2

Z

eu dx .

❊①❡r❝✐s❡ ✶✳✽✳✷

❈❛rr② ♦✉t t❤❡ s✉❜st✐t✉t✐♦♥ v = x3 ✐♥ t❤❡ ❛❜♦✈❡ ✐♥t❡❣r❛❧✳ ❍✐♥t✿ ❨❡s✱ t❤✐r❞ ♣♦✇❡r✳

✶✳✽✳

✻✾

❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s

❊①❛♠♣❧❡ ✶✳✽✳✸✿ s✉❜st✐t✉t✐♦♥ ✐♥ ❖❉❊s

▲❡t✬s r❡❝❛st t❤❡ ❛❜♦✈❡ ❝♦♥str✉❝t✐♦♥ ✐♥ t❡r♠s ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ❚❤❡ ✐♥t❡❣r❛❧ ♠❛② ❜❡ s❡❡♥ ❛s t❤❡ ❛♥s✇❡r✳✳✳ t♦ ✇❤❛t q✉❡st✐♦♥❄ ❆♥ ❖❉❊✳ ❚❤❡ ❖❉❊ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ✐♥t❡❣r❛❧ ✐s✿ 2 y ′ = xex .

▲❡t✬s ✉s❡ t❤❡ ▲❡✐❜♥✐③ ♥♦t❛t✐♦♥✿ ❲❡ ❤❛✈❡ t✇♦

✈❛r✐❛❜❧❡s

dy 2 = xex . dx

r❡❧❛t❡❞ ✈✐❛ ❛♥ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥✿ y = y(x)✳ ❲❡ ✐♥tr♦❞✉❝❡ ❛ ♥❡✇ ✈❛r✐❛❜❧❡✿ u = x2 .

❲❡ ❤❛✈❡ t❤r❡❡ ✈❛r✐❛❜❧❡s r❡❧❛t❡❞ t♦ ❡❛❝❤ ♦t❤❡r ✈✐❛✿ • ❛♥ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥✿ y = y(x) t❤❛t ✐s st✐❧❧ t♦ ❜❡ ❢♦✉♥❞✱ • t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ❢✉♥❝t✐♦♥✿ u = x2 ✱ ❛♥❞ • ❛♥♦t❤❡r ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥✿ y = y(u) t♦ ❜❡ ❢♦✉♥❞ ✜rst✳ ❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ❞✐❛❣r❛♠ ✐s ❜❡❧♦✇✿ x → u ց ↓ y

❚❤❡ t❤r❡❡ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡ t❤r❡❡ ✈❛r✐❛❜❧❡s ❤❛✈❡ t❤❡✐r ❞❡r✐✈❛t✐✈❡s✿ ♦r✐❣✐♥❛❧✿

x

❞❡❝♦♠♣♦s❡❞✿ x

−→

−→

−→

y

u

−→

y

dy 2 = xex dx

−→

du = 2x dx

dy =? du

❚✇♦ ❛r❡ ❦♥♦✇♥ ❛♥❞ ♦♥❡ ✐s ✉♥❦♥♦✇♥✳ ❋♦rt✉♥❛t❡❧②✱ t❤❡s❡ ❞❡r✐✈❛t✐✈❡s ❛r❡ ❝♦♥♥❡❝t❡❞ ✈✐❛ t❤❡ ❈❤❛✐♥ ❘✉❧❡ ✿ dy du dy · = . dx du dx

❚❤❡r❡❢♦r❡✱ ♦✉r ❖❉❊ ❜❡❝♦♠❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❢t❡r t❤✐s s✉❜st✐t✉t✐♦♥✿ dy du dy dy dy 1 2 = xex =⇒ · = xeu =⇒ 2x = xeu =⇒ = eu . dx dx du du du 2

❲❡ ❤❛✈❡ ❛❝❝♦♠♣❧✐s❤❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ❚❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐s ❝♦♠♣❧❡t❡✳ • ❚❤❡ ♥❡✇ ❖❉❊ ✐s s✐♠♣❧❡r✦ ❚❤✐s ✐s t❤❡ s✉♠♠❛r②✿

dy dy dy 2 = xex −→ = . dx du du ❲❡ s♦❧✈❡ t❤❡ ♥❡✇ ❖❉❊ ✭✇✐t❤ r❡s♣❡❝t t♦ u✮ ❡❛s✐❧②✿ Z 1 1 u e du = eu + C . y= 2 2

❙✐♥❝❡ t❤❡ ✈❛r✐❛❜❧❡ u ✇❛s ♠❛❞❡ ✉♣✱ ✇❡ ♥❡❡❞ t❤❡ ❜❛❝❦✲s✉❜st✐t✉t✐♦♥✱ u = x2 ✱ ❣✐✈✐♥❣ ✉s t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ♦r✐❣✐♥❛❧ ❖❉❊ ✭✇✐t❤ r❡s♣❡❝t t♦ x✮✿ 1 2 y = ex + C . 2

✶✳✽✳

❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s

✼✵

❊①❡r❝✐s❡ ✶✳✽✳✹

❈❛rr② ♦✉t t❤❡ s✉❜st✐t✉t✐♦♥ v = x3 ✐♥ t❤❡ ❛❜♦✈❡ ❖❉❊✳ ❊①❡r❝✐s❡ ✶✳✽✳✺

❈❛rr② ♦✉t t❤❡ s✉❜st✐t✉t✐♦♥ v = sin x ✐♥ t❤❡ ❖❉❊ y ′ = cos3 x✳ ❊①❡r❝✐s❡ ✶✳✽✳✻

❊①❡❝✉t❡ t❤❡ s✉❜st✐t✉t✐♦♥ ❚❤❡ s❡❝♦♥❞ ❛♣♣r♦❛❝❤ ✐s t♦ tr② t♦ ❝❤❛♥❣❡ t❤❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✳ ❚❤❡ r❡s✉❧t ✐s ❛ ✈❡rt✐❝❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ ❛♥❞ t❤❡ ❣r❛♣❤s ♦♥ ✐t✿

❚❤❡ s✐♠♣❧❡st s✉❝❤ s✉❜st✐t✉t✐♦♥ ✐s t❤❡ s❤✐❢t ✿

y = z + a,

✇❤❡r❡ z ✐s t❤❡ ♥❡✇ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✳ ■❢ ✇❡ ✇❛♥t t♦ ❝♦♥❝❡♥tr❛t❡ ♦♥ ❛ s✐♥❣❧❡ ♣♦✐♥t y = a ✭❛♥❞ ✐ts ✈✐❝✐♥✐t②✮ ❛t ❛ t✐♠❡✱ t❤❡ s❤✐❢t ❛❧❧♦✇s ✉s t♦ ♠♦✈❡ t❤❛t ♣♦✐♥t t♦ ③❡r♦✳ ❊①❛♠♣❧❡ ✶✳✽✳✼✿ ♠✐❣r❛t✐♦♥

❋♦r ❡①❛♠♣❧❡✱ s✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❧✐♥❡❛r

❖❉❊ ✿ y ′ = my + b, m 6= 0 .

■t ❝❛♥ r❡♣r❡s❡♥t ♣♦♣✉❧❛t✐♦♥ ❣r♦✇t❤✴❞❡❝❛② ❛❝❝♦♠♣❛♥✐❡❞ ❜② ♠✐❣r❛t✐♦♥ ❛t ❛ ❝♦♥st❛♥t r❛t❡✳ ❚❤❡ ❖❉❊ ✐s ❝❧♦s❡ t♦ y ′ = my ✭♥♦ ♠✐❣r❛t✐♦♥✮ t❤❛t ✇❡ ❦♥♦✇ ❤♦✇ t♦ ❤❛♥❞❧❡✳ ❇✉t ❤♦✇ ❞♦ ✇❡ ❣❡t t❤❡r❡❄ ▲❡t✬s ✐♥✈❡st✐❣❛t❡✳ ▲❡t✬s ❝♦♠♣❛r❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✿ ∆y ∆y = my ✈s✳ = my + b . ∆x ∆x

❲❡ ♣❧♦t ❛ ❝♦✉♣❧❡ ♦❢ t❤❡✐r s♦❧✉t✐♦♥s✿

✶✳✽✳

❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s

✼✶

■♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ st❛♥❞❛r❞ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ ✭❧❡❢t✮✱ t❤❡r❡ s❡❡♠s t♦ ❜❡ ❛ ✈❡rt✐❝❛❧ s❤✐❢t ✭r✐❣❤t✮✳ ❇✉t ❤♦✇ ❢❛r❄ ▲❡t✬s ❝❤♦♦s❡ ❛ ♥❡✇ ✈❛r✐❛❜❧❡✿ z = y − a.

❚❤❛t✬s ❛ ✈❡rt✐❝❛❧ s❤✐❢t ✭✇✐t❤ a st✐❧❧ t♦ ❜❡ ❢♦✉♥❞✮✳ ❚❤❡♥✱ ✜rst✱ y ′ = (z + a)′ = z ′ ,

❛♥❞✱ s❡❝♦♥❞✱



b my + b = m(z + a) + b = mz + ma + b = mz + m a + m

❋♦r t❤❡ ❧❛st t❡r♠ t♦ ❞✐s❛♣♣❡❛r ✇❡ ❥✉st s❡❧❡❝t✿ a=−

❲❡ ❤❛✈❡ ❛ ♥❡✇ ❖❉❊ ✇✐t❤ r❡s♣❡❝t t♦ z ✿

b . m

z ′ = mz .

❲❡ ❤❛✈❡ ❛❝❝♦♠♣❧✐s❤❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ❚❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐s ❝♦♠♣❧❡t❡✳ • ❚❤❡ ♥❡✇ ❖❉❊ ✐s s✐♠♣❧❡r✦ ❚❤✐s ✐s t❤❡ s✉♠♠❛r②✿ ❲❡ s♦❧✈❡ ✐t ❛s ❜❡❢♦r❡✿

dz dy = my + b −→ = mz . dx dx z = Cemt .

❆❢t❡r ❛ ❜❛❝❦✲s✉❜st✐t✉t✐♦♥✱ ✇❡ ❤❛✈❡ ❢♦r t❤❡ ♦r✐❣✐♥❛❧ ❖❉❊✿ ♦r ❚❤❡ ✇❤♦❧❡ ♣✐❝t✉r❡ ❥✉st s❤✐❢ts ✉♣✦

y − a = Cemt , y=−

b + Cemt . m



.

✶✳✽✳

❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❖❉❊s

✼✷

❊①❛♠♣❧❡ ✶✳✽✳✽✿ ❜❛♥❦✐♥❣

❲❤❛t ❤❛♣♣❡♥s ✐❢ ✇❡ ❝♦♠❜✐♥❡ r❡❣✉❧❛r ❞❡♣♦s✐ts ✇✐t❤ ❝♦♠♣♦✉♥❞❡❞ ✐♥t❡r❡st❄ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ t❤❡ s❛♠❡ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ ❛s ✐♥ t❤❡ ❝❛s❡ ✇✐t❤♦✉t ❞❡♣♦s✐ts ❜✉t s❤✐❢t❡❞ ❞♦✇♥ ✭♥♦t ✉♣✦✮✳ ❲❤❛t ❤❛♣♣❡♥s ✐❢ ✇❡ ❝♦♠❜✐♥❡ r❡❣✉❧❛r ✇✐t❤❞r❛✇❛❧s ✇✐t❤ ❝♦♠♣♦✉♥❞❡❞ ✐♥t❡r❡st❄ ❚❤❡ s❛♠❡ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ ❛s ✐♥ t❤❡ ❝❛s❡ ✇✐t❤♦✉t ✇✐t❤❞r❛✇❛❧s ❜✉t s❤✐❢t❡❞ ✉♣✳ ❆❧t❡r♥❛t✐✈❡❧②✱ ✇❤❛t ✐❢ ♦✉r ✐♥✈❡st♠❡♥t ✐s ❧♦s✐♥❣ ✈❛❧✉❡ ❜✉t ✇❡ ❝♦♥t✐♥✉❡ t♦ ♣✉t ♠♦♥❡② ✐♥ ✐t❄ ❚❤❡ ❡①♣♦✲ ♥❡♥t✐❛❧ ❞❡❝❛② ♦r ❛ st❛❣♥❛t✐♥❣ ❣r♦✇t❤ t♦✇❛r❞ ❛ ♥♦♥✲③❡r♦ ✈❛❧✉❡✿

❊①❛♠♣❧❡ ✶✳✽✳✾✿ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ❈♦♦❧✐♥❣

▲❡t✬s ✇r✐t❡ t❤❡ ❖❉❊s ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥ ❣r♦✇t❤ ♠♦❞❡❧ ❛♥❞ t❤❡ ❝♦♦❧✐♥❣ ❧❛✇ ♠♦❞❡❧ ♥❡①t t♦ ❡❛❝❤ ♦t❤❡r✿ y ′ = ky ❛♥❞ y ′ = k · (r − y) .

❚❤❡② s❡❡♠ s✐♠✐❧❛r✦ ▲❡t✬s ♣❧♦t t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ♦❢ t❤❡s❡ ♠♦❞❡❧s✿

❚❤❡② s❡❡♠ s✐♠✐❧❛r✦ ❚❤❡ ❝♦♠♣❛r✐s♦♥ s✉❣❣❡sts ❛ s✉❜st✐t✉t✐♦♥✿ ❛ ✈❡rt✐❝❛❧ s❤✐❢t ❛♥❞ ❛ ❤♦r✐③♦♥t❛❧ ✢✐♣✳ ❲❡ ❡①❡❝✉t❡ ♦♥❡ ❛t ❛ t✐♠❡✳ ❚❤❡ s❤✐❢t ✐s ❣✐✈❡♥ ❜②✿ ❚❤❡♥ t❤❡ ✐♥✈❡rs❡ s✉❜st✐t✉t✐♦♥ ✐s✿

z = y −r. y = z +r.

❚❤❡♥ t❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s t♦ ♦✉r ❖❉❊ ❛s ✇❡ s✉❜st✐t✉t❡✿ dy d(z + r) dz = k(r − y) =⇒ = k(r − (z + r)) =⇒ = −kz . dt dt dt

❊①❝❡♣t ❢♦r t❤❡ ♠✐♥✉s s✐❣♥ ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣✱ t❤✐s ✐s t❤❡ ♣♦♣✉❧❛t✐♦♥ ❖❉❊✦ ◆♦✇ t❤❡ ✢✐♣✿ s = −t .

✶✳✾✳

✼✸

❊✉❧❡r✬s ♠❡t❤♦❞✿ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡

❚❤❡♥ t❤❡ ✐♥✈❡rs❡ s✉❜st✐t✉t✐♦♥ ✐s✿ t = −s .

◆♦✇ t❤❡ ❞❡r✐✈❛t✐✈❡s✿

dz dz ds dz = =− . dt ds dt ds

❖✉r ❖❉❊ ❜❡❝♦♠❡s✿

dz = kz . ds

❲❡ ❤❛✈❡ ❝♦♥✜r♠❡❞ t❤❛t t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ t❤❡ ❝♦♦❧✐♥❣ ❖❉❊ ✐s t❤❛t ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥ ❖❉❊ s✐❢t❡❞ ✈❡rt✐❝❛❧❧② ❜② r ❛♥❞ t❤❡♥ ✢✐♣♣❡❞ ❤♦r✐③♦♥t❛❧❧②✦ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❤❛✈❡ t❤❡ ❡①♣❧✐❝✐t s♦❧✉t✐♦♥s t♦♦ ✈✐❛ ❛ ❜❛❝❦✲s✉❜st✐t✉t✐♦♥s✿ z = Ceks =⇒ y − r = Cek(−t) =⇒ y = r + Ce−kt . ❊①❡r❝✐s❡ ✶✳✽✳✶✵

❊①❡❝✉t❡ t❤❡ s✉❜st✐t✉t✐♦♥

✶✳✾✳ ❊✉❧❡r✬s ♠❡t❤♦❞✿ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡

❯♥❢♦rt✉♥❛t❡❧②✱ ❛♥ ❖❉❊ t②♣✐❝❛❧❧② ❤❛s ♥♦ ❛❧❣❡❜r❛✐❝ s♦❧✉t✐♦♥✦ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥s ♦❢ ❛ ❣❡♥❡r❛❧ ■❱P✿ y ′ = f (t, y), y(t0 ) = y0 .

❚❤❡ ■❱P t❡❧❧s ✉s✿ • ✇❤❡r❡ ✇❡ ❛r❡ ✭t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✮✱ ❛♥❞ • t❤❡ ❞✐r❡❝t✐♦♥ ✇❡ ❛r❡ ❣♦✐♥❣ ✭t❤❡ ❖❉❊✮✳

■t ✐s t❤❡♥ ❡❛s② t♦ ♣r♦❞✉❝❡ ❛ s♦❧✉t✐♦♥ ✐❢ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ❝❤❛♥❣❡ ✐s ✐♥❝r❡♠❡♥t❛❧✿ ❛s ✇❡ ❛rr✐✈❡ t♦ ❛ ♥❡✇ ❧♦❝❛t✐♦♥✱ ✇❡✱ ♦♥❝❡ ❛❣❛✐♥✱ ❦♥♦✇ ✇❤❡r❡ ✇❡ ❛r❡ ❛♥❞ ✇❤❡r❡ ✇❡ ❛r❡ ❣♦✐♥❣✳

❚❤❡ ♠❡t❤♦❞ ♣r❡s❡♥t❡❞ ❤❡r❡ ❢♦❧❧♦✇s t❤❡ ✐❞❡❛ ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✻✿ t❤❡ ✉♥❦♥♦✇♥ s♦❧✉t✐♦♥ ✐s r❡♣❧❛❝❡❞ ✇✐t❤ ✐ts ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥✳

✶✳✾✳ ❊✉❧❡r✬s ♠❡t❤♦❞✿ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡

✼✹

❊①❛♠♣❧❡ ✶✳✾✳✶✿ ❢❛♠✐❧② ♦❢ ❝✐r❝❧❡s ▲❡t✬s ❝♦♥s✐❞❡r ❛❣❛✐♥ t❤❡s❡ ❝♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s✿

❚❤❡② ❛r❡ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❖❉❊✿

x y ′ = f (x, y) = − . y

❲❡ ✇✐❧❧ ❜❡ s♦❧✈✐♥❣ ♥✉♠❡r♦✉s ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠s ✇❤✐❧❡ st❛②✐♥❣ ❛✇❛② ❢r♦♠ t❤❡

x✲❛①✐s

✇❤❡r❡ t❤❡r❡

❛r❡ ♥♦ s♦❧✉t✐♦♥s✳ ❲❡ ❝❤♦♦s❡ t❤❡ ✐♥❝r❡♠❡♥t ♦❢

x✿ ∆x = 1 .

❲❡ st❛rt ✇✐t❤ t❤✐s ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✿

x0 = 0,

y0 = 1 .

❲❡ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦ ♥✉♠❜❡rs ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥✿

0 y′ = − = 0 . 1 ❚❤✐s ✐s t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧✐♥❡ ✇❡ ✇✐❧❧ ❢♦❧❧♦✇✳ ❍♦✇ ❢❛r❄ ❆s ❢❛r ❛s t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ✐♥❝r❡♠❡♥t ♦❢

y

✐s

❖✉r ♥❡①t ❧♦❝❛t✐♦♥ ♦♥ t❤❡

xy ✲♣❧❛♥❡

∆y = 0 · ∆x = 0 × 1 = 0 . ✐s t❤❡♥✿

x1 = x0 + ∆x = 0 + 1 = 1,

y1 = y0 + ∆y = 1 + 0 = 1 .

❚❤✐s ❝♦♠♣✉t❛t✐♦♥ ❣✐✈❡s ✉s ❛ ♥❡✇ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✿

x1 = 1,

y1 = 1 .

x

❛❧❧♦✇s✳ ❚❤❡r❡❢♦r❡✱

✶✳✾✳ ❊✉❧❡r✬s ♠❡t❤♦❞✿ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡

✼✺

❲❡ ❛❣❛✐♥ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦ ♥✉♠❜❡rs ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥✿

1 y ′ = − = −1 . 1 ❚❤✐s ✐s t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧✐♥❡ ✇❡ ✇✐❧❧ ❢♦❧❧♦✇✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ✐♥❝r❡♠❡♥t ♦❢

y

✐s

∆y = −1 · ∆x = −1 . ❖✉r ♥❡①t ❧♦❝❛t✐♦♥ ♦♥ t❤❡

xy ✲♣❧❛♥❡

✐s t❤❡♥✿

x2 = x1 + ∆x = 1 + 1 = 2, ❲❡ ❤❛✈❡ ❡♥❞❡❞ ✉♣ ♦♥ t❤❡

x✲❛①✐s

y2 = y1 + ∆y = 1 + (−1) = 0 .

❛♥❞ st♦♣✳ ❚❤❡s❡ t❤r❡❡ ♣♦✐♥ts ❢♦r♠ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥✳

❲❡ st❛rt ❛❧❧ ♦✈❡r ❢r♦♠ ❛♥♦t❤❡r ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✿

x0 = 0,

y0 = 2 .

❲❡ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦ ♥✉♠❜❡rs ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥✿

0 y′ = − = 0 , 2 ♣r♦❞✉❝✐♥❣ t❤❡ s❧♦♣❡ ✇❡ ✇✐❧❧ ❢♦❧❧♦✇✳ ❚❤❡ ✐♥❝r❡♠❡♥t ♦❢

y

✐s

∆y = 0 · ∆x = 0 · 1 = 0 . ❖✉r ♥❡①t ❧♦❝❛t✐♦♥ ♦♥ t❤❡

xy ✲♣❧❛♥❡

✐s t❤❡♥✿

x1 = x0 + ∆x = 0 + 1 = 1, y1 = y0 + ∆y = 2 + 0 = 2 . ❆ ♥❡✇ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❛♣♣❡❛rs✿

x0 = 1,

y0 = 2 .

❲❡ ❛❣❛✐♥ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦ ♥✉♠❜❡rs ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥✿

1 y ′ = − = −1/2 , 2 ♣r♦❞✉❝✐♥❣ t❤❡ s❧♦♣❡ t♦ ❢♦❧❧♦✇✳ ❚❤❡ ✐♥❝r❡♠❡♥t ♦❢

y

✐s

∆y = −1/2 · ∆x = −1/2 . ❖✉r ♥❡①t ❧♦❝❛t✐♦♥ ♦♥ t❤❡

xy ✲♣❧❛♥❡

✐s t❤❡♥✿

x2 = x1 + ∆x = 1 + 1 = 2,

y2 = y1 + ∆y = 2 + (−1/2) = 3/2 .

❖♥❡ ♠♦r❡ ■❱P✿

x2 = 2, y2 = 3/2 . ❚❤❡ ✐♥❝r❡♠❡♥t ♦❢

y

✐s

❖✉r ♥❡①t ❧♦❝❛t✐♦♥ ♦♥ t❤❡

2 x · 1 = −4/3 . ∆y = − · ∆x = − y 3/2 xy ✲♣❧❛♥❡

✐s t❤❡♥✿

x3 = x2 + ∆x = 2 + 1 = 3, ❲❡ ✇♦✉❧❞ ❤❛✈❡ t♦ ♣❛ss t❤❡

x✲❛①✐s

y3 = y2 + ∆y = 3/2 − 4/3 = 1/6 .

✇✐t❤ ♥❡①t st❡♣ ❛♥❞ ✇❡ st♦♣ ♥♦✇✳ ❚❤❡s❡ ❢♦✉r ♣♦✐♥ts✱ ✐♥ ❛❞❞✐t✐♦♥ t♦

t❤❡ ♣r❡✈✐♦✉s t❤r❡❡✱ ❢♦r♠ ❛ ✈❡r② ❝r✉❞❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ♦✉r ❝✐r❝✉❧❛r s♦❧✉t✐♦♥s✿

✶✳✾✳

✼✻

❊✉❧❡r✬s ♠❡t❤♦❞✿ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡

■♥ ❢❛❝t✱ ✇❡ s❡❡ ✇❤❛t ✐s♥✬t s✉♣♣♦s❡❞ t♦ ❤❛♣♣❡♥✿ t❤❡ s♦❧✉t✐♦♥ ❣♦❡s ❜❡❧♦✇ t❤❡ x✲❛①✐s✦ ❋r♦♠ t❤❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦❢ ♠♦t✐♦♥✱ t❤✐s ✐s t❤❡ ♠❡t❤♦❞✬s ✐♥t❡r♣r❡t❛t✐♦♥✿

◮ ❆t ♦✉r ❝✉rr❡♥t ❧♦❝❛t✐♦♥ ❛♥❞ ❝✉rr❡♥t t✐♠❡✱ ✇❡ ❡①❛♠✐♥❡ t❤❡ ❖❉❊ t♦ ✜♥❞ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡♥ ♠♦✈❡ ✇✐t❤ t❤✐s ✈❡❧♦❝✐t② t♦ t❤❡ ♥❡①t ❧♦❝❛t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✶✳✾✳✷✿ ❊✉❧❡r s♦❧✉t✐♦♥ ❚❤❡

❊✉❧❡r s♦❧✉t✐♦♥

✇✐t❤ ✐♥❝r❡♠❡♥t h > 0 ♦❢ t❤❡ ■❱P

y ′ = f (t, y), y(t0 ) = y0 , ✐s t❤❡ s❡q✉❡♥❝❡ {yn } ♦❢ r❡❛❧ ♥✉♠❜❡rs ❣✐✈❡♥ ❜②✿

yn+1 = yn + f (tn , yn ) · h ✇❤❡r❡ tn+1 = tn + h✳ ❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❢❛❝t t♦ ❦♥♦✇ ❛❜♦✉t ❊✉❧❡r✬s ♠❡t❤♦❞ ✐s t❤❛t ✐❢ ✇❡ ❞❡r✐✈❡❞ ♦✉r ❖❉❊ ❢r♦♠ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧ ✕ ❛♥❞ ❢r♦♠ ❛ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ✭❞✐s❝r❡t❡ ❖❉❊✮ ✈✐❛ ∆t → 0 ✕ ❊✉❧❡r✬s ♠❡t❤♦❞ ✇✐❧❧ ❜r✐♥❣ ✉s r✐❣❤t ❜❛❝❦ t♦ ✐t✿ ❖❉❊ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥

ր

same!

←−−−−−−−

ց

❊✉❧❡r✬s ♠❡t❤♦❞

❊①❡r❝✐s❡ ✶✳✾✳✸ ❊①❡❝✉t❡ t❤❡ ♠❡t❤♦❞ ❢♦r t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡ ❛♥❞ h = 1/2✳

❊①❛♠♣❧❡ ✶✳✾✳✹✿ ❝✐r❝❧❡s ▲❡t✬s ♥♦✇ ❝❛rr② ♦✉t t❤✐s ♣r♦❝❡❞✉r❡ ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t ❢♦r t❤❡ ❖❉❊✿

y y′ = − . x ❚❤❡ ❢♦r♠✉❧❛ ❢♦r yn ✐s✿ ❚❤❡ r❡s✉❧ts ❛r❡ ❧❡ss ❝r✉❞❡✿

❂❘❬✲✶❪❈✰❘❬✲✶❪❈❬✶❪✯❘✷❈✶

✶✳✾✳ ❊✉❧❡r✬s ♠❡t❤♦❞✿ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡

✼✼

❍♦✇❡✈❡r✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥s ❛♣♣❡❛r t♦ ❜❡❤❛✈❡ ❡rr❛t✐❝❛❧❧② ✇❤❡♥ t❤❡② ❣❡t ❝❧♦s❡ t♦ t❤❡ r❡❛s♦♥ ✐s t❤❡ ❞✐✈✐s✐♦♥ ❜②

x✲❛①✐s✱

y

x✲❛①✐s✳

❚❤❡

✇❤✐❝❤ ♠❛② ❜❡ ✈❡r② s♠❛❧❧✳ ■♥ ❢❛❝t t❤❡ s♦❧✉t✐♦♥s ❛r❡ s✉♣♣♦s❡❞ t♦ st♦♣ ❛t t❤❡

❜✉t✱ ❜② ❞❡s✐❣♥ ♦❢ ❊✉❧❡r✬s ♠❡t❤♦❞✱ t❤❡② ❝❛♥✬t✳

❊❧s❡✇❤❡r❡✱ t❤❡ ❊✉❧❡r s♦❧✉t✐♦♥s ❛r❡ ❝❧♦s❡ t♦ ♣❡r❢❡❝t✿

❚❤❡ s❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s s♦❧✉t✐♦♥✿

dy x = − =⇒ ydy = −xdx =⇒ dx y

Z

ydy = −

Z

xdx =⇒

y2 x2 =− +C. 2 2

❊①❡r❝✐s❡ ✶✳✾✳✺ ❋✐♥✐s❤ t❤❡ s♦❧✉t✐♦♥✳

❲❛r♥✐♥❣✦

❚❤❡ ❊✉❧❡r s♦❧✉t✐♦♥s ♠✐❣❤t ❜❡ ✉♥❛✛❡❝t❡❞ ❜② t❤❡ ♥♦♥✲❡①✐st❡♥❝❡ ♣r♦♣❡rt② ♦❢ t❤❡ ❖❉❊✳ ❊①❛♠♣❧❡ ✶✳✾✳✻✿ ❤②♣❡r❜♦❧❛s ▲❡t✬s ❝♦♥s✐❞❡r ❛❣❛✐♥ t❤❡s❡ ❤②♣❡r❜♦❧❛s✿

xy = C .

✶✳✾✳ ❊✉❧❡r✬s ♠❡t❤♦❞✿ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡

✼✽

❚❤❡② ❛r❡ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❖❉❊✿

y′ = ❋♦r ❧❛r❣❡r ✈❛❧✉❡s ♦❢

h✱

y . x

t❤❡ ❊✉❧❡r s♦❧✉t✐♦♥ ♠✐❣❤t ❝r♦ss t❤❡

x✲❛①✐s

❞❡♠♦♥str❛t✐♥❣ ♥♦♥✲✉♥✐q✉❡♥❡ss✿

❙❡✈❡r❛❧ ❜❡tt❡r ❧♦♦❦✐♥❣ ❊✉❧❡r s♦❧✉t✐♦♥s ❛r❡ s❤♦✇♥ ❜❡❧♦✇✿

❊✈❡♥ t❤❡♥✱ ✐s t❤✐s ❛s②♠♣t♦t✐❝ ❝♦♥✈❡r❣❡♥❝❡ t♦✇❛r❞ t❤❡

x✲❛①✐s

♦r ❞♦ t❤❡② ♠❡r❣❡❄

❚❤❡ s❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s s♦❧✉t✐♦♥✿

dy y dy dx = − =⇒ =− =⇒ dx x y x

Z

dy =− y

Z

dx =⇒ ln y = − ln x + K . x

❊①❡r❝✐s❡ ✶✳✾✳✼ ❋✐♥✐s❤ t❤❡ s♦❧✉t✐♦♥✳

❲❛r♥✐♥❣✦ ❊✉❧❡r✬s ♠❡t❤♦❞ ♠✐❣❤t ✐♥tr♦❞✉❝❡ ♥♦♥✲✉♥✐q✉❡♥❡ss t♦ ❛♥ ❖❉❊✳

✶✳✶✵✳ ❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄

✼✾

✶✳✶✵✳ ❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄ ❊①❛♠♣❧❡ ✶✳✶✵✳✶✿ ❛♣♣r♦①✐♠❛t✐♦♥s❄

❍♦✇ ❢❛r ❛♣❛rt ❛r❡ t❤❡ s♦❧✉t✐♦♥s ♦❢ ❛ ♣❛✐r ♦❢ ♠❛t❝❤✐♥❣ ❞✐s❝r❡t❡ ❛♥❞ ❝♦♥t✐♥✉♦✉s ❖❉❊s❄ ■♥ ♦t❤❡r ✇♦r❞s✱ ❤♦✇ ❢❛r ✐s t❤❡ ❊✉❧❡r s♦❧✉t✐♦♥ ❢r♦♠ t❤❡ ❛❝t✉❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❖❉❊ ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡s❄ ❋♦r ❡①❛♠♣❧❡✱ ❝♦♥s✐❞❡r✿ ∆y t t = − ❛♥❞ y ′ = − . ∆t

y

y

❚❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❧❛tt❡r ❛r❡ ❝✐r❝❧❡s✳ ❙♦✱ ✐❢ ♦♥❡ s✉❝❤ s♦❧✉t✐♦♥ st❛rts ❛t (0, y0 )✱ ✐✳❡✳✱ t = 0 ❛♥❞ y = y0 ✱ t❤❡♥ t❤❡ ❝✉r✈❡ ✐s s✉♣♣♦s❡❞ t♦ ❡♥❞ ❛t t❤❡ s❛♠❡ ❧♦❝❛t✐♦♥ ♦♥ t❤❡ t✲❛①✐s✱ ✐✳❡✳✱ (y0 , 0)✳ ❚❤✐s ✐s ♥♦t ✇❤❛t ✇❡ s❡❡ ✇❤❡♥ ✇❡ ❧♦t ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❢♦r♠❡r ❡q✉❛t✐♦♥✿

❲❤② ❞♦❡s t❤✐s ❤❛♣♣❡♥❄ ❚❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❢♦r♠❡r ❞❡❝r❡❛s❡ s❧♦✇❡r t❤❛♥ t❤♦s❡ ♦❢ t❤❡ ❧❛tt❡r✳ ❖❢ ❝♦✉rs❡✱ t❤❡ ❞✐✛❡r❡♥❝❡ ❞✐♠✐♥✐s❤❡s ✇✐t❤ t❤❡ ✈❛❧✉❡ ♦❢ h = ∆t✳ ❆♣♣r♦①✐♠❛t✐♥❣ ❢✉♥❝t✐♦♥s ✐s ❧✐❦❡ ❛♣♣r♦①✐♠❛t✐♥❣ ♥✉♠❜❡rs ✭s✉❝❤ ❛s π ♦r t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧✮ ❜✉t ❤❛r❞❡r✳ ❘❡❝❛❧❧ ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✻ t❤❛t ❧✐♥❡❛r✐③❛t✐♦♥ ♠❡❛♥s r❡♣❧❛❝✐♥❣ ❛ ❣✐✈❡♥ ❢✉♥❝t✐♦♥ y = f (x) ✇✐t❤ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ y = L(x) t❤❛t ❜❡st ❛♣♣r♦①✐♠❛t❡s ✐t ❛t ❛ ❣✐✈❡♥ ♣♦✐♥t✳ ■t ✐s ❝❛❧❧❡❞ ✐ts ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ❛♥❞ ✐ts ❤❛♣♣❡♥s t♦ ❜❡ t❤❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ t❤❡ ❣r❛♣❤ ♦❢ ✇❤✐❝❤ ✐s t❤❡ t❛♥❣❡♥t ❧✐♥❡ ❛t t❤❡ ♣♦✐♥t✳ ❚❤❡ r❡♣❧❛❝❡♠❡♥t ✐s ❥✉st✐✜❡❞ ❜② t❤❡ ❢❛❝t t❤❛t ✇❤❡♥ ②♦✉ ③♦♦♠ ✐♥ ♦♥ t❤❡ ♣♦✐♥t✱ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✇✐❧❧ ♠❡r❣❡ ✇✐t❤ t❤❡ ❣r❛♣❤✿

✶✳✶✵✳

❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄

❍♦✇❡✈❡r✱ t❤❡r❡ ✐s ❛ ♠♦r❡ ❜❛s✐❝ ❛♣♣r♦①✐♠❛t✐♦♥✿ ❛ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥✱

✽✵

y = C(x)✳

❊①❛♠♣❧❡ ✶✳✶✵✳✷✿ sq✉❛r❡ r♦♦t

▲❡t✬s r❡✈✐❡✇ t❤✐s ❡①❛♠♣❧❡ ❢r♦♠ ❈❤❛♣t❡r ✶P❈✲✷❈✲✻✿

√ x❄ ❲❡ ❛♣♣r♦①✐♠❛t❡✳ f (x) = √ x ✏❛r♦✉♥❞✑ a = 4✳ ❢✉♥❝t✐♦♥ f (x) =

❡✈❛❧✉❛t✐♥❣ t❤❡

❲❡ ✜rst ❛♣♣r♦①✐♠❛t❡ t❤❡ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛

❆♥❞ t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ♥✉♠❜❡r

❝♦♥st❛♥t

√ √4.1 ✇✐t❤♦✉t ❛❝t✉❛❧❧② ♦❢ 4.1✱ ✇❡ ❛♣♣r♦①✐♠❛t❡

❤♦✇ ❞♦ ✇❡ ❝♦♠♣✉t❡

❢✉♥❝t✐♦♥✿

C(x) = 2 . ❚❤✐s ✈❛❧✉❡ ✐s ❝❤♦s❡♥ ❜❡❝❛✉s❡

f (a) =

√ √

4 = 2✳

❚❤❡♥ ✇❡ ❤❛✈❡✿

4.1 = f (4.1) ≈ C(4.1) = 2 .

■t ✐s ❛ ❝r✉❞❡ ❛♣♣r♦①✐♠❛t✐♦♥✿

❚❤❡ ♦t❤❡r✱ ❧✐♥❡❛r✱ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ✈✐s✐❜❧② ❜❡tt❡r✳ ❲❡ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛

1 L(x) = 2 + (x − 4) . 4 ❚❤✐s ✈❛❧✉❡ ✐s ❝❤♦s❡♥ ❜❡❝❛✉s❡



f (a) =



4=2

❛♥❞

f ′ (a) =

1 ✳ 4

❚❤❡♥ ✇❡ ❤❛✈❡✿

1 4.1 = f (4.1) ≈ L(4.1) = 2 + (4.1 − 4) = 2.025 . 4

❧✐♥❡❛r ❢✉♥❝t✐♦♥✿

✶✳✶✵✳

❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄

✽✶

❲❡ ❤❛✈❡ ❢♦r ❛ ❢✉♥❝t✐♦♥ y = f (x) ❛♥❞ x = a✿ ❆ ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥✿ ❆ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥✿

C(x) = f (a) L(x) = f (a) +f ′ (a)(x − a)

❲❡ s❤♦✉❧❞ ♥♦t✐❝❡ ❡❛r❧② ♦♥ t❤❛t t❤❡ ❧❛tt❡r ❥✉st ❛❞❞s ❛ ♥❡✇ ✭❧✐♥❡❛r✮ t❡r♠ t♦ t❤❡ ❢♦r♠❡r✦ ❆❧s♦✱ t❤❡ ❧❛tt❡r ✐s ❜❡tt❡r t❤❛♥ t❤❡ ❢♦r♠❡r ✕ ❜✉t ♦♥❧② ✇❤❡♥ ✇❡ ♥❡❡❞ ♠♦r❡ ❛❝❝✉r❛❝②✳ ❖t❤❡r✇✐s❡✱ t❤❡ ❢♦r♠❡r ✐s ✇♦rs❡ ❜❡❝❛✉s❡ ✐t r❡q✉✐r❡s ♠♦r❡ ❝♦♠♣✉t❛t✐♦♥✳ ❇❡❧♦✇ ✇❡ ✐❧❧✉str❛t❡ ❤♦✇ ✇❡ ❛tt❡♠♣t t♦ ❛♣♣r♦①✐♠❛t❡ ❛ ❢✉♥❝t✐♦♥ ❛r♦✉♥❞ t❤❡ ♣♦✐♥t (1, 1) ✇✐t❤ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥s ✜rst❀ ❢r♦♠ t❤♦s❡ ✇❡ ❝❤♦♦s❡ t❤❡ ❤♦r✐③♦♥t❛❧ ❧✐♥❡ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t✳ ❚❤✐s ❧✐♥❡ t❤❡♥ ❜❡❝♦♠❡s ♦♥❡ ♦❢ ♠❛♥② ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ t❤❡ ❝✉r✈❡ t❤❛t ♣❛ss t❤r♦✉❣❤ t❤❡ ♣♦✐♥t❀ ❢r♦♠ t❤♦s❡ ✇❡ ❝❤♦♦s❡ t❤❡ t❛♥❣❡♥t ❧✐♥❡✳

◆♦✇✱ ✇❡ s❤❛❧❧ s❡❡ t❤❛t t❤❡s❡ ❛r❡ ❥✉st t❤❡ t✇♦ ✜rst st❡♣s ✐♥ ❛ s❡q✉❡♥❝❡ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s✦ ❚❤❡ t❛♥❣❡♥t ❧✐♥❡ ❜❡❝♦♠❡s ♦♥❡ ♦❢ ♠❛♥② q✉❛❞r❛t✐❝ ❝✉r✈❡s ✕ ♣❛r❛❜♦❧❛s ✕ t❤❛t ♣❛ss t❤r♦✉❣❤ t❤❡ ♣♦✐♥t✳✳✳ ❛♥❞ ❛r❡ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡✳ ❲❤✐❝❤ ♦♥❡ ♦❢ t❤♦s❡ ❞♦ ✇❡ ❝❤♦♦s❡❄ ■♥ ♦r❞❡r t♦ ❛♥s✇❡r t❤❛t✱ ✇❡ ♥❡❡❞ t♦ r❡✈✐❡✇ ❛♥❞ ✉♥❞❡rst❛♥❞ ❤♦✇ t❤❡ ❜❡st ❝♦♥st❛♥t ❛♥❞ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ✇❡r❡ ❝❤♦s❡♥✳ ■♥ ✇❤❛t ✇❛② ❛r❡ t❤❡② t❤❡ ❜❡st❄ ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ y = f (x) ✐s ❣✐✈❡♥ ❛♥❞ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❛♣♣r♦①✐♠❛t❡ ✐ts ❜❡❤❛✈✐♦r ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ ❛ ♣♦✐♥t✱ x = a✱ ✇✐t❤ ❛♥♦t❤❡r ❢✉♥❝t✐♦♥ y = T (x)✳ ❚❤❡ ❧❛tt❡r ✐s t♦ ❜❡ t❛❦❡♥ ❢r♦♠ s♦♠❡ ❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥s t❤❛t ✇❡ ✜♥❞ s✉✐t❛❜❧❡✳ ❲❤❛t ✇❡ ♥❡❡❞ t♦ ❝♦♥s✐❞❡r ✐s t❤❡

❡rr♦r✱ ✐✳❡✳✱ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❢✉♥❝t✐♦♥ f ❛♥❞ ✐ts ❛♣♣r♦①✐♠❛t✐♦♥ T ✿ E(x) = |f (x) − T (x)|.

❲❡ ❛r❡ s✉♣♣♦s❡❞ t♦ ♠✐♥✐♠✐③❡ t❤❡ ❡rr♦r ❢✉♥❝t✐♦♥ ✐♥ s♦♠❡ ✇❛②✳ ❖❢ ❝♦✉rs❡✱ t❤❡ ❡rr♦r ❢✉♥❝t✐♦♥ y = E(x) ✐s ❧✐❦❡❧② t♦ ❣r♦✇ ✇✐t❤ ♥♦ ❧✐♠✐t ❛s ✇❡ ♠♦✈❡ ❛✇❛② ❢r♦♠ ♦✉r ♣♦✐♥t ♦❢ ✐♥t❡r❡st✱ x = a✳✳✳ ❜✉t ✇❡ ❞♦♥✬t ❝❛r❡✳ ❲❡ ✇❛♥t t♦ ♠✐♥✐♠✐③❡ t❤❡ ❞✐✛❡r❡♥❝❡ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ a ✇❤✐❝❤ ♠❡❛♥s ♠❛❦✐♥❣ s✉r❡ t❤❛t t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❡rr♦r ❛s x → a ❣♦❡s t♦ 0✦

✶✳✶✵✳ ❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄

✽✷

❚❤❡♦r❡♠ ✶✳✶✵✳✸✿ ❇❡st ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥

❙✉♣♣♦s❡ f ✐s ❝♦♥t✐♥✉♦✉s ❛t x = a ❛♥❞ C(x) = k

✐s ❛♥② ♦❢ ✐ts ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥s ✭✐✳❡✳✱ ❛r❜✐tr❛r② ❝♦♥st❛♥t ❢✉♥❝t✐♦♥s✮✳ ❚❤❡♥✱ t❤❡ ❡rr♦r E ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❛♣♣r♦❛❝❤❡s 0 ❛t x = a ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦♥st❛♥t ✐s ❡q✉❛❧ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f ❛t x = a❀ ✐✳❡✳✱ lim (f (x) − C(x)) = 0 ⇐⇒ k = f (a)

x→a

❚❤❛t✬s t❤❡ ❛♥❛❧♦❣ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠ ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✻✳ ❚❤❡♦r❡♠ ✶✳✶✵✳✹✿ ❇❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥

❙✉♣♣♦s❡ f ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a ❛♥❞ L(x) = f (a) + m(x − a)

✐s ❛♥② ♦❢ ✐ts ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❚❤❡♥✱ t❤❡ ❡rr♦r E ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❛♣♣r♦❛❝❤❡s 0 ❛t x = a ❢❛st❡r t❤❛♥ x − a ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ ❧✐♥❡❛r t❡r♠ ✐s ❡q✉❛❧ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f ❛t x = a❀ ✐✳❡✳✱ f (x) − L(x) = 0 ⇐⇒ m = f ′ (a) x→a x−a lim

❈♦♠♣❛r✐♥❣ t❤❡s❡ t✇♦ ❝♦♥❞✐t✐♦♥s✿ f (x) − C(x) → 0 ❛♥❞

f (x) − L(x) → 0, x−a

r❡✈❡❛❧s t❤❡ s✐♠✐❧❛r✐t② ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ✐♥ ❤♦✇ ✇❡ ♠✐♥✐♠✐③❡ t❤❡ ❡rr♦r✦ ❚❤❡ ❞✐✛❡r❡♥❝❡ ✐s ✐♥ t❤❡ ❞❡❣r❡❡✿ ❤♦✇ ❢❛st t❤❡ ❡rr♦r ❢✉♥❝t✐♦♥ ❣♦❡s t♦ ③❡r♦✳ ■♥❞❡❡❞✱ ✇❡ ❧❡❛r♥❡❞ ✐♥ ❈❤❛♣t❡r ✷❉❈✲✻ t❤❛t t❤❡ ❧❛tt❡r ❝♦♥❞✐t✐♦♥ ♠❡❛♥s t❤❛t f (x) − L(x) ❝♦♥✈❡r❣❡s t♦ 0 ❢❛st❡r t❤❛♥ x − a✱ ✐✳❡✳✱ f (x) − L(x) = o(x − a),

t❤❡r❡ ✐s ♥♦ s✉❝❤ r❡str✐❝t✐♦♥ ❢♦r t❤❡ ❢♦r♠❡r✳ ❙♦ ❢❛r✱ t❤✐s ✐s ✇❤❛t ✇❡ ❤❛✈❡ ❞✐s❝♦✈❡r❡❞✿ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ❛r❡ ❜✉✐❧t ❢r♦♠ t❤❡ ❜❡st ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ❜② ❛❞❞✐♥❣ ❛ ❧✐♥❡❛r t❡r♠✳ ❚❤❡ ❜❡st ♦♥❡ ♦❢ t❤♦s❡ ❤❛s t❤❡ s❧♦♣❡ ✭✐ts ♦✇♥ ❞❡r✐✈❛t✐✈❡✮ ❡q✉❛❧ t♦ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f ❛t a✳ ❍♦✇ t❤❡ s❡q✉❡♥❝❡ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s ✇✐❧❧ ♣r♦❣r❡ss ✐s ♥♦✇ ❝❧❡❛r❡r✿ q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ❛r❡ ❜✉✐❧t ❢r♦♠ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ❜② ❛❞❞✐♥❣ ❛ q✉❛❞r❛t✐❝ t❡r♠✳

✶✳✶✵✳

❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄

✽✸

❇✉t ✇❤✐❝❤ ♦♥❡ ♦❢ t❤♦s❡ ✐s t❤❡ ❜❡st❄

❚❤❡♦r❡♠ ✶✳✶✵✳✺✿ ❇❡st q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥

❙✉♣♣♦s❡ f ✐s t✇✐❝❡ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a ❛♥❞ Q(x) = f (a) + f ′ (a)(x − a) + p(x − a)2

✐s ❛♥② ♦❢ ✐ts q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❚❤❡♥✱ t❤❡ ❡rr♦r E ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❛♣♣r♦❛❝❤❡s 0 ❛t x = a ❢❛st❡r t❤❛♥ (x − a)2 ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ q✉❛❞r❛t✐❝ t❡r♠ ✐s ❡q✉❛❧ t♦ ❤❛❧❢ ♦❢ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f ❛t x = a❀ ✐✳❡✳✱ f (x) − Q(x) 1 = 0 ⇐⇒ p = f ′′ (a) 2 x→a (x − a) 2 lim

❖♥❝❡ ❛❣❛✐♥✱ t❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡ t❤❡♦r❡♠ ♠❡❛♥s t❤❛t

f (x) − Q(x)

❝♦♥✈❡r❣❡s t♦

0

❢❛st❡r t❤❛♥

(x − a)2 ✱

f (x) − Q(x) = o((x − a)2 ) . ❲❡ st❛rt t♦ s❡❡ ❛ ♣❛tt❡r♥✿



❚❤❡ ❞❡❣r❡❡s ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♥❣ ♣♦❧②♥♦♠✐❛❧s ❛r❡ ❣r♦✇✐♥❣✳



❚❤❡ ❞❡❣r❡❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s ❜❡✐♥❣ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ❛r❡ ❣r♦✇✐♥❣ t♦♦✳

❊①❛♠♣❧❡ ✶✳✶✵✳✻✿ s✐♥❡ ▲❡t✬s ❛♣♣r♦①✐♠❛t❡

f (x) = sin x

❛t

x = 0✳

❋✐rst✱ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡s✿

f (x) = sin x =⇒ f (0) = 0 =⇒ C(x) = 0 ′ f (x) = cos x =⇒ f ′ (0) = 1 =⇒ L(x) = x f ′′ (x) = − sin x =⇒ f ′′ (0) = 0 =⇒ Q(x) = ?

♦r

✶✳✶✵✳

❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄

✽✹

❚❤❡r❡❢♦r❡✱ t❤❡ ❜❡st q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s✿

0 Q(x) = 0 + 1(x − 0) − (x − 0)2 = x . 2 ❙❛♠❡ ❛s t❤❡ ❧✐♥❡❛r✦ ❲❤②❄ ❇❡❝❛✉s❡

♥✉♠❜❡rs

❲❡ ✉s❡❞ s❡q✉❡♥❝❡s ♦❢

❢✉♥❝t✐♦♥s

sin

✐s ♦❞❞✳

t♦ ❛♣♣r♦①✐♠❛t❡ ♦t❤❡r ♥✉♠❜❡rs ✭❈❤❛♣t❡r ✺✮❀ ♥♦✇ ✇❡ ✇✐❧❧ ✉s❡ s❡q✉❡♥❝❡s ♦❢

t♦ ❛♣♣r♦①✐♠❛t❡ ♦t❤❡r ❢✉♥❝t✐♦♥s✳ ■♥ ♦r❞❡r t♦ ❜❡ ❛❜❧❡ t♦ ❣♦ ❜❡②♦♥❞ q✉❛❞r❛t✐❝ ✐♥ ♦✉r s❡q✉❡♥❝❡ ♦❢

♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s✱ ✇❡ r❡♥❛♠❡ t❤❡♠ ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r

❞❡❣r❡❡s ✿

T0 (x) = C(x) , T1 (x) = L(x) , T2 (x) = Q(x) , ... ❊①❛♠♣❧❡ ✶✳✶✵✳✼✿ sq✉❛r❡ r♦♦t

❇❛❝❦ t♦ t❤❡ ♦r✐❣✐♥❛❧ ❡①❛♠♣❧❡ ♦❢

f (x) =



x

❛t

❝♦♥st❛♥t✿ ❧✐♥❡❛r✿

❝✉❜✐❝✿

1 (x − 4)2 2 · 32 1 (x − 4)2 2 · 32



q✉❛❞r❛t✐❝✿

(?)(x − 4)3 − ✳ ✳ ✳

✳ ✳ ✳

a = 4✳

❖♥❡ ❝❛♥ ❣✉❡ss ✇❤❡r❡ t❤✐s ✐s ❣♦✐♥❣✿

2 = T0 (x) 1 (x − 4) + 2 = T1 (x) 4 1 (x − 4) + 2 = T2 (x) + 4 1 + (x − 4) + 2 = T3 (x) 4 ✳ ✳ ✳

✳ ✳ ✳

❲❡ ❛❞❞ ❛ t❡r♠ ❡✈❡r② t✐♠❡ ✇❡ ♠♦✈❡ ❞♦✇♥ t♦ t❤❡ ♥❡①t ❞❡❣r❡❡❀ ✐t✬s ❛

✳ ✳ ✳

f − T0 = o(1)

f − T1 = o(x − a) f − T2 = o((x − a)2 ) f − T3 = o((x − a)3 ) ✳ ✳ ✳

r❡❝✉rs✐♦♥ ✦

❙✉❝❤ ❛ s❡q✉❡♥❝❡ ✭❛

s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s✦✮ ✐s ❝❛❧❧❡❞ ❛ ✏s❡r✐❡s✑✳

❲❡ ❝♦♥s✐❞❡r ❛ s♦❧✉t✐♦♥ ♦❢ ❛ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ✭❞✐s❝r❡t❡ ❖❉❊✮ ❛♥❞ t❤❛t ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦♥t✐♥✉♦✉s ❖❉❊ ✇✐t❤ t❤❡ s❛♠❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✱

y(t0 ) = y0 ✳ t > t0 ✐s

✈❛❧✉❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ❡✈❛❧✉❛t❡❞ ❛t s♦♠❡

❚❤❡✐r ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ t✇♦ ✐s ❛ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ❛❜s♦❧✉t❡ r❡❢❡rr❡❞ t♦ ❛s t❤❡

❡rr♦r✳

✶✳✶✵✳ ❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄

✽✺

❚♦ ❡st✐♠❛t❡ t❤❡ ❡rr♦r✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❡rr♦r ❜♦✉♥❞ ♦❢ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✻✳ ❙✉♣♣♦s❡ y ✐s t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t = t0 ❛♥❞ L(t) = y(t0 ) + y ′ (t0 )(t − t0 ) ✐s ✐ts ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ❛t t0 ✳ ❚❤❡♥✱ t❤❡ ❡rr♦r s❛t✐s✜❡s✿ E(t) = |y(t) − L(t)| ≤ 12 K(t − t0 )2 ,

✇❤❡r❡ K ✐s ❛ ❜♦✉♥❞ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦♥ t❤❡ ✐♥t❡r✈❛❧ ❢r♦♠ t0 t♦ t✿ |y ′′ (c)| ≤ K ❢♦r ❛❧❧ c ✐♥ t❤✐s ✐♥t❡r✈❛❧ .

❙✉♣♣♦s❡ ♥♦✇ t❤❛t y ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ■❱P✿ y ′ = f (t, y), y(t0 ) = y0 .

Pr♦✈✐❞❡❞ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s ❜♦✉♥❞❡❞ ❛s ❛❜♦✈❡✱ t❤❡ ❡rr♦r ♦❢ ❛ s✐♥❣❧❡ st❡♣ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ✭✐✳❡✳✱ ❊✉❧❡r ♠❡t❤♦❞✮ ✐s ❜♦✉♥❞❡❞ ❜②✿ E(t0 + h) = |y(t0 + h) − y1 | ≤ 12 Kh2 ,

✇❤❡r❡ y1 = L1 (t0 + h) ✐s t❤❡ ✜rst st❡♣ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ t❤❛t ❝♦♠❡s ❢r♦♠ L1 (t) = y0 + f (t0 , y0 )(t − t0 ) ,

t❤❡ ❧✐♥❡❛r✐③❛t✐♦♥ ♦❢ y ❛t t0 ✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❡rr♦r ♦❢ ❛ s✐♥❣❧❡ st❡♣ ✐s ❞❡❝r❡❛s✐♥❣ q✉❛❞r❛t✐❝❛❧❧② ✇✐t❤ h✳ ◆♦✇✱ ❧❡t✬s ❝♦♥s✐❞❡r ♠✉❧t✐♣❧❡ st❡♣s✳ ❚❤✐s ✐s ✇❤❛t ✇✐❧❧ ❤❛♣♣❡♥✿

✶✳✶✵✳ ❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄

✽✻

❲❤❛t ❝❛♥ ✇❡ ❞♦ ❛❜♦✉t t❤✐s ♣r♦♣❛❣❛t✐♦♥ ♦❢ ❡rr♦r ❄ ■❢ y ′′ ✐s ❜♦✉♥❞❡❞ ❜② t❤❡ s❛♠❡ ❝♦♥st❛♥t K ✱ ✇❡ ❝❛♥ ❛♣♣❧② t❤✐s ❢♦r♠✉❧❛ t♦ t❤❡ s❡❝♦♥❞ st❡♣ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥✿ |y(t1 + h) − y2 | ≤ 12 Kh2 ,

✇❤❡r❡ y2 = L2 (t1 + h) ✐s t❤❡ s❡❝♦♥❞ st❡♣ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ t❤❛t ❝♦♠❡s ❢r♦♠ L2 (t) = y1 + f (t1 , y1 )(t − t1 ) ,

t❤❡ ❧✐♥❡❛r✐③❛t✐♦♥ ♦❢ y ❛t t1 ✳ ◆♦t✐❝❡ t❤❛t t❤❡ t✇♦ ❡rr♦r ❜♦✉♥❞s ❛♣♣❧② t♦ t✇♦ ❞✐✛❡r❡♥t s♦❧✉t✐♦♥s y ♦❢ t✇♦ ❞✐✛❡r❡♥t ■❱Ps✿

❚❤❡r❡❢♦r❡✱ t❤❡ ❜❡st ❜♦✉♥❞ ❢♦r t❤❡ ❡rr♦r ♦❢ t❤❡ t✇♦ st❡♣s ✐s t❤❡ s✉♠ ♦❢ t❤❡ t✇♦ ✭✐♥ ♦✉r ❝❛s❡✱ ✐❞❡♥t✐❝❛❧✮ ❜♦✉♥❞s✿ E(t0 + 2h) = |y(t0 + 2h) − y2 | ≤ 2 · 12 Kh2 .

❆♥❞ s♦ ♦♥✳✳✳ ❚❤❡ ❜♦✉♥❞ ❢♦r t❤❡ ❡rr♦r ♦❢ t❤❡ n st❡♣s ✐s n t✐♠❡s t❤❡ ❜♦✉♥❞✿ E(t0 + nh) = |y(t0 + nh) − yn | ≤ n · 21 Kh2 .

❙✉♣♣♦s❡ ♥♦✇ t❤❛t ✇❡ ❡①t❡♥❞ t❤❡ ❞✐s❝r❡t❡ s♦❧✉t✐♦♥ ✇✐t❤ n st❡♣s t♦ t0 + H ✳ ❚❤❡♥ n=

H . h

❚❤❡r❡❢♦r❡✱ E(t0 + nh) = |y(t0 + nh) − yn | ≤

1 H 1 · Kh2 = KHh . h 2 2

❲❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❡rr♦r ✐s ❞❡❝r❡❛s✐♥❣ ❧✐♥❡❛r❧② ✇✐t❤ h✳ ❚❤❡♦r❡♠ ✶✳✶✵✳✽✿ ❊rr♦r ❇♦✉♥❞

❙✉♣♣♦s❡ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ z = f (t, y) s❛t✐s✜❡s ∂f ∂f (t, y) + (t, y) f (t, y) ≤ K , ∂t ∂y

❢♦r ❡✈❡r② (t, y) ✐♥ t❤❡ r❡❝t❛♥❣❧❡ t0 ≤ t ≤ t0 + H, A ≤ y ≤ B ✳ ❙✉♣♣♦s❡ y ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ■❱P y ′ = f (t, y), y(t0 ) = y0 ,

✇✐t❤ t❤❡ ❞♦♠❛✐♥ [t0 , t0 + H] ❛♥❞ t❤❡ r❛♥❣❡ ✇✐t❤✐♥ [A, B]✳ ❙✉♣♣♦s❡ yn ✐s t❤❡ n st❡♣s ♦❢ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❞✐s❝r❡t❡ ❝♦✉♥t❡r♣❛rt ♦❢ t❤✐s ■❱P ✇✐t❤ h = H/n✳ ❚❤❡♥

✶✳✶✵✳ ❍♦✇ ❧❛r❣❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s❄

✽✼

t❤❡✐r ❞✐✛❡r❡♥❝❡ s❛t✐s✜❡s✿ |y(t0 + nh) − yn | ≤ 12 KHh . Pr♦♦❢✳

❚❤❡ ♣r♦♦❢ ✐s ❛❜♦✈❡❀ ✇❡ ❥✉st ✉s❡ t❤❡ ❈❤❛✐♥ ❘✉❧❡ t♦ ❞❡r✐✈❡ t❤✐s✿ y ′′ (t) =

∂f ∂f (t, y(t)) + (t, y(t)) f (t, y(t)) . ∂t ∂y

❋♦r ❡❛❝❤ h✱ t❤❡ ❡rr♦r ❜♦✉♥❞ ♣r♦✈✐❞❡s ❛ t✉♥♥❡❧✱ ♦❢ ❝♦♥st❛♥t ✇✐❞t❤✱ ❛r♦✉♥❞ t❤❡ ❞✐s❝r❡t❡ s♦❧✉t✐♦♥ t❤❛t ❝♦♥t❛✐♥s t❤❡ ✭✉♥❦♥♦✇♥✮ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❝♦♥t✐♥✉♦✉s ❖❉❊✱ ❛♥❞ ✈✐❝❡ ✈❡rs❛✿

❊①❡r❝✐s❡ ✶✳✶✵✳✾

Pr♦✈❡ t❤❡ ❧❛st st❛t❡♠❡♥t✳ ❍✐♥t✿ ②♦✉ ♥❡❡❞ ♠♦r❡ t❤❛♥ ❥✉st n ♣♦✐♥ts ❛s ✐♥ t❤❡ t❤❡♦r❡♠✳ ❊①❡r❝✐s❡ ✶✳✶✵✳✶✵

❆♣♣❧② t❤❡ t❤❡♦r❡♠ t♦ t❤❡ ❡①❛♠♣❧❡ ♦❢ ❝♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ s❡❝t✐♦♥✳ ❊①❛♠♣❧❡ ✶✳✶✵✳✶✶✿ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ ❝✐r❝❧❡s

■♥ ♦✉r ❡①❛♠♣❧❡ ♦❢ ❝♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s✱ ✇❡ ❛❧s♦ s❛✇ t❤❛t t❤❡ ❞✐s❝r❡t❡ s♦❧✉t✐♦♥s ♦✈❡rs❤♦♦t ✇❤❡♥ t❤❡② r❡❛❝❤ t❤❡ t✲❛①✐s✳ ❚❤❡ r❡❛s♦♥ ✐s ❝♦♥❝❛✈✐t②✳ ❙✉♣♣♦s❡ y ✐s ❛ s♦❧✉t✐♦♥ ❛❜♦✈❡ t❤❡ t✲❛①✐s✱ t❤❡♥ y > 0 ❛♥❞ y ′ < 0✳ ❚❤❡♥✱ ✇❡ ❝❛♥ ✜♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ y ✿ d d y = f (t, y) = dt dt ′′



t − y



=−

1 · y − t · y′ t · y′ − y = < 0. y2 y2

■t✬s ♥❡❣❛t✐✈❡✦ ■t✬s ♣♦s✐t✐✈❡ ❜❡❧♦✇ t❤❡ t✲❛①✐s✳ ❚❤❡r❡❢♦r❡✱ t❤❡ s♦❧✉t✐♦♥s ❛r❡ ❝♦♥❝❛✈❡ ❞♦✇♥ ✐♥ t❤❡ ❢♦r♠❡r ❛♥❞ ❝♦♥❝❛✈❡ ✉♣ ✐♥ t❤❡ ❧❛tt❡r ❝❛s❡✳ ❚❤❡♥✱ t❤❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s t❤❛t ✇❡ ✉s❡ ❢♦r ❊✉❧❡r✬s ♠❡t❤♦❞ ♦✈❡r❡st✐♠❛t❡ t❤❡ s♦❧✉t✐♦♥s ✐♥ t❤❡ ❢♦r♠❡r ❝❛s❡ ❛♥❞ ✉♥❞❡r❡st✐♠❛t❡ ✐♥ t❤❡ ❧❛tt❡r✳

❚❤❡ ❜❡♥❡✜t ✐s t❤❛t ✇❡ ❛r❡ ❝✉tt✐♥❣ t❤❡ t✉♥♥❡❧ ✐♥ ❤❛❧❢✳

✶✳✶✶✳

◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s

✽✽

❈♦r♦❧❧❛r② ✶✳✶✵✳✶✷✿ ❖♥❡✲s✐❞❡❞ ❊rr♦r ❇♦✉♥❞ ❯♥❞❡r t❤❡ ❝♦♥❞✐t♦♥s ♦❢ t❤❡ ❧❛st t❤❡♦r❡♠✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿



❲❤❡♥

f ′ > 0✱

✇❡ ❤❛✈❡ ❢♦r ❡❛❝❤

t

✐♥

[t0 , t0 + H]✿

yn ≤ y(t0 + nh) ≤ yn + 12 KHh . •

❲❤❡♥

f ′ < 0✱

✇❡ ❤❛✈❡ ❢♦r ❡❛❝❤

t

✐♥

[t0 , t0 + H]✿

yn − 21 KHh ≤ y(t0 + nh) ≤ yn . ❊①❡r❝✐s❡ ✶✳✶✵✳✶✸

❆♣♣❧② t❤❡ ❝♦r♦❧❧❛r② t♦ t❤❡ ❡①❛♠♣❧❡ ♦❢ ❝♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ s❡❝t✐♦♥✳

✶✳✶✶✳ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s

❯♥❢♦rt✉♥❛t❡❧②✱ ❊✉❧❡r✬s ♠❡t❤♦❞ ❞❡♣❡♥❞s ♦♥ t❤❡ ✈❛❧✉❡ ♦❢ h ❛♥❞✱ ❢r♦♠ s✐♠♣❧❡ ❡①♣❡r✐♠❡♥t❛t✐♦♥✱ ♦♥❡ ❝❛♥♥♦t ❦♥♦✇ ✇❤❡t❤❡r h ✐s s♠❛❧❧ ❡♥♦✉❣❤✳ ■t✬s ❥✉st ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥✦ ❚❤❡ ♠❡t❤♦❞ ❝❛♥ ❛❧s♦ ❢❛✐❧ ♥❡❛r t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ❞♦♠❛✐♥✳ ❲✐t❤ t❤❡s❡ ❧✐♠✐t❛t✐♦♥ ♦❢ t❤❡ q✉❛♥t✐t❛t✐✈❡ ♠❡t❤♦❞s✱ q✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♣r♦✈✐❞❡s ❢✉❧❧② ❛❝❝✉r❛t❡ ✐❢ ❜r♦❛❞ ❞❡s❝r✐♣t✐♦♥s ♦❢ t❤❡ s♦❧✉t✐♦♥s✳ ❚❤❡ ♠❡t❤♦❞ ❛♠♦✉♥ts t♦ ❣❛t❤❡r✐♥❣ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ s♦❧✉t✐♦♥s ✇✐t❤♦✉t s♦❧✈✐♥❣ t❤❡ ❖❉❊ ✕ ❡✐t❤❡r ❛♥❛❧②t✐❝❛❧❧② ♦r ♥✉♠❡r✐❝❛❧❧②✳ ❆s ✇❡ s❛✇ ❡❛r❧② ✐♥ t❤✐s ❝❤❛♣t❡r✱ s♦♠❡ ❖❉❊s y ′ = f (y) ❤❛✈❡ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❢✉♥❝t✐♦♥ ✐♥❞❡♣❡♥❞❡♥t ♦❢ t ❛s ✐♥ f (y) = y 2 ✱ ❡t❝✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ✜❡❧❞ ♦❢ s❧♦♣❡s ♦♥ t❤❡ xy ✲♣❧❛♥❡ ✉♥❞❡r ❤♦r✐③♦♥t❛❧ s❤✐❢t ✇✐❧❧ ❧❛♥❞ ♦♥ ✐ts❡❧❢✿

❚❤❡♥ s♦ ✇✐❧❧ ✐ts s♦❧✉t✐♦♥ s❡t✦ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐❧❧ ❜❡ ✈❡r② ✐♠♣♦rt❛♥t✳ ❚❤❡♦r❡♠ ✶✳✶✶✳✶✿ ❚✐♠❡ ■♥❞❡♣❡♥❞❡♥t ❖❉❊s

y ′ = f (y) ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t✳ ❚❤❡♥✱ s♦ ✐s y = y(t + s) ❢♦r ❛♥② r❡❛❧ s✳

❙✉♣♣♦s❡ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ❛♥ ❖❉❊

y = y(t) ❊①❡r❝✐s❡ ✶✳✶✶✳✷

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❖❉❊ t❤❡♥

✐❢

✶✳✶✶✳

◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s

✽✾

❊①❛♠♣❧❡ ✶✳✶✶✳✸✿ ❧♦❣✐st✐❝ ❡q✉❛t✐♦♥

❈♦♥s✐❞❡r t❤❡ ❧♦❣✐st✐❝ ❡q✉❛t✐♦♥✿

❈♦♥s✐❞❡r ❛❧s♦ t❤❡ ♣♦♣✉❧❛t✐♦♥ ♠♦❞❡❧✿

◆♦t ♦♥❧② ✐t ✐s ♣r❡s❡r✈❡❞ ✉♥❞❡r ❤♦r✐③♦♥t❛❧ s❤✐❢ts✱ ❜✉t ❛❧s♦ ✉♥❞❡r ✈❡rt✐❝❛❧ str❡t❝❤❡s✦ ❚❤❡ ♠❛✐♥ t♦♦❧s ❝♦♠❡ ❢r♦♠ ❱♦❧✉♠❡ ✷ ✭t❤❡ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠✱ ❈❤❛♣t❡r ✷❉❈✲✺✮✳ ❋✐rst✱ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ y ❞❡✜♥❡❞ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ ✐s ✐♥❝r❡❛s✐♥❣ ✇❤❡♥ ✐ts ❞❡r✐✈❛t✐✈❡ ✐s ♣♦s✐t✐✈❡ ❛♥❞ ❞❡❝r❡❛s✐♥❣ ✇❤❡♥ ✐t✬s ♥❡❣❛t✐✈❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ y ′ > 0 =⇒ y ր y ′ < 0 =⇒ y ց ❚❤❡♦r❡♠ ✶✳✶✶✳✹✿ ▼♦♥♦t♦♥✐❝✐t② ♦❢ ❙♦❧✉t✐♦♥s

■❢

y

✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❖❉❊

❤❛✈❡ ♦♥ t❤✐s s❡t✿

y ′ = f (t, y)

✇✐t❤✐♥ ❛♥ ♦♣❡♥ s❡t ✐♥ t❤❡

f (t, y) > 0 =⇒ y f (t, y) < 0 =⇒ y

ty ✲♣❧❛♥❡✱

✇❡

✐s ✐♥❝r❡❛s✐♥❣. ✐s ❞❡❝r❡❛s✐♥❣.

❚❤❡ s❡❝♦♥❞ t♦♦❧ ✐s ❛❧s♦ ❡❧❡♠❡♥t❛r②✳ ❚❤❡♦r❡♠ ✶✳✶✶✳✺✿ ❙t❛t✐♦♥❛r② ❙♦❧✉t✐♦♥s

■❢

f (t, y0 ) = 0

❢♦r ❛❧❧

t

✇✐t❤✐♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ ❛♥❞ s♦♠❡

✭❝♦♥st❛♥t✮ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❖❉❊

y ′ = f (t, y)✳

y0 ✱

t❤❡♥

y(t) = y0

✐s ❛

❊①❛♠♣❧❡ ✶✳✶✶✳✻✿ tr✐❣♦♥♦♠❡tr✐❝

❈♦♥s✐❞❡r✿

y ′ = − tan(y) .

❋✐rst✱ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✱ t❤❡ t❛♥❣❡♥t✱ ✐s ✉♥❞❡✜♥❡❞ ❛t y = π/2 + kπ ❢♦r ❛❧❧ ✐♥t❡❣❡r k✳ ❚❤❡r❡❢♦r❡✱ t❤❡s❡

✶✳✶✶✳

◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s

❤♦r✐③♦♥t❛❧ ❧✐♥❡s ♦♥ t❤❡ ty ✲♣❧❛♥❡ ✇✐❧❧ ❝✉t t❤❡ ❞♦♠❛✐♥ ✐♥t♦ str✐♣s s♦ t❤❛t ❡✈❡r② s♦❧✉t✐♦♥ ✇✐❧❧ st❛② ✐♥s✐❞❡ ♦♥❡ ♦❢ t❤❡♠✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ s✐❣♥ ♦❢ t❤❡ t❛♥❣❡♥t ❝❤❛♥❣❡s ❛t t❤♦s❡ ✈❛❧✉❡s ♦❢ y ❛s ✇❡❧❧ ❛s ❛t y = kπ ✱ ✐♥ t❤❡ ♠✐❞❞❧❡ ♦❢ ❡❛❝❤ str✐♣✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ✇❤❛t ✇❡ ❝♦♥❝❧✉❞❡ ❛❜♦✉t t❤❡ str✐♣ (−π/2, π/2)✿ • ❋♦r −π/2 < y < 0✱ ✇❡ ❤❛✈❡ y ′ = − tan y > 0 ❛♥❞✱ t❤❡r❡❢♦r❡✱ y ր ✳ • ❋♦r y = 0✱ ✇❡ ❤❛✈❡ y ′ = − tan y = 0 ❛♥❞✱ t❤❡r❡❢♦r❡✱ y ✐s ❛ ❝♦♥st❛♥t s♦❧✉t✐♦♥✳ • ❋♦r 0 < y < π/2✱ ✇❡ ❤❛✈❡ y ′ = − tan y < 0 ❛♥❞✱ t❤❡r❡❢♦r❡✱ y ց ✳

❚❤❡ r❡s✉❧ts ♦❢ t❤❡ ❛♥❛❧②s✐s ❛r❡ s✉♠♠❛r✐③❡❞ ❜❡❧♦✇ ✇✐t❤ t❤❡ ♠♦♥♦t♦♥✐❝✐t② ♦❢ t❤❡ s♦❧✉t✐♦♥ ♠❛t❝❤❡❞ ✇✐t❤ t❤❡ s✐❣♥ ♦❢ t❤❡ t❛♥❣❡♥t✿

❙♦✱ ❡✈❡r② s♦❧✉t✐♦♥ y ✐s ❞❡❝r❡❛s✐♥❣ ✭♦r ✐♥❝r❡❛s✐♥❣ r❡s♣❡❝t✐✈❡❧②✮ t❤r♦✉❣❤♦✉t ✐ts ❞♦♠❛✐♥✳ ❚❤❡r❡ ❛r❡ ♠♦r❡ s✉❜t❧❡ ❝♦♥❝❧✉s✐♦♥s✳ ❆❝❝♦r❞✐♥❣ t♦ ❛ t❤❡♦r❡♠✱ t❤❡ ❡①✐st❡♥❝❡ ✐s s❛t✐s✜❡❞ ✇✐t❤✐♥ t❤❡ str✐♣✳ ❚❤❡r❡❢♦r❡✱ t❤❡ s♦❧✉t✐♦♥s ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ ❢✉rt❤❡r ❛♥❞ ❢✉rt❤❡r✳ ❆❝❝♦r❞✐♥❣ t♦ ❛♥♦t❤❡r t❤❡♦r❡♠✱ t❤❡ ✉♥✐q✉❡♥❡ss ✐s s❛t✐s✜❡❞ ✇✐t❤✐♥ t❤❡ str✐♣✳ ❚❤❡r❡❢♦r❡✱ t❤❡♥ t❤❡ s♦❧✉t✐♦♥s ❝❛♥✬t ❝r♦ss t❤❡ ❧✐♥❡ y = 0✳ ❲❡ ❝♦♥❝❧✉❞❡✿ ◮ ❊✈❡r② s♦❧✉t✐♦♥ ✐s ❛♣♣r♦❛❝❤✐♥❣ t❤❡ ♠✐❞❞❧❡ ❧✐♥❡ ♦❢ t❤❡ str✐♣✱ y = 0✱ ❛s t → +∞✱ ❛♥❞ t❤✐s ❧✐♥❡ ✐s ❛ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡ ♦❢ t❤❡ s♦❧✉t✐♦♥✱ ✐✳❡✳✱ y → 0+ ✳

❇✉t ❤♦✇ ❞♦❡s ✐t ❛♣♣r♦❛❝❤ t❤❡ ♥♦✲s♦❧✉t✐♦♥ ❧✐♥❡s y = π/2 + kπ ❄ ❆s t❤❡ t❛♥❣❡♥t ✐s ❛♣♣r♦❛❝❤✐♥❣ ✐♥✜♥✐t②✱ s♦ ✐s t❤❡ s❧♦♣❡ ♦❢ t❤❡ s♦❧✉t✐♦♥✳ ❲❡ ❝♦♥❝❧✉❞❡✿ ◮ ❊✈❡r② s♦❧✉t✐♦♥ ✐s ❛♣♣r♦❛❝❤✐♥❣ t❤❡ ❡❞❣❡ ❧✐♥❡s ♦❢ t❤❡ str✐♣✱ y = ±π/2✱ ❛♥❞ ✐t ❜❡❝♦♠❡s ♠♦r❡ ❛♥❞ ♠♦r❡ ✈❡rt✐❝❛❧✱ ✐✳❡✳✱ y ′ → ±∞✳

❚❤✐s ✐s ✇❤❛t t❤❡ s♦❧✉t✐♦♥ s❡t ♠✐❣❤t ❧♦♦❦ ❧✐❦❡ ✇❤❡♥ ❞r❛✇♥ ❜② ❤❛♥❞✿

◆♦✇ ✇❡ ✉t✐❧✐③❡ s♦♠❡ ❢❛❝ts ❛❜♦✉t t❤❡ ❢✉♥❝t✐♦♥ f ✳ ✶✳ ❲❡ ✉s❡ t❤❛t ❢❛❝t t❤❛t f ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t t♦ ♣r♦❞✉❝❡ ✭❛❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠ ❛❜♦✈❡✮ ♠♦r❡ s♦❧✉t✐♦♥s ❜② ❛ ❤♦r✐③♦♥t❛❧ s❤✐❢ts✳ ✷✳ ❲❡ ✉s❡ t❤❡ ❢❛❝t t❤❛t f ✐s ♦❞❞ t♦ ♣r♦❞✉❝❡ ♠♦r❡ s♦❧✉t✐♦♥s ❜② ❛ ✈❡rt✐❝❛❧ ✢✐♣✳ ✸✳ ❲❡ ✉s❡ t❤❡ ❢❛❝t t❤❛t f ✐s ♣❡r✐♦❞✐❝ t♦ ♣r♦❞✉❝❡ ♠♦r❡ s♦❧✉t✐♦♥s ❜② ✈❡rt✐❝❛❧ s❤✐❢ts✳ ❚❤✐s ✐s t❤❡ s♦❧✉t✐♦♥ s❡t✿

✾✵

✶✳✶✶✳

◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s

✾✶

❚❤❡ ❝♦♥❝❧✉s✐♦♥s ❛r❡ ❝♦♥✜r♠❡❞ ✇✐t❤ ❊✉❧❡r✬s ♠❡t❤♦❞✿

❲❡ ♥♦t✐❝❡ ♥♦✇ t❤❛t t❤❡ s♦❧✉t✐♦♥s ❛r❡ r✉♥♥✐♥❣ ❛✇❛② ❢r♦♠ t❤❡ ❡❞❣❡ ♦❢ t❤❡ ❞♦♠❛✐♥ ❛♥❞ t♦✇❛r❞ t❤❡ ❝♦♥st❛♥t s♦❧✉t✐♦♥✳ ❲❡ ❛❧s♦ ♥♦t✐❝❡ t❤❛t t❤❡ s♦❧✉t✐♦♥s ❛r❡ r❡♣❡❛t❡❞ ✉♥❞❡r ❛ ❤♦r✐③♦♥t❛❧ s❤✐❢t✳

❲❛r♥✐♥❣✦

❲❡ ❝♦♥✜r♠ ♦✉r q✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ✇✐t❤ ❊✉❧❡r✬s ♠❡t❤♦❞✳✳✳ ❛♥❞ ✈✐❝❡ ✈❡rs❛✦ ❊①❡r❝✐s❡ ✶✳✶✶✳✼ ■♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ ✐s y = π/2 ❛❧s♦ ❛ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡❄ ❈♦♥✜r♠ ②♦✉r ❝♦♥❝❧✉s✐♦♥ ✇✐t❤ ❊✉❧❡r✬s y ′ ♥❡❛r t❤✐s ❧✐♥❡❄

♠❡t❤♦❞✳ ❲❤❛t ✐s

◆♦t✐❝❡ t❤❛t ❜♦t❤ ❛❜♦✈❡ ❛♥❞ ❜❡❧♦✇ ❊✉❧❡r✬s ♠❡t❤♦❞ ✐s r✉♥ ❢♦r ✈❛r✐♦✉s ✈❛❧✉❡s ♦❢

t0 = 0, 1, 2, 3...

✐♥ ♦r❞❡r t♦

❛✈♦✐❞ ❡♠♣t② s♣❛❝❡s ❝r❡❛t❡❞ ❜② t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ t❤❡ s♦❧✉t✐♦♥s✳

❊①❛♠♣❧❡ ✶✳✶✶✳✽✿ ♠♦r❡ ❝♦♠♣❧❡① ❈♦♥s✐❞❡r ♥❡①t✿

y ′ = cos y ·

p

❲❡ st❛rt ✇✐t❤ t❤❡ ✏s✐❣♥ ❛♥❛❧②s✐s✑ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✱

|y| . f (y) = cos y ·

p |y|✱

✇❤✐❝❤ ✐s ❢✉❧❧② ❞❡t❡r♠✐♥❡❞

✶✳✶✶✳

✾✷

◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s

❜② cos y ✉♥❧❡ss y = 0✿ y f (y) y = y(t) 3π/2 0 y(t) = 3π/2, ❝♦♥st❛♥t − yց 0 y(t) = π/2, ❝♦♥st❛♥t π/2 + yր 0 y(t) = 0, ❝♦♥st❛♥t 0 + yր 0 y(t) = −π/2, ❝♦♥st❛♥t −π/2 − yց 0 y(t) = −3π/2, ❝♦♥st❛♥t −3π/2

→→→→ ցցցց →→→→ րրրր →→→→ րրրր →→→→ ցցցց →→→→

❚❤❡ r❡s✉❧ts ❛r❡ ❝♦♥✜r♠❡❞ ✇✐t❤ ❊✉❧❡r✬s ♠❡t❤♦❞✿

❚❤❡ ♣❧♦tt❡❞ ❊✉❧❡r s♦❧✉t✐♦♥s ❝r♦ss t❤❡ ♣❛t❤ ♦❢ y = 0 ✐♥❞✐❝❛t✐♥❣ ♥♦♥✲✉♥✐q✉❡♥❡ss✳ ❈♦♥s✐❞❡r✐♥❣ ♥♦♥✲ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ♦❢ f (y) ❛t y = 0✱ t❤✐s ✐s ❛ ♣♦ss✐❜✐❧✐t②✳ ❊❧s❡✇❤❡r❡✱ ❤♦✇❡✈❡r✱ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❞✐✛❡r❡♥t✐❛❧❜❧❡ ❛♥❞ t❤❡ ✉♥✐q✉❡♥❡ss ❤♦❧❞s✳ ❚❤❡ ❜❡❤❛✈✐♦r ✐s ❛s②♠♣t♦t✐❝ ❥✉st ❛s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✳ ❊①❛♠♣❧❡ ✶✳✶✶✳✾✿ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s

❚❤❡ ♥❡①t ♦♥❡ ✐s ❞❡♣❡♥❞❡♥t ♦♥ t✿ y′ =

1 . t−y

❚❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s ✉♥❞❡✜♥❡❞ ✇❤❡♥❡✈❡r t − y = 0✳ ❚❤❡r❡❢♦r❡✱ ♥♦ s♦❧✉t✐♦♥ ❝r♦ss❡s t❤❡ ❞✐❛❣♦♥❛❧ ❧✐♥❡ y = t✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ s♦❧✉t✐♦♥s ❛❜♦✈❡ ✐t ❛r❡ ❞❡❝r❡❛s✐♥❣ ❛♥❞ t❤❡ ♦♥❡s ❜❡❧♦✇ ✐t ❛r❡ ✐♥❝r❡❛s✐♥❣ ❛s ❢♦❧❧♦✇s✿

❈❤❡❝❦✿

✶✳✶✶✳

◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s

✾✸

❊①❛♠♣❧❡ ✶✳✶✶✳✶✵✿ tr✐❣ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s

❚❤❡ ♥❡①t ♦♥❡ ✐s ❞❡✜♥❡❞ ♦♥ t❤❡ ✇❤♦❧❡ ♣❧❛♥❡✿ y ′ = cos(t + y) .

❚❤❡ s✐❣♥ ✐s ❝❤❛♥❣✐♥❣ ✇❤❡♥❡✈❡r t + y = π/2 + kπ ❢♦r ❛❧❧ ✐♥t❡❣❡r k ✳ ❚❤✉s t❤❡ s♦❧✉t✐♦♥ ❝❤❛♥❣❡ t❤❡✐r ♠♦♥♦t♦♥✐❝✐t② ✇❤❡♥❡✈❡r t❤❡② r❡❛❝❤ ♦♥❡ ♦❢ t❤❡ ❧✐♥❡s y = −t + π/2 + kπ

♦❢ s❧♦♣❡ −1✳ ❖♥❡ ♦❢ t❤❡s❡ ❧✐♥❡s ❤♦✇❡✈❡r ✐s s♣❡❝✐❛❧ ❛♥❞ ✇❡ ✇♦✉❧❞♥✬t t❤✐♥❦ ♦❢ ✐t ✇✐t❤♦✉t ❞♦✐♥❣ ❊✉❧❡r✬s ♠❡t❤♦❞ ✜rst✿

■♥❞❡❡❞✱ y = π − t ✐s ❛ s♦❧✉t✐♦♥✦ ❚❤❡ r❡st ♦❢ t❤❡♠ ❛r❡ ✇❛✈❡s ♠♦✈✐♥❣ ❞✐❛❣♦♥❛❧❧②✳ ❊①❛♠♣❧❡ ✶✳✶✶✳✶✶✿ ❞✐s❝♦♥t✐♥✉♦✉s ❘❍❙

❚❤❡ ♥❡①t ♦♥❡ ✐s ❞✐s❝♦♥t✐♥✉♦✉s✿

y ′ = [t + y] .

❙✐♥❝❡ t❤❡ ❢✉♥❝t✐♦♥ ❝❤❛♥❣❡s ❛❜r✉♣t❧②✱ s♦ ❞♦❡s t❤❡ ❞❡r✐✈❛t✐✈❡ ✭t❤❡ s❧♦♣❡✮ ♦❢ ❡❛❝❤ s♦❧✉t✐♦♥✳ ❋♦r ❊✉❧❡r✬s ♠❡t❤♦❞ ✇❡ ✉s❡ t❤❡ ❋▲❖❖❘ ❢✉♥❝t✐♦♥✿

✶✳✶✶✳

◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s

✾✹

❚❤❡ ❝♦r♥❡rs ✐♥❞✐❝❛t❡ t❤❛t t❤❡s❡ ❛r❡ ✇❡❛❦ s♦❧✉t✐♦♥s✳

❙✉♣♣♦s❡ t❤❡ ❖❉❊ ✐s

t✐♠❡✲✐♥❞❡♣❡♥❞❡♥t✱

y ′ = f (y) .

❚❤❡♥ ✐t ✐s t❤♦✉❣❤t ♦❢ ❛ ❧✐q✉✐❞ ✢♦✇✿ ✐♥ ❛ ❝❛♥❛❧ ♦r ❛ ♣✐♣❡✳

❊s♣❡❝✐❛❧❧② ✐♥ t❤❡ ❧❛tt❡r ❝❛s❡✱ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s r❡❝♦❣♥✐③❡❞ ❛s ❛

♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ✈❡❝t♦r ✜❡❧❞✳

❲✐t❤ s✉❝❤ ❛♥ ❡q✉❛t✐♦♥✱ t❤❡ q✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ✐s ♠✉❝❤ s✐♠♣❧❡r✳ ■♥ ❢❛❝t✱ ♦♥❡ ♦❢ t❤❡ ❖❉❊s ❛❜♦✈❡✱

p cos y · |y|,

❡①❤✐❜✐ts ❛❧❧ ♣♦ss✐❜❧❡ ♣❛tt❡r♥s ♦❢

❧♦❝❛❧

f (y) =

❜❡❤❛✈✐♦r✳

❲❡ ❝♦♥❝❡♥tr❛t❡ ♦♥ ✇❤❛t ✐s ❣♦✐♥❣ ♦♥ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ ❛ ❣✐✈❡♥ ❧♦❝❛t✐♦♥

y = a✳

f (a) 6= 0✳ ❚❤❡♥✱ ❢r♦♠ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ f ✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t f (y) > 0 ♦r f (y) < 0 ❝♦♥t❛✐♥s a✳ ❚❤❡♥✱ t❤❡ s♦❧✉t✐♦♥s ❧♦❝❛t❡❞ ✇✐t❤✐♥ t❤❡ ❜❛♥❞ (−∞, +∞) × I ❛r❡ ❡✐t❤❡r

❚❤❡ ✜rst ♠❛✐♥ ♣♦ss✐❜✐❧✐t② ✐s ✐♥ s♦♠❡ ✐♥t❡r✈❛❧

I

t❤❛t

❛❧❧ ✐♥❝r❡❛s✐♥❣ ♦r ❛❧❧ ❞❡❝r❡❛s✐♥❣ r❡s♣❡❝t✐✈❡❧②✿

❚❤❡ ❜❡❤❛✈✐♦r ✐s ✏❣❡♥❡r✐❝✑✳ ▼♦r❡ ✐♥t❡r❡st✐♥❣ ❜❡❤❛✈✐♦rs ❛r❡ s❡❡♥ ❛r♦✉♥❞ ❛ ③❡r♦ ♦❢

◮ f (a) = 0 =⇒ y = a

f✿

✐s ❛ st❛t✐♦♥❛r② s♦❧✉t✐♦♥ ✭❛♥ ❡q✉✐❧✐❜r✐✉♠✮✳

❚❤❡♥ t❤❡ ♣❛tt❡r♥ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤❡ ♣♦✐♥t✱ ✐✳❡✳✱ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ ♦❢

f✱

I ✱ ❞❡♣❡♥❞s ♦♥ ✇❤❡t❤❡r t❤✐s ✐s ❛ ♠❛①✐♠✉♠

❛ ♠✐♥✐♠✉♠✱ ♦r ♥❡✐t❤❡r✳

❋✐rst✱ ✐❢

f

❝❤❛♥❣❡s ✐ts s✐❣♥ ❢r♦♠ ♣♦s✐t✐✈❡ t♦ ♥❡❣❛t✐✈❡✱ ✇❡ ❤❛✈❡ ❛

st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✱ ♦r ❛ ✏s✐♥❦✑✿

✶✳✶✶✳

◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ ❖❉❊s

✾✺

■♥❞❡❡❞✱ ✇❡ ❤❛✈❡ ❛❧❧ s♦❧✉t✐♦♥s ✇✐t❤✐♥ t❤❡ ❜❛♥❞

(−∞, +∞)×I ✱ ✉♥❞❡r t❤❡ ✉♥✐q✉❡♥❡ss ❝♦♥❞✐t✐♦♥✱ ❛s②♠♣t♦t✐❝❛❧❧②

y = a✿

❛♣♣r♦❛❝❤

y→a ❙❡❝♦♥❞✱ ✐❢

f

❛s

❝❤❛♥❣❡s ✐ts s✐❣♥ ❢r♦♠ ♥❡❣❛t✐✈❡ t♦ ♣♦s✐t✐✈❡✱ ✇❡ ❤❛✈❡ ❛♥

■t ♠❡❛♥s t❤❛t t❤❡r❡ ❛r❡ ♥♦ s♦❧✉t✐♦♥s ✇✐t❤✐♥ t❤❡ ❜❛♥❞ ❛s②♠♣t♦t✐❝❛❧❧② ❛♣♣r♦❛❝❤

y=a

❛s

x → +∞✳

f

❞♦❡s ♥♦t ❝❤❛♥❣❡ ✐ts s✐❣♥ ❛t

y = a✱

✉♥st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✱ ♦r ❛ ✏s♦✉r❝❡✑✿

(−∞, +∞) × I ✱

✉♥❞❡r t❤❡ ✉♥✐q✉❡♥❡ss ❝♦♥❞✐t✐♦♥✱ t❤❛t

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✇❡ ❤❛✈❡

y→a ❚❤✐r❞✱ ✐❢

x → +∞ .

❛s

x → −∞ .

✇❡ ❤❛✈❡ ❛♥

s❡♠✐✲st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✱ ♦r ❛ ✏♣❛ss✑✿

■t ♠❡❛♥s t❤❛t t❤❡ s♦❧✉t✐♦♥s ✇✐t❤✐♥ ♦♥❡ ♦❢ t❤❡ t✇♦ s✐❞❡s ♦❢ t❤❡ ❧✐♥❡

y=a

✇✐t❤✐♥ t❤❡ ❜❛♥❞

(−∞, +∞) × I ✱

✉♥❞❡r t❤❡ ✉♥✐q✉❡♥❡ss ❝♦♥❞✐t✐♦♥✱ ❛s②♠♣t♦t✐❝❛❧❧② ❛♣♣r♦❛❝❤ t❤✐s ❧✐♥❡ ❛♥❞ t❤❡ ♦♥❡s ♦♥ t❤❡ ♦t❤❡r s✐❞❡ ❞♦ ♥♦t✳ ❚❤❡ ❧❛st ♣♦ss✐❜✐❧✐t② ✐s

f (y) = 0

❞❡❣❡♥❡r❛t❡ ❡q✉✐❧✐❜r✐✉♠✳ ❚❤✐s

❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❡q✉✐❧✐❜r✐❛

♦♥ t❤❡ ✇❤♦❧❡

I✳

❚❤❡♥ ❛❧❧ t❤❡ s♦❧✉t✐♦♥s ✐♥ t❤❡ ❜❛♥❞ ❛r❡ st❛t✐♦♥❛r②✳ ❚❤✐s ✐s ❛

✐s ❡❛s② t♦ ✉♥❞❡rst❛♥❞ ✐♥ t❡r♠s ♦❢ t❤❡ ✢♦✇ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❖❉❊✿



s✐♥❦✿ ✢♦✇ ✐♥ ♦♥❧② ✭st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✮❀



s♦✉r❝❡✿ ✢♦✇ ♦✉t ♦♥❧② ✭✉♥st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✮❀



♣❛ss✿ ✢♦✇✱ st♦♣✱ ✢♦✇ ✭s❡♠✐✲st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✮✳

✶✳✶✷✳

▲✐♥❡❛r✐③❛t✐♦♥ ♦❢ ❖❉❊s

✾✻ ❲❛r♥✐♥❣✦ ■♥ s♣✐t❡ ♦❢ t❤❡ ♥❛♠❡s✱ ❛ ♣❛rt✐❝❧❡ ♣❧❛❝❡❞ ❛t ❛ s♦✉r❝❡ ✇✐❧❧ ♥❡✈❡r ❧❡❛✈❡ ✐t ❛♥❞ ❛ ♣❛rt✐❝❧❡ ♣❧❛❝❡❞ ♥❡❛r ❛ s✐♥❦ ✇✐❧❧ ♥❡✈❡r r❡❛❝❤ ✐t✱ ✉♥❞❡r t❤❡ ✉♥✐q✉❡♥❡ss ❝♦♥❞✐t✐♦♥✳

❚❤❡r❡ ❛r❡ ♦♥❧② t❤❡s❡ t❤r❡❡ ♠❛✐♥ ♣♦ss✐❜✐❧✐t✐❡s ✐♥ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✳ ❲❡ ✇✐❧❧ s❡❡ ✐♥ ❈❤❛♣t❡r ✸ ❛ ✇✐❞❡ ✈❛r✐❡t② ♦❢ ❜❡❤❛✈✐♦rs ❛r♦✉♥❞ ❛♥ ❡q✉✐❧✐❜r✐✉♠ ✇❤❡♥ t❤❡ s♣❛❝❡ ♦❢ ❧♦❝❛t✐♦♥ ✐s 2✲❞✐♠❡♥s✐♦♥❛❧✱ ❛ ♣❧❛♥❡✳

✶✳✶✷✳ ▲✐♥❡❛r✐③❛t✐♦♥ ♦❢ ❖❉❊s

❊✉❧❡r✬s ♠❡t❤♦❞ ❛♣♣r♦①✐♠❛t❡s s♦❧✉t✐♦♥s ♦❢ ❛♥ ❖❉❊ ❜② ❧✐♥❡❛r✐③✐♥❣ t❤❡♠✱ ♦♥❡ ❧♦❝❛t✐♦♥ ❛t ❛ t✐♠❡✳ ❲❡ ❝❛♥ ❛❧s♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❖❉❊s ❜② ❧✐♥❡❛r✐③✐♥❣✱ ❧♦❝❛❧❧②✱ ✐ts r✐❣❤t✲❤❛♥❞ s✐❞❡✳ ❚❤❡ ♠❛✐♥ ✐❞❡❛ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❆s ✇❡ ❛r❡ ♦❢t❡♥ ✉♥❛❜❧❡ t♦ s♦❧✈❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✇❡ ❡♥❝♦✉♥t❡r✱ ❧❡t✬s tr② t♦ r❡♣❧❛❝❡ ✐t ✇✐t❤ ❛ s✐♠♣❧❡r ♦♥❡ t❤❛t ❝❛♥ ❜❡ s♦❧✈❡❞✳ ❍♦✇ ❛❝❝✉r❛t❡ ❞♦ ✇❡ ♥❡❡❞ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ t♦ ❜❡❄ ❆t t❤❡ ✈❡r② ❧❡❛st✱ t❤❡ ♥❡✇ ❖❉❊ s❤♦✉❧❞ ❤❛✈❡ t❤❡ s❛♠❡ q✉❛❧✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ✕ ✐♥❝r❡❛s✐♥❣✱ ❞❡❝r❡❛s✐♥❣✱ ❛♥❞ st❛t✐♦♥❛r② s♦❧✉t✐♦♥s ✕ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤❡ ❝❤♦s❡♥ ♣♦✐♥t✳ ❲❡ ✇✐❧❧ ❢♦❝✉s ♦♥ t❤❡ t✐♠❡✲✐♥❞❡♣❡♥❞❡♥t ❖❉❊ ✇✐t❤ ❛ ❝♦♥t✐♥✉♦✉s r✐❣❤t ❤❛♥❞✲s✐❞❡✱

y ′ = f (y) , ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ ❛ ❝❤♦s❡♥ ♣♦✐♥t y = a✳ ❲❡ ✇✐❧❧ r❡❧② ♦♥ t❤❡ r❡s✉❧ts st❛t❡❞ ♣r❡✈✐♦✉s❧②✳ ❇❡❢♦r❡ ✇❡ ❣❡t t♦ t❤❡ ❧✐♥❡❛r✱ ❧❡t✬s tr② t❤❡

❜❡st ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥✳

❲❤❛t ✐s t❤❡ ❜❡st ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ❛t x = a ♦❢ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ y = f (x)❄ ■t✬s y = f (a)✦ ❊①❛♠♣❧❡ ✶✳✶✷✳✶✿ s✐♠♣❧❡st ❛♣♣r♦①✐♠❛t✐♦♥

❙♦❧✈❡ ❛♣♣r♦①✐♠❛t❡❧②✿

y ′ = y 2 ❛r♦✉♥❞ y = 1 .

❲❡ ❡✈❛❧✉❛t❡ t❤❡ ❢✉♥❝t✐♦♥ ❛t t❤❛t ♣♦✐♥t✿

y 2

= 1. y=1

❚❤❡♥✱ ✇❡ r❡♣❧❛❝❡ ♦✉r ❢✉♥❝t✐♦♥ ✇✐t❤ t❤✐s ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ g(y) = 1✳ ❲❡ ❤❛✈❡ ❛ ♥❡✇✱ ❛♥❞ ✈❡r② s✐♠♣❧❡✱ ❖❉❊✿ y′ = 1 . ■ts s♦❧✉t✐♦♥ ✐s✿

y =x+C. ■t✬s ❛ ♣♦♦r s✉❜st✐t✉t❡✳ ❍♦✇❡✈❡r✱ t❤❡ q✉❛❧✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ✐s t❤❡ s❛♠❡✦ ❍♦✇❡✈❡r✱ ❧❡t✬s s♦❧✈❡ t❤❡ s❛♠❡ ❖❉❊ ❛r♦✉♥❞ ❛♥♦t❤❡r ♣♦✐♥t✿

y ′ = y 2 ❛r♦✉♥❞ y = 0 . ❲❡ ❡✈❛❧✉❛t❡ t❤❡ ❢✉♥❝t✐♦♥ ❛t t❤❛t ♣♦✐♥t✿

y 2

= 0. y=0

❚❤❡♥✱ ✇❡ r❡♣❧❛❝❡ ♦✉r ❢✉♥❝t✐♦♥ ✇✐t❤ t❤✐s ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ g(y) = 0✳ ❲❡ ❤❛✈❡ ❛ ♥❡✇✱ ❛♥❞ ✈❡r② s✐♠♣❧❡✱ ❖❉❊✿ y′ = 0 .

✶✳✶✷✳

▲✐♥❡❛r✐③❛t✐♦♥ ♦❢ ❖❉❊s

✾✼

■ts s♦❧✉t✐♦♥ ✐s ❝♦♥st❛♥t✿ y =C.

■t✬s ❛♥ ✉♥❛❝❝❡♣t❛❜❧② ❜❛❞ s✉❜st✐t✉t❡ ❜❡❝❛✉s❡ t❤❡ q✉❛❧✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ✐s t❤❡ ❞✐✛❡r❡♥t✦ ■♥ ❣❡♥❡r❛❧✱ ❢♦r ❛ ❝❤♦s❡♥ ♣♦✐♥t y = a✱ ✇❡ r❡♣❧❛❝❡ z = f (y) ✇✐t❤ z = f (a)✿ ◮ ■♥st❡❛❞ ♦❢ y ′ = f (y)✱ ✇❡ s♦❧✈❡ y ′ = f (a)✳

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✈❛❧✉❡ f (a) ♦❢ f ❛t a ✐s ✉s❡❞ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ♥❡✇ ❖❉❊✿ y ′ = f (a) ,

❢♦r ❛❧❧ y ✐♥ s♦♠❡ ♦♣❡♥ ✐♥t❡r✈❛❧ I t❤❛t ❝♦♥t❛✐♥s a✳ ■ts s♦❧✉t✐♦♥s ❛r❡ t❤❡s❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ♦❢ t✿ y = f (a)(t + C) ,

❢♦r ❡❛❝❤ r❡❛❧ C ✳ ❉♦❡s t❤❡✐r ❜❡❤❛✈✐♦r ♠❛t❝❤ t❤❡ ♦r✐❣✐♥❛❧❄ ❚❤❡r❡ ❛r❡ t✇♦ ♠❛✐♥ ❝❛s❡s✿ ✶✳ f (a) 6= 0

✷✳ f (a) = 0

❚❤✐s ✐s ✇❤❛t ✇❡ ❝♦♥❝❧✉❞❡ ❛❜♦✉t ❛❧❧ s♦❧✉t✐♦♥s ♦❢ t❤❡ s✐♠♣❧✐✜❡❞ ❖❉❊✳ ■♥ t❤❡ ❢♦r♠❡r ❝❛s❡✱ ✇❡ ❤❛✈❡✿ f (a) > 0 =⇒ y ր ❛♥❞ f (a) < 0 =⇒ y ց ,

❢♦r ❛❧❧ s♦❧✉t✐♦♥s ✐♥ t❤❡ ❜❛♥❞ (−∞, +∞) × I ✳ ❚❤❡② ❛r❡ ❝✉r✈❡❞ ♦♥ t❤❡ ❧❡❢t ❛♥❞ str❛✐❣❤t ♦♥ t❤❡ r✐❣❤t✿

■♥❞❡❡❞✱ t♦ ❝♦♠♣❛r❡ t♦ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❖❉❊✱ ✇❡ ❝❛♥ ❥✉st ❛♣♣❧② t❤❡ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠ ❢r♦♠ ❝❛❧❝✉❧✉s✳ ■t✬s ❛ ♠❛t❝❤✦ ❇❡❧♦✇ ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤✐s ❝❛s❡✳ ❚❤❡♦r❡♠ ✶✳✶✷✳✷✿ ❈♦♥st❛♥t ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❖❉❊s

❙✉♣♣♦s❡ z = f (y) ✐s ❝♦♥t✐♥✉♦✉s ❛t y = a✳ ❙✉♣♣♦s❡ f (a) 6= 0 .

❚❤❡♥ t❤❡ q✉❛❧✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❖❉❊ y ′ = f (y) ♠❛t❝❤ t❤♦s❡ ♦❢ ✐ts ❝♦♥st❛♥t s✉❜st✐t✉t❡ y ′ = f (a)✱ ❛s ❢♦❧❧♦✇s✿ t❤❡r❡ ✐s s✉❝❤ ❛♥ ε > 0 t❤❛t ❡❛❝❤ s♦❧✉t✐♦♥ y ♦❢ y ′ = f (y) t❤❡ ❣r❛♣❤ ♦❢ ✇❤✐❝❤ ❧✐❡s ✇✐t❤✐♥ t❤❡ ❜❛♥❞ (−∞, +∞) × (a − ε, a + ε) s❛t✐s✜❡s✿ f (a) > 0 =⇒ y ✐s ✐♥❝r❡❛s✐♥❣. f (a) < 0 =⇒ y ✐s ❞❡❝r❡❛s✐♥❣. Pr♦♦❢✳

❚❤❡ ❝♦♥t✐♥✉✐t② ♦❢ f ✐♠♣❧✐❡s t❤❛t t❤❡r❡ ✐s ε > 0 s♠❛❧❧ ❡♥♦✉❣❤ t❤❛t ❢♦r ❡❛❝❤ y ✇✐t❤✐♥ (a − ε, a + ε)✱ ✇❡

✶✳✶✷✳

▲✐♥❡❛r✐③❛t✐♦♥ ♦❢ ❖❉❊s

✾✽

❤❛✈❡

f (a) > 0 =⇒ f (y) > 0 f (a) < 0 =⇒ f (y) < 0

■♥ t❤❡ ❧❛tt❡r ❝❛s❡✱ ✇❡ ❤❛✈❡✿

f (a) = 0 =⇒ y

.

✐s ❝♦♥st❛♥t

❚❤❡♥ ❛❧❧ t❤❡ s♦❧✉t✐♦♥s ✐♥ t❤❡ ❜❛♥❞ ❛r❡ st❛t✐♦♥❛r②✳ ❇✉t t❤❡ ♦r✐❣✐♥❛❧ ❖❉❊ ♠✐❣❤t ❤❛✈❡ ❛ s❡♠✐✲st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✱ ❛ ♠✐s♠❛t❝❤✦ ❈♦♥❝❧✉s✐♦♥✿



❚❤❡ ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢

❚♦ ✐♠♣r♦✈❡ ♦✉r ❝❤❛♥❝❡s✱ ✇❡ tr② ✐s t❤❡

y = f (y)

❛t

x=a

❢❛✐❧s ✇❤❡♥

❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥

♦❢

f (a) = 0✳

f✳

❲❡ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ❝❛s❡✱ ♦❢ ❝♦✉rs❡✱ t❤❛t ❤❛s ♥♦t ❜❡❡♥✱ q✉❛❧✐t❛t✐✈❡❧②✱ s♦❧✈❡❞ ❜② t❤❡ ❜❡st ❝♦♥st❛♥t ❛♣♣r♦①✲ ✐♠❛t✐♦♥✱ ✐✳❡✳✱

f (a) = 0 . ❊①❛♠♣❧❡ ✶✳✶✷✳✸✿ ❛r♦✉♥❞ st❛t✐♦♥❛r② ♣♦✐♥t ▲❡t✬s s♦❧✈❡ ❛♣♣r♦①✐♠❛t❡❧②✿

y′ = y2 + y

❛r♦✉♥❞

y = 1.

❲❡ ❡✈❛❧✉❛t❡ t❤❡ ❢✉♥❝t✐♦♥ ❛t t❤❛t ♣♦✐♥t✿

y + y 2

= 2. y=1

g(y) = 2✳

❚❤❡♥✱ ✇❡ r❡♣❧❛❝❡ ♦✉r ❢✉♥❝t✐♦♥ ✇✐t❤ t❤✐s ❝♦♥st❛♥t ❢✉♥❝t✐♦♥

❲❡ ❤❛✈❡ ❛ ♥❡✇✱ ❛♥❞ ✈❡r② s✐♠♣❧❡✱

❖❉❊✿

y′ = 2 . ■ts s♦❧✉t✐♦♥ ✐s✿

y = 2x + C . ❚❤❡ q✉❛❧✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ✐s t❤❡ s❛♠❡ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠✳ ❚❤❡ ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❣♦♦❞ ❡♥♦✉❣❤✦

▲❡t✬s s♦❧✈❡ ❛♣♣r♦①✐♠❛t❡❧② ❡❧s❡✇❤❡r❡✿

y′ = y2 + y

❛r♦✉♥❞

y = 0.

❲❡ ❡✈❛❧✉❛t❡ t❤❡ ❢✉♥❝t✐♦♥ ❛t t❤❛t ♣♦✐♥t✿

y + y 2

= 0. y=0

❚❤❡♥✱ ✇❡ r❡♣❧❛❝❡ ♦✉r ❢✉♥❝t✐♦♥ ✇✐t❤ t❤✐s ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ❖❉❊✿

y′ = 0 . ■ts s♦❧✉t✐♦♥ ✐s✿

y =C.

g(y) = 0✳

❲❡ ❤❛✈❡ ❛ ♥❡✇✱ ❛♥❞ ✈❡r② s✐♠♣❧❡✱

✶✳✶✷✳

▲✐♥❡❛r✐③❛t✐♦♥ ♦❢ ❖❉❊s

✾✾

❚❤❡ q✉❛❧✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ✐s ♥♦t t❤❡ s❛♠❡✳ ❚❤❡ ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❣♦♦❞ ❡♥♦✉❣❤✦

◆♦✇ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥✳ ❲❡ ❡✈❛❧✉❛t❡ t❤❡ ❞❡r✐✈❛t✐✈❡✿

(y + y) 2



y=0

= 2y + 1

❚❤❡♥✱ ✇❡ r❡♣❧❛❝❡ ♦✉r ❢✉♥❝t✐♦♥ ✇✐t❤ t❤✐s ❧✐♥❡❛r ❢✉♥❝t✐♦♥

= 1. y=0

g(y) = y ✳

❲❡ ❤❛✈❡ ❛ ♥❡✇✱ ❛♥❞ ✈❡r② s✐♠♣❧❡✱

❖❉❊✿

y′ = y . ■ts s♦❧✉t✐♦♥ ✐s✿

y = Cex . ❚❤❡ q✉❛❧✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ✐s t❤❡ s❛♠❡✳ ❚❤❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❣♦♦❞ ❡♥♦✉❣❤✦

◆♦✇ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡✳ ❋♦r s✐♠♣❧✐❝✐t②✱ t❤❡ ♣r♦❝❡ss ♦❢ ❧✐♥❡❛r✐③❛t✐♦♥ st❛rts ✇✐t❤ ♠♦✈✐♥❣ t❤❡ ♣♦✐♥t ♦❢ ✐♥t❡r❡st✱ ❝✉rr❡♥t❧②

y = a✱

t♦

0✳

❙✉❝❤ ❛ s✉❜st✐t✉t✐♦♥ ✐s ❞❡♠♦♥str❛t❡❞ ❡❛r❧✐❡r ✐♥ t❤❡ ❝❤❛♣t❡r✳ ❚❤✐s ✐s t❤❡ ♥❡✇ ❖❉❊✿

y ′ = f (y)

✇✐t❤

f (0) = 0 .

❚❤❡ ❧✐♥❡❛r✐③❡❞ ❖❉❊ ✐s

y ′ = f ′ (0)y , ❢♦r ❛❧❧

y

✐♥ s♦♠❡ ♦♣❡♥ ✐♥t❡r✈❛❧

I

t❤❛t ❝♦♥t❛✐♥s

0✳

■ts s♦❧✉t✐♦♥s ❛r❡ t❤❡s❡

❡①♣♦♥❡♥t✐❛❧

❢✉♥❝t✐♦♥s ♦❢

t✿



y = Cef (0)t , ❢♦r ❡❛❝❤ r❡❛❧

C✳

❉♦❡s t❤❡✐r ❜❡❤❛✈✐♦r ♠❛t❝❤ t❤❡ ♦r✐❣✐♥❛❧❄ ❚❤❡r❡ ❛r❡ t✇♦ ♠❛✐♥ ❝❛s❡s✿ ✶✳ ✷✳

f ′ (0) 6= 0 f ′ (0) = 0

❚❤✐s ✐s ✇❤❛t ✇❡ ❝♦♥❝❧✉❞❡ ❛❜♦✉t

❛❧❧

s♦❧✉t✐♦♥s ♦❢ t❤❡ ❧✐♥❡❛r✐③❡❞ ❖❉❊✳ ❋✐rst ✇❡ ❤❛✈❡✿

f ′ (0) < 0 =⇒ f ց =⇒ f (y) > 0 ❢♦r ❛❧❧ s♦❧✉t✐♦♥s ✐♥ t❤❡ ❜❛♥❞ ❛♥ ✉♥st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✿

(−∞, +∞) × I ✳

❚❤❡♥✱

f

❢♦r

y 0,

❝❤❛♥❣❡s ✐ts s✐❣♥ ❢r♦♠ ♣♦s✐t✐✈❡ t♦ ♥❡❣❛t✐✈❡ ❛♥❞ ✇❡ ❤❛✈❡

❯♥❧✐❦❡ t❤❡ ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥✱ t❤❡ ❧✐♥❡❛r✐③❛t✐♦♥ ❤❛s ♣r♦❞✉❝❡❞ ❛ ♠❛t❝❤✦ ❙❡❝♦♥❞✱ ✇❡ ❤❛✈❡ ✇✐t❤✐♥ t❤❡ ❜❛♥❞✿

f ′ (0) > 0 =⇒ f ր =⇒ f (y) < 0 ❚❤❡♥✱

f

❢♦r

y 0

❢♦r

y > 0.

❝❤❛♥❣❡s ✐ts s✐❣♥ ❢r♦♠ ♥❡❣❛t✐✈❡ t♦ ♣♦s✐t✐✈❡ ❛♥❞ ✇❡ ❤❛✈❡ ❛ st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✳ ❖♥❝❡ ❛❣❛✐♥✱ ❛ ♠❛t❝❤✦

❇❡❧♦✇ ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤✐s ❝❛s❡✳

✶✳✶✸✳

✶✵✵

▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥

❚❤❡♦r❡♠ ✶✳✶✷✳✹✿ ▲✐♥❡❛r ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❖❉❊s

❙✉♣♣♦s❡ z = f (y) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t y = 0✳ ❙✉♣♣♦s❡ f (0) = 0 ❛♥❞ f ′ (0) 6= 0 .

❚❤❡♥ t❤❡ q✉❛❧✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❖❉❊ y ′ = f (y) ♠❛t❝❤ t❤♦s❡ ♦❢ ✐ts ❧✐♥❡❛r s✉❜st✐t✉t❡ y ′ = f ′ (0)y ✱ ❛s ❢♦❧❧♦✇s✿ t❤❡r❡ ✐s s✉❝❤ ❛♥ ε > 0 t❤❛t ❡❛❝❤ s♦❧✉t✐♦♥ y ♦❢ y ′ = f (y) t❤❡ ❣r❛♣❤ ♦❢ ✇❤✐❝❤ ❧✐❡s ✇✐t❤✐♥ t❤❡ ❜❛♥❞ (−∞, +∞)×(ε, ε) s❛t✐s✜❡s✿ f ′ (0) > 0 =⇒ y ✐s ❞❡❝r❡❛s✐♥❣ ✇❤❡♥ y < 0 ❛♥❞ y ✐s ✐♥❝r❡❛s✐♥❣ ✇❤❡♥ y > 0 . f ′ (0) < 0 =⇒ y ✐s ✐♥❝r❡❛s✐♥❣ ✇❤❡♥ y < 0 ❛♥❞ y ✐s ❞❡❝r❡❛s✐♥❣ ✇❤❡♥ y > 0 . ❊①❡r❝✐s❡ ✶✳✶✷✳✺

▲✐♥❡❛r✐③❡ t❤❡ ❖❉❊ y ′ = yey ❛t y = 0✳ ❲❤❛t ❛❜♦✉t t❤❡ ❝❛s❡ f ′ (0) = 0❄ ❲❡ ❛r❡ ✐♥ ❛ s✐♠✐❧❛r ♣❧❛❝❡ t♦ t❤❡ ♦♥❡ ❢♦r t❤❡ ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥✳ ❍❡r❡✱ t❤❡ ❧✐♥❡❛r✐③❛t✐♦♥ ❣✐✈❡s ✉s t❤❡ ❖❉❊ y ′ = 0 ✇✐t❤ st❛t✐♦♥❛r② s♦❧✉t✐♦♥s ♦♥❧②✳ ■♥ t❤❡ ♠❡❛♥t✐♠❡✱ f ♠✐❣❤t ❤❛✈❡ ❛ s❡♠✐✲st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✿

❆ ♠✐s♠❛t❝❤✦ ❈♦♥❝❧✉s✐♦♥✿ ◮ ❚❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ y = f (y) ❛t x = 0 ❢❛✐❧s ✇❤❡♥ f (0) = 0 ❛♥❞ f ′ (0) = 0✳

❚❤❡ ❛♥s✇❡r ✐s t❤❡ ❜❡st q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥✱ ✐✳❡✳✱ t❤❡ s❡❝♦♥❞ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧ T2 ♦❢ f ❛r♦✉♥❞ y = 0✱ ♣r♦✈✐❞❡❞ f ✐s t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ♠❛② st✐❧❧ ❜❡ ❡①❝❡♣t✐♦♥s t❤❛t ✇✐❧❧ ❝❛❧❧ ❢♦r ✉s✐♥❣ t❤❡ ❝✉❜✐❝ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧ T3 ♦❢ f ✳ ❆♥❞ s♦ ♦♥✳ ❲❡ ✇✐❧❧ ♥❡❡❞ ❛❧❧ t❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s✱ ✐✳❡✳✱ t❤❡ ❚❛②❧♦r s❡r✐❡s✳ ❚❤❡ ✐❞❡❛ ✐s ❞❡✈❡❧♦♣❡❞ ✐♥ ❈❤❛♣t❡r ✷✳

✶✳✶✸✳ ▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥

❊①❛♠♣❧❡ ✶✳✶✸✳✶✿ ♠♦✈✐♥❣ ❜❛❧❧

❲❡ ❦♥♦✇ t❤❛t ❛ ❜❛❧❧

r♦❧❧✐♥❣

♦♥ ❛ ❤♦r✐③♦♥t❛❧ ♣❧❛♥❡ ✇✐❧❧ ❤❛✈❡ ❛ ❝♦♥st❛♥t ✈❡❧♦❝✐t②✿

✶✳✶✸✳

✶✵✶

▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥

❲❤❛t ✐❢ t❤❡ ❜❛❧❧ ✐s ♥♦✇ t❤r♦✇♥ ✉♣ ✐♥ t❤❡ ❛✐r ❄ ❚❤❡ ❞②♥❛♠✐❝s ✐s ✈❡r② ❞✐✛❡r❡♥t✳ ■♥ t❤❡ ❢♦r♠❡r✱ ❛s t❤❡r❡ ✐s ♥♦ ❢♦r❝❡ ❝❤❛♥❣✐♥❣ t❤❡ ✈❡❧♦❝✐t②✱ t❤❡ ❧❛tt❡r r❡♠❛✐♥s ❝♦♥st❛♥t✳ ■♥ t❤❡ ❧❛tt❡r✱ t❤❡ ✈❡❧♦❝✐t② ✐s ❝♦♥st❛♥t❧② ❝❤❛♥❣❡❞ ❜② t❤❡ ❣r❛✈✐t②✳ ■♠❛❣✐♥❡ t❤❛t ✇❡ ❤❛✈❡ t❤✐s ❡①♣❡r✐♠❡♥t❛❧ ❞❛t❛ ♦❢ t❤❡ ❤❡✐❣❤ts ♦❢ ❛ ♣✐♥❣✲♣♦♥❣ ❜❛❧❧ ❢❛❧❧✐♥❣ ❞♦✇♥ r❡❝♦r❞❡❞ ❛❜♦✉t ❡✈❡r② .1 s❡❝♦♥❞ ♠❡❛s✉r❡❞ ✐♥ ✐♥❝❤❡s✿

❲❡ ✉s❡ ❛ s♣r❡❛❞s❤❡❡t t♦ ♣❧♦t t❤❡ ✐✳❡✳✱ t❤❡ ✈❡❧♦❝✐t②✱ vn ✭❣r❡❡♥✮✿

❧♦❝❛t✐♦♥

s❡q✉❡♥❝❡✱ pn ✭r❡❞✮✳ ❲❡ t❤❡♥ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ pn ✱

■t ❧♦♦❦s ❧✐❦❡ ❛ str❛✐❣❤t ❧✐♥❡✳ ❇✉t t❤✐s t✐♠❡✱ ✇❡ t❛❦❡ ♦♥❡ ♠♦r❡ st❡♣✿ ❲❡ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ✈❡❧♦❝✐t② s❡q✉❡♥❝❡✳ ■t ✐s t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✱ an ✭❜❧✉❡✮✳ ■t ❛♣♣❡❛rs ❝♦♥st❛♥t✦ ❚❤❡r❡ ♠✐❣❤t ❜❡ ❛ ❧❛✇ ♦❢ ♥❛t✉r❡ ❤❡r❡✳ ▲❡t✬s ❛❝❝❡♣t t❤❡ ♣r❡♠✐s❡ ✇❡✬✈❡ ♣✉t ❢♦r✇❛r❞✿





❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ ❢r❡❡ ❢❛❧❧ ✐s ❝♦♥st❛♥t

❚❤❡♥ ✇❡ ❝❛♥ tr② t♦ ♣r❡❞✐❝t t❤❡ ❜❡❤❛✈✐♦r ♦❢ ❛♥ ♦❜❥❡❝t t❤r♦✇♥ ✐♥ t❤❡ ❛✐r ✕ ❢r♦♠ ❛♥② ✐♥✐t✐❛❧ ❤❡✐❣❤t ❛♥❞ ✇✐t❤ ❛♥② ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②✳ ❚❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ♦✉r ❝♦♠♣✉t❛t✐♦♥ ✐s ♦♣♣♦s✐t❡ t♦ t❤❛t ♦❢ t❤❡ ❧❛st ❡①❛♠♣❧❡✿

✶✳✶✸✳

▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ◮

✶✵✷

❲❡ ✉s❡ ♦✉r ❦♥♦✇❧❡❞❣❡ ♦❢ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ t♦ ❞❡r✐✈❡ t❤❡ ✈❡❧♦❝✐t②✱ ❛♥❞ t❤❡♥ ❞❡r✐✈❡ t❤❡ ♣♦s✐t✐♦♥

♦❢ t❤❡ ♦❜❥❡❝t ✐♥ t✐♠❡✳ ❲❤✐❧❡ ✇❡ ✉s❡❞

❞✐✛❡r❡♥❝❡s

✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ ✇❡ ✉s❡

s✉♠s

✭❈❤❛♣t❡r ✶P❈✲✶✮ ♥♦✇✳

❲❡ ♣❧♦t t❤❡s❡ ♣♦s✐t✐♦♥s ❛❣❛✐♥st t✐♠❡✿

❚❤❡ ❣r❛♣❤ ✏❧♦♦❦s ❧✐❦❡✑ ❛ ♣❛r❛❜♦❧❛✿

❲❡ ✉s❡ t❤❡s❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❢♦r♠✉❧❛s t♦ ✜♥❞ t❤❡ ✈❡❧♦❝✐t② ❢r♦♠ t❤❡ ♣♦s✐t✐♦♥ ❛♥❞ t❤❡♥ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❢r♦♠ t❤❡ ✈❡❧♦❝✐t②✿

vn = ✇❤❡r❡

h

∆p pn+1 − pn = ∆t h

❛♥❞

an =

∆v vn+1 − vn = , ∆t h

✐s t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t✐♠❡✳ ❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ✈❡❧♦❝✐t② ♦♥ t❤❡ ♣♦s✐t✐♦♥ ❛♥❞ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦♥

t❤❡ ✈❡❧♦❝✐t② ✐s✱ ♦❢ ❝♦✉rs❡✱ ✐❞❡♥t✐❝❛❧✳ ❊①❛♠♣❧❡ ✶✳✶✸✳✷✿ s♣r❡❛❞s❤❡❡t

❚❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ✐s ✉s❡❞ ❛❣❛✐♥✿

❂✭❘❈❬✲✶❪✲❘❬✲✶❪❈❬✲✶❪✮✴❘✷❈✶ ❚❤❡s❡ ❛r❡ t❤❡ r❡s✉❧ts✿

✶✳✶✸✳

✶✵✸

▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥

◆♦✇ ✐♥ r❡✈❡rs❡✦ ❋♦r s✐♠✉❧❛t✐♦♥✱ t❤❡ ❞❡r✐✈❛t✐♦♥ ❣♦❡s ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✿ • t❤❡ ✈❡❧♦❝✐t② ❢r♦♠ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❛♥❞ t❤❡♥ • t❤❡ ❧♦❝❛t✐♦♥ ❢r♦♠ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✳

❚❤❡ ❢♦r♠✉❧❛s ❛r❡ s♦❧✈❡❞ ❢♦r pn+1 ❛♥❞ vn+1 r❡s♣❡❝t✐✈❡❧②✿ pn+1 − pn =⇒ vn+1 = vn + han h vn+1 − vn an = =⇒ pn+1 = pn + hvn h vn =

❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ✈❡❧♦❝✐t② ♦♥ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❛♥❞ t❤❡ ♣♦s✐t✐♦♥ ♦♥ t❤❡ ✈❡❧♦❝✐t② ✐s✱ ♦❢ ❝♦✉rs❡✱ ✐❞❡♥t✐❝❛❧✳ ❙✉♠♠❛r②✿ ♣♦s✐t✐♦♥ ✈❡❧♦❝✐t② ❛❝❝❡❧❡r❛t✐♦♥

✈❡rt✐❝❛❧ pn ❣✐✈❡♥

pn+1 − pn h vn+1 − vn an = h vn =

=⇒

❛❝❝❡❧❡r❛t✐♦♥ ✈❡❧♦❝✐t② ♣♦s✐t✐♦♥

✈❡rt✐❝❛❧ an ❣✐✈❡♥ vn+1 = vn + han pn+1 = pn + hvn

❊①❛♠♣❧❡ ✶✳✶✸✳✸✿ ♠♦✈✐♥❣ ❜❛❧❧

▲❡t✬s ❝♦♥s✐❞❡r ❛ s♣❡❝✐✜❝ ♣r♦❜❧❡♠✳ ◮ P❘❖❇▲❊▼✿ ❋r♦♠ ❛ 100 ❢❡❡t ❜✉✐❧❞✐♥❣✱ ❛ ❜❛❧❧ ✐s t❤r♦✇♥ ✉♣ ❛t 50 ❢❡❡t ♣❡r s❡❝♦♥❞ s♦ t❤❛t ✐t ❢❛❧❧s ♦♥ t❤❡ ❣r♦✉♥❞✳ ❍♦✇ ❤✐❣❤ ✇✐❧❧ t❤❡ ❜❛❧❧ ❣♦❄ ❲❡ ✉s❡ t❤❡ s❛♠❡ s♣r❡❛❞s❤❡❡t ❢♦r♠✉❧❛ ❢♦r t❤❡ ✈❡❧♦❝✐t② ❛♥❞ ♣♦s✐t✐♦♥✿ ❂❘❬✲✶❪❈✰❘❬✲✶❪❈❬✲✶❪✯❘✷❈✶

◆♦✇ t❤❡ s♣❡❝✐✜❝ ❝❛s❡ ♦❢ ❢r❡❡ ❢❛❧❧✱ t❤❡r❡ ✐s ❥✉st ♦♥❡ ❢♦r❝❡✱ t❤❡ ❣r❛✈✐t②✱ ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❦♥♦✇♥ t♦ ❜❡ a = −g ✱ ✇❤❡r❡ g ✐s t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❝♦♥st❛♥t✿ g = 32 ❢t/s❡❝2 .

◆❡①t✱ ✇❡ ❛❝q✉✐r❡ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✿ • ❚❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ ✐s ❣✐✈❡♥ ❜②✿ p0 = 100✳ • ❚❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t② ✐s ❣✐✈❡♥ ❜②✿ v0 = 50✳ ❲❡ ✉s❡ t❤❡ ❢♦r♠✉❧❛s t♦ ❡✈❛❧✉❛t❡ t❤❡ ❧♦❝❛t✐♦♥ ❡✈❡r② h = .20 s❡❝♦♥❞✳ ❚❤✐s ✐s ✇❤❛t t❤❡ ❣r❛♣❤s ❧♦♦❦ ❧✐❦❡✿

✶✳✶✸✳

✶✵✹

▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥

❲✐t❤ t❤❡ s♣r❡❛❞s❤❡❡t✱ ✇❡ ❝❛♥ ❛s❦ ❛♥❞ ❛♥s✇❡r ❛ ✈❛r✐❡t② ♦❢ ♦t❤❡r q✉❡st✐♦♥s ❛❜♦✉t s✉❝❤ ♠♦t✐♦♥ ✭❤♦✇ ❤❛r❞ ✐t ❤✐ts t❤❡ ❣r♦✉♥❞✱ ❡t❝✳✮✳ ❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ ♦♥❧② ❞♦ ♦♥❡ ❡①❛♠♣❧❡ ❛t ❛ t✐♠❡✦ ❚❤❡ ❝♦♥❝❧✉s✐♦♥s ✇❡ ❞r❛✇ ❛r❡ ❛❧s♦ s♣❡❝✐✜❝ t♦ t❤❡s❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳ ❲❡ ♥♦✇ ❝♦♥s✐❞❡r t❤❡

❝♦♥t✐♥✉♦✉s

❝❛s❡✱ ✐✳❡✳✱ ✇❡ t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ ❡✈❡r②t❤✐♥❣ ❛❜♦✈❡✿ h = ∆x → 0 .

■♥st❡❛❞ ♦❢ s❡q✉❡♥❝❡s✱ ✇❡ ❤❛✈❡ t❤✐s t✐♠❡ t❤❡s❡ ❢✉♥❝t✐♦♥s

♦❢ t✐♠❡



• p ✐s t❤❡ ❤❡✐❣❤t✱ t❤❡ ✈❡rt✐❝❛❧ ❧♦❝❛t✐♦♥✳ • v = p′ ✐s t❤❡ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t②✳

• a = v ′ ✐s t❤❡ ✈❡rt✐❝❛❧ ❛❝❝❡❧❡r❛t✐♦♥✳

◆♦✇ t❤❡ s♣❡❝✐✜❝ ❝❛s❡ ♦❢ ❢r❡❡ ❢❛❧❧✿

a = −g .

❲❡ ❦♥♦✇ t❤❛t✿ ✶✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s ❧✐♥❡❛r✳ ✷✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧ ✐s ❝♦♥st❛♥t✳ ❈♦♥✈❡rs❡❧②✿ ✶✳ ❚❤❡ ♦♥❧② ❢✉♥❝t✐♦♥ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✇❤✐❝❤ ✐s ❧✐♥❡❛r ✐s ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✳ ✷✳ ❚❤❡ ♦♥❧② ❢✉♥❝t✐♦♥ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✇❤✐❝❤ ✐s ❝♦♥st❛♥t ✐s ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ◮ p = p(t) ✐s q✉❛❞r❛t✐❝✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ❲❤❛t ♠❛❦❡s t❤❡s❡ s♣❡❝✐✜❝ ❛r❡ t❤❡ ✐♥✐t✐❛❧

p(t) = ax2 + bx + c . ❝♦♥❞✐t✐♦♥s



✶✳✶✸✳

✶✵✺

▼♦t✐♦♥ ✉♥❞❡r ❢♦r❝❡s✿ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥

• p0 ✐s t❤❡ ✐♥✐t✐❛❧ ❤❡✐❣❤t✱ p0 = p(0)✳ • v0 ✐s t❤❡ ✐♥✐t✐❛❧ ✈❡rt✐❝❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ✈❡❧♦❝✐t②✱ v(0) = ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

dp ✳ dt t=0

1 p(t) = p0 +v0 t − gt2 2 ❊①❛♠♣❧❡ ✶✳✶✸✳✹✿ ♠♦✈✐♥❣ ❜❛❧❧

■♥ t❤❡ ♣r♦❜❧❡♠ ♦❢ ♦✉rs✱ ✇❡ ❤❛✈❡✿

p0 = 100, v0 = 50 . ❖✉r ❡q✉❛t✐♦♥s ❜❡❝♦♠❡✿

p = 100 +50t −16t2 .

■♥ ❝♦♥tr❛st t♦ t❤❡ ❞✐s❝r❡t❡ ❝❛s❡✱ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ♣♦s✐t✐♦♥ ✐s♥✬t r❡❝✉rs✐✈❡ ❜✉t ❞✐r❡❝t ❛♥❞ ❡①♣❧✐❝✐t✦ ❇❡❢♦r❡ ✇❡ ✉t✐❧✐③❡ t❤❡ ❡①♣❧✐❝✐t ❛❧❣❡❜r❛✐❝ r❡♣r❡s❡♥t❛t✐♦♥✱ ✇❡ ✈✐s✉❛❧✐③❡ t❤❡ r❡s✉❧ts ❜② ♣❧♦tt✐♥❣ t❤❡ ❣r❛♣❤ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ♥❡①t t♦ t❤❡ ♦♥❡ ♦❜t❛✐♥❡❞ r❡❝✉rs✐✈❡❧②✿

❊①❡r❝✐s❡ ✶✳✶✸✳✺

❲❤❛t ❤❛♣♣❡♥❡❞ t♦ ±❄ ❊①❡r❝✐s❡ ✶✳✶✸✳✻

❍♦✇ ❤✐❣❤ ❞♦❡s t❤❡ ♣r♦❥❡❝t✐❧❡ ❣♦ ✐♥ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡❄ ❊①❡r❝✐s❡ ✶✳✶✸✳✼

❯s✐♥❣ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡✱ ❤♦✇ ❧♦♥❣ ✇✐❧❧ ✐t t❛❦❡ ❢♦r t❤❡ ♣r♦❥❡❝t✐❧❡ t♦ r❡❛❝❤ t❤❡ ❣r♦✉♥❞ ✐❢ ✜r❡❞

❞♦✇♥



❊①❡r❝✐s❡ ✶✳✶✸✳✽

❯s❡ t❤❡ ❛❜♦✈❡ ♠♦❞❡❧ t♦ ❞❡t❡r♠✐♥❡ ❤♦✇ ❧♦♥❣ ✐t ✇✐❧❧ t❛❦❡ ❢♦r ❛♥ ♦❜❥❡❝t t♦ r❡❛❝❤ t❤❡ ❣r♦✉♥❞ ✐❢ ✐t ✐s ❞r♦♣♣❡❞✳ ▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ q✉❡st✐♦♥s ❛❜♦✉t t❤❡ s✐t✉❛t✐♦♥ ❛♥❞ ❛♥s✇❡r t❤❡♠✳ ❘❡♣❡❛t✳ ❊①❛♠♣❧❡ ✶✳✶✸✳✾✿ ❛❝❝❡❧❡r❛t✐♦♥ t❤❛t ❞❡♣❡♥❞s ♦♥ t✐♠❡

❲❤❛t ②♦✉r ❛❧❣❡❜r❛✐❝ ❛♥❛❧②s✐s ✐s ✐♥❝❛♣❛❜❧❡ ♦❢ ❞♦✐♥❣ ✭❢♦r ♥♦✇✮ ✐s t♦ ❤❛♥❞❧❡ t❤❡ ❝❛s❡ ♦❢ t✐♠❡✲❞❡♣❡♥❞❡♥t ❛❝❝❡❧❡r❛t✐♦♥✳ ❋♦r ❡①❛♠♣❧❡✱ ✐♠❛❣✐♥❡ t❤❛t t❤❡ ❊❛rt❤ ♠♦✈❡s ✐♥ ❛♥❞ ♦✉t ✕ ♣❡r✐♦❞✐❝❛❧❧② ✕ s♦ ❝❧♦s❡ t♦ t❤❡ ❙✉♥ t❤❛t t❤❡ ❣r❛✈✐t② ♦❢ t❤❡ ❧❛tt❡r ✐s str♦♥❣❡r t❤❛♥ t❤❛t ♦❢ t❤❡ ❢♦r♠❡r✳ ❚❤❡♥ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✇✐❧❧ ❜❡

✶✳✶✹✳

❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r

✶✵✻

❛❧s♦ ❝❤❛♥❣✐♥❣ ♣❡r✐♦❞✐❝❛❧❧②✳ ❲❤❛t ✇✐❧❧ ❜❡ t❤❡ ♠♦t✐♦♥ ♦❢ t❤❡ s❛♠❡ ❜❛❧❧❄ ❚❤❡ s♣r❡❛❞s❤❡❡t ✇♦r❦s ✇✐t❤ ♥♦ ❝❤❛♥❣❡✦ ❲❡ ❥✉st ✐♥s❡rt ❛ ♣❡r✐♦❞✐❝ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✿ ❂✶✵✵✯❙■◆✭❘❈❬✲✶❪✮

❚❤❡♥ t❤❡ r❡st ✐s t❛❦❡♥ ❝❛s❡ ♦❢✿

❊①❡r❝✐s❡ ✶✳✶✸✳✶✵

❉❡✈✐s❡ s✉❝❤ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ❧❛st ❡①❛♠♣❧❡ t❤❛t t❤❡ ♦❜❥❡❝t ♦s❝✐❧❧❛t❡s ❜❡t✇❡❡♥ t❤❡ ❊❛rt❤ ❛♥❞ t❤❡ ❙✉♥✳

✶✳✶✹✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r

❲❡ ❤❛✈❡ ❛❧r❡❛❞② s❡❡♥ t❤❡ s✐♠♣❧❡st ✭✜rst ♦r❞❡r✮ ❖❉❊s ❢♦r ♠♦t✐♦♥✿ ✐❢ ✇❡ ❦♥♦✇ t❤❡ ✈❡❧♦❝✐t② ♦❢ ❛ ♠♦✈✐♥❣ ♦❜❥❡❝t✱ ✇❡ ❝❛♥ ♣r❡❞✐❝t ✐ts ❞②♥❛♠✐❝s✳ ❙✉♣♣♦s❡ ✐♥st❡❛❞ ✇❤❛t ✇❡ ❦♥♦✇ ✐s t❤❡ ✇❡ ♣r❡❞✐❝t ✐ts ❞②♥❛♠✐❝s❄

❢♦r❝❡s

❛✛❡❝t✐♥❣ t❤❡ ♦❜❥❡❝t ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✐ts ❛❝❝❡❧❡r❛t✐♦♥✳ ❍♦✇ ❝❛♥

▲❡t✬s r❡✈✐❡✇ t❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧✳ ❲❡ st❛rt ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦✉r q✉❛♥t✐t✐❡s t❤❛t ❝♦♠❡ ❢r♦♠ t❤❡ s❡t✉♣ ♦❢ t❤❡ ♠♦t✐♦♥✿ • t❤❡ ✐♥✐t✐❛❧ t✐♠❡ t0

• t❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t② v0

• t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ p0

• t❤❡ ❝✉rr❡♥t ❛❝❝❡❧❡r❛t✐♦♥ a1

✶✳✶✹✳

✶✵✼

❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r

❲❡ ❝❤♦♦s❡✿ h = ∆t .

❆s ✇❡ ♣r♦❣r❡ss ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡✱ ♥❡✇ ♥✉♠❜❡rs ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ♥❡①t r♦✇ ♦❢ ♦✉r s♣r❡❛❞s❤❡❡t✿ tn , an , vn , pn , n = 1, 2, 3, ...

✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s✿ • tn+1 = tn + h • vn+1 = vn + an · h • pn+1 = pn + vn · h

❚❤❡ r❡s✉❧t ✐s ❛ ❣r♦✇✐♥❣ t❛❜❧❡ ♦❢ ✈❛❧✉❡s✿ ✐♥✐t✐❛❧✿

✐t❡r❛t✐♦♥ n t✐♠❡ tn 0 1 ... 1000 ...

3.5 3.6 ... 103.5 ...

❛❝❝❡❧❡r❛t✐♦♥ an ✈❡❧♦❝✐t② vn ❧♦❝❛t✐♦♥ pn −− 66 ... 666 ...

33 38.5 ... 4 ...

22 25.3 ... 336 ...

❲❤❡r❡ ❞♦❡s t❤❡ ❝✉rr❡♥t ❛❝❝❡❧❡r❛t✐♦♥ ❝♦♠❡ ❢r♦♠❄ ❖✉r ♠❛✐♥ ✐♥t❡r❡st ✐s t❤❡ ❝❛s❡ ✇❤❡♥ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❞❡♣❡♥❞s ♦♥ t❤❡ ❧❛st ❧♦❝❛t✐♦♥✱ ❡✳❣✳✱ an+1 = 1/p2n ✱ s✉❝❤ ❛s ✇❤❡♥ t❤❡ ❣r❛✈✐t② ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐st❛♥❝❡ t♦ t❤❡ ♣❧❛♥❡t ♦r t❤❡ ❢♦r❝❡ ♦❢ t❤❡ s♣r✐♥❣ ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐st❛♥❝❡ ♦❢ ✐ts ❡♥❞ ❢r♦♠ t❤❡ ❡q✉✐❧✐❜r✐✉♠✳ ❊①❛♠♣❧❡ ✶✳✶✹✳✶✿ ♥♦ ❢♦r❝❡s

❘❡❝❛❧❧ s♦♠❡ ❡①❛♠♣❧❡s✳ ❆ r♦❧❧✐♥❣ ❜❛❧❧ ✐s ✉♥❛✛❡❝t❡❞ ❜② ❤♦r✐③♦♥t❛❧ ❢♦r❝❡s ❛♥❞ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ❤♦r✐③♦♥t❛❧ ♠♦t✐♦♥ s✐♠♣❧✐❢② ❛s ❢♦❧❧♦✇s✿ • ❚❤❡ ✈❡❧♦❝✐t② vn+1 = vn + an · h = vn = v0 ✐s ❝♦♥st❛♥t✳ • ❚❤❡ ♣♦s✐t✐♦♥ pn+1 = pn + vn · h = pn + v0 · h ❣r♦✇s ❛t ❡q✉❛❧ ✐♥❝r❡♠❡♥ts✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♣♦s✐t✐♦♥ ❞❡♣❡♥❞s ❧✐♥❡❛r❧② ♦♥ t❤❡ t✐♠❡✳ ❆s ∆x → 0✱ ✇❡ ❤❛✈❡ ❛♥ ❡q✉❛t✐♦♥✿ y ′′ = 0 .

❚❤❡ s♦❧✉t✐♦♥ s❡t ❧♦♦❦s ✐s✿ y = p0 + v0 t ,

t❛❦❡♥ ♦✈❡r ❛❧❧ ♣♦ss✐❜❧❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳ ■t ❝❛♥ ❜❡ ✐❧❧✉str❛t❡❞ ❛s ❢♦❧❧♦✇s✿

❆❧❧ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ❛r❡ ❤❡r❡✦ ❚❤✐s ✐s ❛ t✇♦ ✲♣❛r❛♠❡t❡r ❢❛♠✐❧②✿ t❤❡ y ✲✐♥t❡r❝❡♣ts ❛♥❞ t❤❡ s❧♦♣❡s✳ ❊①❛♠♣❧❡ ✶✳✶✹✳✷✿ ❢r❡❡ ❢❛❧❧

❆ ❢❛❧❧✐♥❣ ❜❛❧❧ ✐s ✉♥❛✛❡❝t❡❞ ❜② ❤♦r✐③♦♥t❛❧ ❢♦r❝❡s ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ❢♦r❝❡ ✐s ❝♦♥st❛♥t✿ an = a ❢♦r ❛❧❧ n✳ ❚❤❡ ✜rst ♦❢ t❤❡ t✇♦ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ✈❡rt✐❝❛❧ ♠♦t✐♦♥ s✐♠♣❧✐✜❡s ❛s ❢♦❧❧♦✇s✿ • ❚❤❡ ✈❡❧♦❝✐t② vn+1 = vn + an · h = vn + a · h ❣r♦✇s ❛t ❡q✉❛❧ ✐♥❝r❡♠❡♥ts✳ • ❚❤❡ ♣♦s✐t✐♦♥ pn+1 = pn + vn · h ❣r♦✇s ❛t ❧✐♥❡❛r❧② ✐♥❝r❡❛s✐♥❣ ✐♥❝r❡♠❡♥ts✳

✶✳✶✹✳

❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r

❋✐♥❛❧❧②✱ ✐❢ ✇❡ t❤✐♥❦ ♦❢ y ❛s ❛ s❛♠♣❧❡❞ t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥✱ ✇❡ ❤❛✈❡ ❛ s✐♥❣❧❡ ❖❉❊✿ y ′′ = −g .

❚❤❡ s♦❧✉t✐♦♥ s❡t ❧♦♦❦s ✐s✿

1 y = p0 + v0 t − gt2 , 2

t❛❦❡♥ ♦✈❡r ❛❧❧ ♣♦ss✐❜❧❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳ ❚❤✐s ✐s ❛ s❦❡t❝❤ ♦❢ t❤❡ s♦❧✉t✐♦♥ s❡t✿

❚❤✐s ✐s ❛❧s♦ ❛ t✇♦ ✲♣❛r❛♠❡t❡r ❢❛♠✐❧②✿ t❤❡ y ✲✐♥t❡r❝❡♣ts ❛♥❞ t❤❡ s❧♦♣❡s ❛t x = 0✳ P❘❖❇▲❊▼✿

●✐✈❡♥

• t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❧♦❝❛t✐♦♥ z = f (y)✱

r❡♣r❡s❡♥t

• t❤❡ ✈❡❧♦❝✐t② ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ v = v(t) ❛♥❞

• t❤❡ ❧♦❝❛t✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ y = y(t)✳

❚❤❡♥ ♦✉r t✇♦ ❢✉♥❝t✐♦♥s ❤❛✈❡ t♦ s❛t✐s❢②✿

an = f (pn ), vn = v(tn ), ❛♥❞ pn = y(tn ) .

❲❡ ❛ss✉♠❡ t❤❛t t❤❡r❡ ✐s ❛ ✈❡rs✐♦♥ ♦❢ ♦✉r ♣❛✐r r❡❝✉rs✐✈❡ r❡❧❛t✐♦♥s✱ vn+1 = vn +an · h pn+1 = pn +vn · h

❢♦r ❡✈❡r② h > 0 s♠❛❧❧ ❡♥♦✉❣❤✳ ❲❡ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦✱ ❛s ✇❡❧❧ ❛s t = tn ✱ ✐♥t♦ ♦✉r r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s✿

❚❤❡♥✱

v(t + h) = v(t) +f (y(t + h)) · h y(t + h) = y(t) +v(t + h) · h v(t + h) − v(t) = f (y(t + h)) h y(t + h) − y(t) = v(t + h) h

❚❛❦✐♥❣ t❤❡ ❧✐♠✐t ♦✈❡r h → 0 ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ ♦✉r ❢✉♥❝t✐♦♥s✿ v ′ (t) = f (y(t)), y ′ (t) = v(t),

✶✵✽

✶✳✶✹✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r

✶✵✾

♣r♦✈✐❞❡❞ y = y(t) ❛♥❞ v = v(t) ❛r❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t ❛♥❞ z = f (y) ✐s ❝♦♥t✐♥✉♦✉s ❛t y(t)✳ ❚❤❡ ♦✉t❝♦♠❡ ✐s tr❡❛t❡❞ ❡✐t❤❡r ❛s ❛ s②st❡♠ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ♦❢ ✜rst ♦r❞❡r ❞✐s❝✉ss❡❞ ❧❛t❡r✿ 

y ′ = v, v ′ = f (y),

✇❤✐❝❤ ✐s ❛ ✈❡❝t♦r ✜❡❧❞✱ ♦r ❛s ❛ s✐♥❣❧❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ♦❢ s❡❝♦♥❞ ♦r❞❡r y ′′ = f (y) . ❊①❛♠♣❧❡ ✶✳✶✹✳✸✿ r♦❧❧✐♥❣ ❞♦✇♥ ❛ s❧♦♣❡

❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ♥♦t ❦♥♦✇♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✦ ❲❡ ❥✉st ❦♥♦✇ ❤♦✇ ✐t ❞❡♣❡♥❞s ♦♥ ❧♦❝❛t✐♦♥✳ ❆s t❤❡ ❜♦❧❧ ✐s r♦❧❧✐♥❣ ❞♦✇♥ ❛ s❧♦♣❡✱ t❤❡ ♣♦rt✐♦♥ ♦❢ t❤❡ ❣r❛✈✐t② ❛❝❝❡❧❡r❛t✐♦♥ ✈❛r✐❡s✿

❚❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ✈❡❧♦❝✐t② ∆v ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❧♦❝❛t✐♦♥ y ❛♥❞ t♦ ∆t ❛s ✐♥ t❤✐s ❡q✉❛t✐♦♥✿ ∆v = −f (y) · h, f (y) > 0 .

❚❤❡♥ t❤❡ ❡q✉❛t✐♦♥ ❣✐✈❡s ✉s t❤✐s r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿ v(tn+1 ) = v(tn ) − f (y(tn )) · h .

❋♦r t❤❡ ❧♦❝❛t✐♦♥✱ ✇❡ ❤❛✈❡ t❤✐s ❡q✉❛t✐♦♥ ❢♦r ❛♥② ✐♥t❡r✈❛❧ ♦❢ t✐♠❡✿ ∆y = v · h ,

✇❤❡r❡ v ✐s ❦♥♦✇♥ ❢r♦♠ ❛❜♦✈❡✳ ❲❡ ❤❛✈❡ ❛ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿ y(tn+1 ) = y(tn ) + v(tn ) · h .

▲❡t✬s ❝♦♥s✐❞❡r

f (y) = y 2 .

❋♦r t❤❡ s♣r❡❛❞s❤❡❡t✱ ✇❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ t❤❛t ♠❛❦❡s ❛ r❡❢❡r❡♥❝❡ t♦ t❤❡ ❧♦❝❛t✐♦♥✿ ❂✲✭❘❬✲✶❪❈❬✷❪✮✂ ✷

❚❤❡ r❡s✉❧t ✐s ❛❝❝❡❧❡r❛t❡❞ r♦❧❧ ✿

✶✳✶✹✳

✶✶✵

❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r

❋✐♥❛❧❧②✱ ✐❢ ✇❡ t❤✐♥❦ ♦❢ y ❛♥❞ v ❛s s❛♠♣❧❡❞ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s✱ t❤❡ t✇♦ ❡q✉❛t✐♦♥s ❛r❡ ❝♦♥✈❡rt❡❞ t♦ ❖❉❊s ❛s ❢♦❧❧♦✇s✿ y ′ = v, ❛♥❞ v ′ = −f (y) .

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ❛ s✐♥❣❧❡ ❖❉❊✿

y ′′ = −f (y) .

❲❤❡♥ f ✐s ❝♦♥st❛♥t✱ ✇❡ ❛r❡ ❜❛❝❦ t♦ ❢r❡❡ ❢❛❧❧✦ ❆ ♠♦r❡ ✐♥t❡r❡st✐♥❣ ❡①❛♠♣❧❡ ✐s f ❧✐♥❡❛r✳ ❋♦r ❡①❛♠♣❧❡✿ y ′′ = y .

❲❤❛t ❛r❡ t❤❡ s♦❧✉t✐♦♥s❄ ❋✐rst✱ ✐❢ y ′ = y ✱ t❤✐s ✐s ❛❧s♦ ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❛❜♦✈❡✳ ❖♥❡ ❝❧❛ss ✐s✱ t❤❡r❡❢♦r❡✿ y = Aet .

▼♦r❡❄ ■❢ y ′ = −y ✱ t❤✐s ✐s ❛❧s♦ ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❛❜♦✈❡✿ y ′′ = (y ′ )′ = (−y)′ = −y ′ = −(−y) = y .

❆♥♦t❤❡r ❝❧❛ss ✐s✱ t❤❡r❡❢♦r❡✿

y = Be−t .

▼♦r❡❄ ❆❢t❡r s♦♠❡ ❣✉❡ss✐♥❣✱ ✇❡ ❛rr✐✈❡ ❛t t❤❡ s✉♠ ♦❢ t❤❡ ❛❜♦✈❡✿ y = Aet + Be−t .

❚❤✐s ✐s ❛ t✇♦ ✲♣❛r❛♠❡t❡r ❢❛♠✐❧②✦ ❊①❡r❝✐s❡ ✶✳✶✹✳✹

❲❤❛t s❤❛♣❡ ♦❢ ❛ s❧✐❞❡ ✇✐❧❧ ♣r♦✈✐❞❡ t❤❡ ❢❛st❡st tr✐♣ ❢r♦♠ ♣♦✐♥t A t♦ ♣♦✐♥t B ❄ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ ♠♦r❡ ❣❡♥❡r❛❧ r❡s✉❧t✳ ❚❤❡♦r❡♠ ✶✳✶✹✳✺✿ ●❡♥❡r❛❧ ❙♦❧✉t✐♦♥ ♦❢ ❘♦❧❧ ❖❉❊ ❚❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❖❉❊✿

y ′′ = ky, k > 0 , ❛r❡ ❣✐✈❡♥ ❜②✿

y(t) = Ae ✇❤❡r❡

A, B



kt

+ Be−



kt

,

❛r❡ t✇♦ ♣❛r❛♠❡t❡rs t♦ ❜❡ ❞❡t❡r♠✐♥❡❞ ❢r♦♠ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳

✶✳✶✹✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r

✶✶✶

❊①❡r❝✐s❡ ✶✳✶✹✳✻

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❊①❛♠♣❧❡ ✶✳✶✹✳✼✿ s♣r✐♥❣

❚❤❡ ❢♦r❝❡ ♦❢ ❛ s♣r✐♥❣ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ♥❡❣❛t✐✈❡ ♦❢ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❢r♦♠ t❤❡ ❡q✉✐❧✐❜r✐✉♠✳

❉❡s❝r✐♣t✐♦♥ ✭❍♦♦❦❡✬s ▲❛✇✮✿



✏❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ ❛♥ ♦❜❥❡❝t ♦♥ ❛ s♣r✐♥❣ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ♥❡❣❛t✐✈❡ ♦❢ t❤❡ ❝✉rr❡♥t

❧♦❝❛t✐♦♥✑✳ ❚❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ✈❡❧♦❝✐t②

∆v

✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❧♦❝❛t✐♦♥

y

❛♥❞ t♦

∆t

❛s ✐♥ t❤✐s ❡q✉❛t✐♦♥✿

∆v = −k · y · h, k > 0 . ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ✈❡❧♦❝✐t②✿

v(t0 ) = v0 . ❚❤❡♥ t❤❡ ❡q✉❛t✐♦♥ ❣✐✈❡s ✉s t❤✐s r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿

v(tn+1 ) = v(tn ) − ky(tn ) · h . ❋♦r t❤❡ ❧♦❝❛t✐♦♥✱ ✇❡ ❤❛✈❡ t❤✐s ❡q✉❛t✐♦♥ ❢♦r ❛♥② ✐♥t❡r✈❛❧ ♦❢ t✐♠❡✿

∆y = v · h , ✇❤❡r❡

v

✐s ❦♥♦✇♥ ❢r♦♠ ❛❜♦✈❡✳ ❙✉♣♣♦s❡ ✇❡ ❛❧s♦ ❤❛✈❡ ❛♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ❧♦❝❛t✐♦♥✿

y(t0 ) = y0 . ❚❤✐s s❡ts ✉♣ ❛ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿

y(tn+1 ) = y(tn ) + v(tn ) · h . ❚❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❣✐✈❡s ✉s ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧✳ ❲❡ ♠♦❞✐❢② t❤❡ s♣r❡❛❞s❤❡❡t ❜② ❣✐✈✐♥❣ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❛ ❢♦r♠✉❧❛ t❤❛t ♠❛❦❡s ❛ r❡❢❡r❡♥❝❡ t♦ t❤❡ ❧♦❝❛t✐♦♥✿

❂✲❘❬✲✶❪❈❬✷❪ ❚❤❡ r❡s✉❧t ✐s ♦s❝✐❧❧❛t✐♦♥ ✿

✶✳✶✹✳

✶✶✷

❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r

❋✐♥❛❧❧②✱ ✐❢ ✇❡ t❤✐♥❦ ♦❢ y ❛♥❞ v ❛s s❛♠♣❧❡❞ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s✱ t❤❡ t✇♦ ❡q✉❛t✐♦♥s ❛r❡ ❝♦♥✈❡rt❡❞ t♦ ❖❉❊s ❛s ❢♦❧❧♦✇s✿ y ′ = v, ❛♥❞ v ′ = −ky . ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ❛ s✐♥❣❧❡ ❖❉❊✿

❚❤✐s ✐s ❛ s❦❡t❝❤ ♦❢ t❤❡ s♦❧✉t✐♦♥ s❡t✿

y ′′ = −ky, k > 0 .

❲❤❡♥ k = 1✱ ❛ ❝♦✉♣❧❡ ♦❢ s♦❧✉t✐♦♥s ♦❢ y ′′ = −y ❛r❡ ✇♦rt❤ r❡♠❡♠❜❡r✐♥❣✿ y1 (t) = cos t ❛♥❞ y2 (t) = sin t .

▼♦r❡ s♦❧✉t✐♦♥s❄ ❆❢t❡r s♦♠❡ ❣✉❡ss✐♥❣✱ ✇❡ ❛rr✐✈❡ ❛t t❤❡ ❧✐♥❡❛r

❝♦♠❜✐♥❛t✐♦♥

♦❢ t❤❡ ❛❜♦✈❡✿

y = A cos t + B sin t .

❚❤✐s ✐s ❛ t✇♦ ✲♣❛r❛♠❡t❡r ❢❛♠✐❧②✦ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ ♠♦r❡ ❣❡♥❡r❛❧ r❡s✉❧t✳ ❚❤❡♦r❡♠ ✶✳✶✹✳✽✿ ●❡♥❡r❛❧ ❙♦❧✉t✐♦♥ ♦❢ ❙♣r✐♥❣ ❖❉❊ ❚❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❖❉❊✿

y ′′ = −ky, k > 0 ,

✶✳✶✹✳

✶✶✸

❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r

❛r❡ ❣✐✈❡♥ ❜②✿

y(t) = A cos ✇❤❡r❡

A, B



kt + B sin



kt

❛r❡ t✇♦ ♣❛r❛♠❡t❡rs t♦ ❜❡ ❞❡t❡r♠✐♥❡❞ ❢r♦♠ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳

❊①❡r❝✐s❡ ✶✳✶✹✳✾

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❆♥♦t❤❡r ❡①❛♠♣❧❡ ♦❢ s✉❝❤ ♠♦t✐♦♥ ✐s t❤❡ ♣❡♥❞✉❧✉♠ ✭❈❤❛♣t❡r ✹✮✳ ❋♦r ♠♦r❡ ❡①❛♠♣❧❡s ♦❢ ✇❛✈❡ ❢✉♥❝t✐♦♥s✱ s❡❡ ❈❤❛♣t❡r ✶P❈✲✺✳ ❊①❛♠♣❧❡ ✶✳✶✹✳✶✵✿ s♣r✐♥❣ ✇✐t❤ ❞❛♠♣❡♥✐♥❣

❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❝❛♥ ❛❧s♦ ❞❡♣❡♥❞ ♦♥ t❤❡ ✈❡❧♦❝✐t②✦ ❋♦r ❡①❛♠♣❧❡✱ ♦♥❡ ♠✐❣❤t ✇❛♥t t♦ st♦♣ t❤❡ ✉♣ ❛♥❞ ❞♦✇♥ ♠♦t✐♦♥ ♦❢ t❤❡ s✉s♣❡♥s✐♦♥ ♦❢ ❛ ❝❛r✳ ❍♦✇❄ ❇② ✐♥tr♦❞✉❝✐♥❣ s♦♠❡ ❢r✐❝t✐♦♥✳ ❆ ♠♦❞❡❧ ♦❢ ❢r✐❝t✐♦♥ ♠❛② ❜❡ t❤❛t ✐ts ❢♦r❝❡ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ s♣❡❡❞ ♦❢ t❤❡ ♠♦t✐♦♥ ❜✉t ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✱ ✐✳❡✳✱ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ♥❡❣❛t✐✈❡ ♦❢ t❤❡ ✈❡❧♦❝✐t②✿ −my ′ .

❲❡ ❥✉st ♠♦❞✐❢② t❤❡ s♣r❡❛❞s❤❡❡t ❜② ❣✐✈✐♥❣ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❛ ❢♦r♠✉❧❛ t❤❛t ♠❛❦❡s ❛ r❡❢❡r❡♥❝❡ t♦ ❜♦t❤ t❤❡ ❧♦❝❛t✐♦♥ ❛♥❞ t❤❡ ✈❡❧♦❝✐t②✿ ❂✲❘❬✲✶❪❈❬✷❪✲❘❬✲✶❪❈❬✶❪

❚❤❡ r❡s✉❧t ✐s ❛ q✉✐❝❦ ❞✐s❛♣♣❡❛r❛♥❝❡ ♦❢ t❤❡ ♦s❝✐❧❧❛t✐♦♥✿

❚❤❡ ❖❉❊ ✐s ❛s ❢♦❧❧♦✇s✿ ■t ♥❡❡❞s ❢✉rt❤❡r ❛♥❛❧②s✐s✳

y ′′ = −my ′ − ky, m, k > 0 .

❊①❛♠♣❧❡ ✶✳✶✹✳✶✶✿ ❢r❡❡ ❢❛❧❧ ✇✐t❤ ❛✐r r❡s✐st❛♥❝❡

❚❤❡ ❡✛❡❝t ♦❢ ❛✐r r❡s✐st❛♥❝❡ ♦♥ ❛ ❢❛❧❧✐♥❣ ❜❛❧❧ ✐s t❤❡ s❛♠❡✿ y ′′ = −my ′ − g .

❚❤❡s❡ ❛r❡ t❤❡ r❡s✉❧ts ♦❢ s✐♠✉❧❛t✐♦♥s✿

✶✳✶✹✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r

✶✶✹

❚❤❡ t❡r♠✐♥❛❧ ✈❡❧♦❝✐t② ✐s ❜❡✐♥❣ ❛♣♣r♦❛❝❤❡❞✦

❉❡✜♥✐t✐♦♥ ✶✳✶✹✳✶✷✿ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❙✉♣♣♦s❡ h ✐s s♦♠❡ ❢✉♥❝t✐♦♥ ♦❢ t❤r❡❡ ✈❛r✐❛❜❧❡s✳ ❚❤❡♥ ❛ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ♦❢ s❡❝♦♥❞ ♦r❞❡r ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿   ∆2 y ∆y = h t, y, ∆t2 ∆t

❛♥❞ ❛♥ ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ♦❢ s❡❝♦♥❞ ♦r❞❡r ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✲ ✐♥❣✿   dy d2 y ♦r y ′′ = h(t, y, y ′ ) = h t, y, dt2 dt

■t ✐s ♠♦r❡ ♣r♦❞✉❝t✐✈❡ t♦ ❛❞❞r❡ss t❤❡s❡ ❡q✉❛t✐♦♥s ❛s s②st❡♠s ♦❢ ❖❉❊s ♦❢ ✜rst ♦r❞❡r ✭❈❤❛♣t❡r ✸✮✿ (

y′ = v v ′ = h(t, y, v)

❉❡✜♥✐t✐♦♥ ✶✳✶✹✳✶✸✿ ❧✐♥❡❛r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❚❤❡s❡ ❖❉❊s ❛r❡ ❝❛❧❧❡❞ ❧✐♥❡❛r ✇❤❡♥ h ✐s ❧✐♥❡❛r✿ ay ′′ + by ′ + cy = 0 , a 6= 0 .

❆❧♠♦st ❛❧❧ ❡①❛♠♣❧❡s ❤❛✈❡ ❜❡❡♥ ❧✐♥❡❛r✳ ❚❤❡② ❤❛✈❡ s♣❡❝✐❛❧ ♣r♦♣❡rt✐❡s✳ ❋♦r ❡①❛♠♣❧❡✱ ❛tt❛❝❤✐♥❣ t♦ ♦✉r s♣r✐♥❣ ❛♥♦t❤❡r✱ ✐❞❡♥t✐❝❛❧ s♣r✐♥❣ ✇✐❧❧ ❝r❡❛t❡ ❛♥ ♦s❝✐❧❧❛t✐♦♥ ❡q✉✐✈❛❧❡♥t t♦ t❤❛t ♦❢ ❛ s✐♥❣❧❡ s♣r✐♥❣✿

✶✳✶✹✳ ❉✐s❝r❡t❡ ♠♦❞❡❧s✿ ❤♦✇ t♦ s❡t ✉♣ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r

✶✶✺

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ s✉♠ ♦❢ t✇♦ s♦❧✉t✐♦♥s ✐s ❛❧s♦ ❛ s♦❧✉t✐♦♥✦ ▼♦r❡ ❣❡♥❡r❛❧ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡♦r❡♠ ✶✳✶✹✳✶✹✿ ▲✐♥❡❛r✐t② ♦❢ ✷♥❞ ❖r❞❡r ▲✐♥❡❛r ❖❉❊ ■❢

y1

❛♥❞

y2

❛r❡ s♦❧✉t✐♦♥s ♦❢ ❛ ✷♥❞ ❖r❞❡r ▲✐♥❡❛r ❖❉❊✱ t❤❡♥ s♦ ✐s ❛♥② ♦❢ t❤❡✐r

❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s❀ ✐✳❡✳✱ ✐❢

y1

❛♥❞

y2

s❛t✐s❢②

y ′′ + by ′ + cy = 0 , t❤❡♥ s♦ ❞♦❡s

y = Ay1 + By2 , ✇❤❡r❡

A, B

❛r❡ ❛♥② r❡❛❧ ♥✉♠❜❡rs✳

Pr♦♦❢✳

■t ❢♦❧❧♦✇s ❢r♦♠ ❧✐♥❡❛r✐t② ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥✳

❚❤❡ r❡s✉❧t ✐s ❛❧s♦ ❦♥♦✇♥ ❛s t❤❡ ✏❙✉♣❡r♣♦s✐t✐♦♥ Pr✐♥❝✐♣❧❡✑✳ ■♥ t✇♦ t❤❡♦r❡♠s✱ ✇❡ s❤♦✇❡❞ t❤❛t ❢♦r♠✉❧❛s ❢♦r t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❘♦❧❧ ❖❉❊✱

y ′′ = ky (k > 0), t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❙♣r✐♥❣ ❖❉❊✱

y ′′ = −ky (k > 0), s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥ ❜② ❞✐r❡❝t s✉❜st✐t✉t✐♦♥s✳ ❇✉t ❤♦✇ ✇♦✉❧❞ ♦♥❡ ❞✐s❝♦✈❡r t❤❡♠❄ ❚❤❡ ✐❞❡❛ ♦❢ t❤❡ ❢♦r♠❡r ✐s t❤❛t t❤❡ s♦❧✉t✐♦♥ ✐s ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✿

y = ert . ❲❡ s✉❜st✐t✉t❡✿

(ert )′′ = kert . ❈♦♠♣✉t❡✿

r2 ert = kert . ❲❡ ❞✐s❝♦✈❡r t❤❛t

√ r = ± k.

✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞

✶✶✻

❚❤❡ r❡s✉❧t ♠❛t❝❤❡s t❤❡ t❤❡♦r❡♠✳ ■❢ ✇❡ ❛♣♣❧② t❤✐s ✐❞❡❛ t♦ t❤❡ ❧❛tt❡r✱ ✇❡ ❛rr✐✈❡ t♦✿

√ r = ± −k . ❚❤❡s❡ ❛r❡ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs✦ ❍♦✇ ✐♠❛❣✐♥❛r② ✭❛♥❞ ❝♦♠♣❧❡①✮ ♥✉♠❜❡rs ♣r♦❞✉❝❡ s✐♥❡ ❛♥❞ ❝♦s✐♥❡ ✐s s❤♦✇♥ ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✳

✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞

▲❡t✬s r❡✈✐❡✇✳ ❲❡ ❞✐✈✐❞❡ t❤❡

x✲❛①✐s ✭✐✳❡✳✱ t❤❡ r❡❛❧ ❧✐♥❡ R✮ ✐♥t♦ ❞✐s❝r❡t❡ ♣✐❡❝❡s Q✳ ❚❤❡s❡ ❛r❡ t❤❡ t✇♦ t②♣❡s ♦❢ ♣✐❡❝❡s✿

✐♥t♦ ✐♥t❡r✈❛❧s ♦❢ ❡q✉❛❧ ❧❡♥❣t❤

❢r♦♠ s♦♠❡ ❧♦❝❛t✐♦♥



❚❤❡ ♥♦❞❡s ❛r❡

0✲❝❡❧❧s✱ x = ..., Q − 2h , Q − h, Q, Q + h, Q + 2h, ...✳



❚❤❡ ❡❞❣❡s ❛r❡ t❤❡

■♥ t❤❡ ♠❡❛♥t✐♠❡✱ t❤❡

1✲❝❡❧❧s✱ [x, x + h] = ... [Q − h, Q], [Q, Q + h], [Q + h, Q + 2h], ...✳

y ✲❛①✐s

✐s ❥✉st t❤❡ r❡❛❧s✿

◆♦✇✱ t❤❡ ♠❛✐♥ ♦❜❥❡❝ts ✐♥ ♦✉r st✉❞② ✇✐❧❧ ❜❡ t❤❡s❡ t✇♦ t②♣❡s ♦❢ ❞✐s❝r❡t❡ ❢✉♥❝t✐♦♥s✿



❚❤❡ ❢✉♥❝t✐♦♥s t❤❛t ❤❛✈❡ ♥♦❞❡s ❛s ✐♥♣✉ts ❛r❡ ❝❛❧❧❡❞

0✲❢♦r♠s✳



❚❤❡ ❢✉♥❝t✐♦♥s t❤❛t ❤❛✈❡ ❡❞❣❡s ❛s ✐♥♣✉ts ❛r❡ ❝❛❧❧❡❞

1✲❢♦r♠s✳

❚❤❡ ♦✉t♣✉ts ❛r❡ r❡❛❧ ♥✉♠❜❡rs✳ ❲❡ ✉s❡ ❛rr♦✇s t♦ ♣✐❝t✉r❡ t❤❡s❡ ❢✉♥❝t✐♦♥s ❛s ❝♦rr❡s♣♦♥❞❡♥❝❡s✿

h>0

st❛rt✐♥❣

✶✳✶✺✳

✶✶✼

❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞

❲❡ ❝❛♥

❧✐st t❤❡ ✈❛❧✉❡s

♦❢ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✿

• ❛ 0✲❢♦r♠ ✭♥♦❞❡ ❢✉♥❝t✐♦♥✮ f ✇✐t❤ f (0) = 2, f (1) = 4, f (2) = 3, ...       • ❛ 1✲❢♦r♠ ✭❡❞❣❡ ❢✉♥❝t✐♦♥✮ s ✇✐t❤ s [0, 1] = 3, s [1, 2] = .5, s [2, 3] = 1, ...

●✐✈❡♥ ❛ ❢✉♥❝t✐♦♥ f ✱ ✐ts ❣r❛♣❤ ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♣♦✐♥ts ♦♥ t❤❡ xy ✲♣❧❛♥❡ s♦ t❤❛t ✇❡ ❛❧✇❛②s ❤❛✈❡ y = f (x)✿

❆❜♦✈❡ ✇❡ s❡❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ✶✳ ❋♦r ❛ ♥♦❞❡ ❢✉♥❝t✐♦♥✱ x ✐s ❛ ♥♦❞❡✱ ❛ ♥✉♠❜❡r✱ ❛♥❞ y = f (x) ✐s ❛❧s♦ ❛ ♥✉♠❜❡r✳ ❚♦❣❡t❤❡r✱ t❤❡② ♣r♦❞✉❝❡ (x, y)✱ ❛ ♣♦✐♥t ♦♥ t❤❡ xy ✲♣❧❛♥❡ ✭✇✐t❤ t❤❡ x✲❛①✐s s♣❧✐t ✐♥t♦ ❝❡❧❧s ❛s s❤♦✇♥ ❛❜♦✈❡✮✳ ✷✳ ❋♦r ❛♥ ❡❞❣❡ ❢✉♥❝t✐♦♥✱ [A, B] ✐s ❛♥ ❡❞❣❡✱ ❛♥ ✐♥t❡r✈❛❧ ✐♥ t❤❡ x✲❛①✐s✱ ❛♥❞ y = g([A, B]) ✐s ❛ ♥✉♠❜❡r✳ ❚♦❣❡t❤❡r✱ t❤❡② ♣r♦❞✉❝❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♣♦✐♥ts ♦♥ t❤❡ xy ✲♣❧❛♥❡ s✉❝❤ ❛s (x, y) ❢♦r ❡✈❡r② x ✐♥ [A, B]✳ ❚❤❡ r❡s✉❧t ✐s ❛ ❤♦r✐③♦♥t❛❧ s❡❣♠❡♥t✳ ❲❡ ♠✐❣❤t ❞✐s❝♦✈❡r s✉❝❤ ❢✉♥❝t✐♦♥s ✐❢ ✇❡ ③♦♦♠ ✐♥ ♦♥ ❛ ❝♦♥t✐♥✉♦✉s

◆♦✇ ❛ ♥❡✇ q✉❡st✐♦♥✿ ❲❤❛t ❝❛♥ ✇❡ s❛② ❛❜♦✉t t❤❡

❝✉r✈❡



❝❤❛♥❣❡ ♦❢ t❤❡ ❝❤❛♥❣❡



❋♦r ❡①❛♠♣❧❡✱ ✇❡ ♣r♦❣r❡ss ❢r♦♠ t❤❡ ❧♦❝❛t✐♦♥s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ t✐♠❡ ❧✐♥❡ t♦ t❤❡ ❝❤❛♥❣❡s ♦❢ ❧♦❝❛t✐♦♥s ✭❞✐s♣❧❛❝❡♠❡♥ts✮ ❞❡✜♥❡❞ ♦♥ t❤❡ ❡❞❣❡s t♦ t❤❡ ❝❤❛♥❣❡s ♦❢ ❞✐s♣❧❛❝❡♠❡♥ts ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ❛❣❛✐♥✿ f: − 2 −−− 5 −−− 10 − ∆f : − − • − 5 − 2 = 3 −•− 10 − 5 = 5 − • − − ∆∆f : − −?− −−− 5−3=2 −−− −?− − t: 1 2 3 4 5

●❡♥❡r❛❧❧②✱ ✐❢ ✇❡ ❦♥♦✇ ♦♥❧②

t❤r❡❡

✈❛❧✉❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ✭✜rst ❧✐♥❡✮ ❛t t❤❡ ❡♥❞s ♦❢ ❛♥ ✐♥t❡r✈❛❧✱ ✇❡ ❝♦♠♣✉t❡ t❤❡

✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞

✶✶✽

❞✐✛❡r❡♥❝❡s ❛❧♦♥❣ t❤❡ t✇♦ ✐♥t❡r✈❛❧s ✭s❡❝♦♥❞ ❧✐♥❡✮ ❛♥❞ ♣❧❛❝❡ t❤❡ r❡s✉❧ts ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡❞❣❡✿ − f (x1 )

− − −  − − • − ∆ f [x1 , x2 ]

− −•− x1

−−−

f (x2 ) −•−    ∆f [x2 , x3 ] − ∆ f [x1 , x2 ] x2 

−  − −  f (x3 ) − ∆f [x2 , x3 ] − • − − −−−

−•− − x3

❚♦ ✜♥❞ t❤❡ ❝❤❛♥❣❡ ♦❢ t❤✐s ♥❡✇ ❢✉♥❝t✐♦♥✱ ✇❡ ❝❛rr② ♦✉t t❤❡ s❛♠❡ ♦♣❡r❛t✐♦♥ ❛♥❞ ♣❧❛❝❡ t❤❡ r❡s✉❧t ✐♥ t❤❡ ♠✐❞❞❧❡ ✭t❤✐r❞ ❧✐♥❡✮✳ ❲❤❛t ❝❛♥ ✇❡ s❛② ❛❜♦✉t t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ❄ ❆❞❞ ❞✐✈✐s✐♦♥✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ♣r♦❣r❡ss ❢r♦♠ t❤❡ ❧♦❝❛t✐♦♥s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ t✐♠❡ ❧✐♥❡ t♦ t❤❡ ✈❡❧♦❝✐t✐❡s ❞❡✜♥❡❞ ♦♥ t❤❡ ❡❞❣❡s t♦ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ❛❣❛✐♥✿ ❧♦❝❛t✐♦♥✿ − 2 −−− 5−2 = 3/2 ✈❡❧♦❝✐t②✿ − − • − ❛❝❝❡❧❡r❛t✐♦♥✿ − −?− t✐♠❡✿ 1

5

−−− 10 − 10 − 5 −•− = 5/2 − • − − 2 5/2 − 3/2 = 1/2 −−− −?− − 2 3 4 5

2 −−− 2

●❡♥❡r❛❧❧②✱ ✐❢ ✇❡ ❦♥♦✇ ♦♥❧② t❤r❡❡ ✈❛❧✉❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ✭✜rst ❧✐♥❡✮ ❛t t❤❡ ❡♥❞s ♦❢ ❛♥ ✐♥t❡r✈❛❧✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛❧♦♥❣ t❤❡ t✇♦ ✐♥t❡r✈❛❧s ✭s❡❝♦♥❞ ❧✐♥❡✮ ❛♥❞ ♣❧❛❝❡ t❤❡ r❡s✉❧ts ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡❞❣❡✿ − f (x1 ) − − − ∆f − −•− h − −•− −−− x1

f (x2 ) −•−

∆f h

− ∆f h h x2

− − − f (x3 ) − ∆f −•− − h −−− −•− − x3

❚♦ ✜♥❞ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤✐s ♥❡✇ ❢✉♥❝t✐♦♥✱ ✇❡ ❝❛rr② ♦✉t t❤❡ s❛♠❡ ♦♣❡r❛t✐♦♥ ❛♥❞ ♣❧❛❝❡ t❤❡ r❡s✉❧t ✐♥ t❤❡ ♠✐❞❞❧❡ ✭t❤✐r❞ ❧✐♥❡✮ ▲❡t✬s r❡✈✐❡✇ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐♥ ❢✉❧❧ ❣❡♥❡r❛❧✐t②✳ ❋✐rst✱ ✇❡ ❤❛✈❡ ❛ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b]✳ ❲❡ ♣❛rt✐t✐♦♥ ✐t ✐♥t♦ n ❡❞❣❡s ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡ ♥♦❞❡s✿ a = x0 , x1 , x2 , ..., xn−1 , xn = b .

❚❤❡s❡ ❛r❡ t❤❡ ❡❞❣❡s✿ c1 = [x0 , x1 ], c2 = [x1 , x2 ], ..., cn = [xn−1 , xn ] .

■❢ ❛ ❢✉♥❝t✐♦♥ y = f (x) ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s xk , k = 0, 1, 2, ..., n✱ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ f ✐s ❞❡✜♥❡❞ ❛t t❤❡ ❡❞❣❡s ❜②✿ ∆f (ck ) = f (xk+1 ) − f (xk )

✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞

✶✶✾

❢♦r ❡❛❝❤ k = 1, 2, ..., n ✳ ❆❧s♦✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ✐s ❞❡✜♥❡❞ ❛t t❤❡ ❡❞❣❡s✿ f (xk+1 ) − f (xk ) ∆f (ck ) = ∆x xk+1 − xk

❢♦r ❡❛❝❤ k = 1, 2, ..., n ✳ ❚❤❡ ❢♦r♠❡r ❢✉♥❝t✐♦♥ r❡♣r❡s❡♥ts t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛❧✉❡ ❢r♦♠ ♥♦❞❡ t♦ ♥♦❞❡✱ ✇❤✐❧❡ t❤❡ ❧❛tt❡r ❢✉♥❝t✐♦♥ r❡♣r❡s❡♥ts t❤❡ s❧♦♣❡s ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡s ♦✈❡r t❤❡ ♥♦❞❡s ♦❢ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥✿

■♥ ♦r❞❡r t♦ r❡♣❡❛t t❤❡ ❞✐✛❡r❡♥❝❡ ♦r ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❝♦♥str✉❝t✐♦♥ ♦♥ t❤✐s ♥❡✇ ❢✉♥❝t✐♦♥✱ ✇❡ ✇✐❧❧ ♥❡❡❞ ♥♦✇ ❛ ♥❡✇ ❞❡❝♦♠♣♦s✐t✐♦♥ ✿

❲❡ ♥❡❡❞ ❝♦♥✈❡rt ❡❞❣❡ ❢✉♥❝t✐♦♥s t♦ ♥♦❞❡ ❢✉♥❝t✐♦♥s ❛♥❞ ✈✐❝❡ ✈❡rs❛ ❜② s✇✐t❝❤✐♥❣ ❜❡t✇❡❡♥ ♥♦❞❡s ❛♥❞ ❡❞❣❡s✳ ❚♦ s❡t t❤✐s ✉♣ ✇❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛t❝❤ ❜❡t✇❡❡♥ t❤❡ ❝❡❧❧s✿

❚❤✉s t❤❡r❡ ❛r❡ ❛ ♥❡✇ ♥♦❞❡ ❢♦r ❡❛❝❤ ❡❞❣❡ ✐♥ t❤❡ ❞♦♠❛✐♥ ❛♥❞ ❛ ♥❡✇ ❡❞❣❡ ❢♦r ❡❛❝❤ ♥♦❞❡✳ ❚♦❣❡t❤❡r✱ t❤❡s❡ ♥❡✇ ♥♦❞❡s ❛♥❞ ❡❞❣❡s ❢♦r♠ ❛ ♥❡✇ ❝♦♣② ♦❢ t❤❡ ❞♦♠❛✐♥✳ ❚❤✐s ✐s ❤♦✇ t❤❡ ❝❡❧❧s ❛❜♦✈❡ ❛r❡ ♠❛t❝❤❡❞ ❛❜♦✈❡✿ • ❆♥ ❡❞❣❡ [x, x + h] ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ♥♦❞❡ x + h/2✳

• ❆ ♥♦❞❡ x ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❡❞❣❡ [x − h/2, x + h/2]✳

❋♦r t❤❡ ❣❡♥❡r❛❧ s✐t✉❛t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳

❉❡✜♥✐t✐♦♥ ✶✳✶✺✳✶✿ ♣r✐♠❛❧ ❛♥❞ ❞✉❛❧ ❞♦♠❛✐♥s ❆ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❝♦♣② ♦❢ t❤❡ r❡❛❧ ❧✐♥❡ R ✐s ❝❛❧❧❡❞ t❤❡ ♣r✐♠❛❧ ❞♦♠❛✐♥✳ ❆ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛♥♦t❤❡r ❝♦♣② ♦❢ t❤❡ r❡❛❧ ❧✐♥❡ R ✐s ❝❛❧❧❡❞ t❤❡ ❞✉❛❧ ❞♦♠❛✐♥✳ ❆ ❢✉♥❝t✐♦♥ ❢r♦♠ 0✲❝❡❧❧s ♦❢ t❤❡ ❢♦r♠❡r t♦ t❤❡ 1✲❝❡❧❧s ♦❢ t❤❡ ❧❛tt❡r ❛♥❞ ❢r♦♠ 1✲❝❡❧❧s ♦❢ t❤❡ ❢♦r♠❡r t♦ t❤❡ 0✲❝❡❧❧s ♦❢ t❤❡ ❧❛tt❡r t❤❛t ✐s ♠♦♥♦t♦♥❡ ✐s ❝❛❧❧❡❞ t❤❡ st❛r ♦♣❡r❛t♦r ♦r t❤❡ ⋆✲♦♣❡r❛t♦r✳

✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞

✶✷✵

❚❤✐s ✐s ❤♦✇ t❤❡② ❛r❡ ❞❡♥♦t❡❞✿

❉✉❛❧✐t② ♦❢ ❝❡❧❧s ❆ ♣r✐♠❛❧ ❝❡❧❧ ❛ ❞✉❛❧ ❝❡❧❧

a



a ❝♦rr❡s♣♦♥❞s t♦ ✳

❚❤✐s ✐s ❤♦✇ t❤❡② ♠❛t❝❤ ✉♣✿

❚❤❡ r❡❧❛t✐♦♥ ❝❛♥ ❜❡ r❡✈❡rs❡❞✿



❆ ❞✉❛❧ ❡❞❣❡

a = [x − h/2, x + h/2]



❆ ❞✉❛❧ ♥♦❞❡

x + h/2

❝♦rr❡s♣♦♥❞s t♦ t❤❡ ♣r✐♠❛❧ ♥♦❞❡

❝♦rr❡s♣♦♥❞s t♦ t❤❡ ♣r✐♠❛❧ ❡❞❣❡

a⋆ = x✳

x⋆ = [x, x + h]✳

❉❡✜♥✐t✐♦♥ ✶✳✶✺✳✷✿ ❞✉❛❧ ❝❡❧❧s ❋♦r ❜♦t❤ ♣r✐♠❛❧ ❛♥❞ ❞✉❛❧ ❛♥❞ ❢♦r ❜♦t❤ ♥♦❞❡s ❛♥❞ ❡❞❣❡s✱

a ❛♥❞ a⋆

❛r❡ ❝❛❧❧❡❞ ❞✉❛❧

♦❢ ❡❛❝❤ ♦t❤❡r✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿



❆ ♣r✐♠❛❧

k ✲❝❡❧❧

❝♦rr❡s♣♦♥❞s t♦ ❛ ❞✉❛❧

(1 − k)✲❝❡❧❧✳

◆♦✇ t❤❡ ❞✉❛❧✐t② ♦❢ ❢♦r♠s✳ ❚❤❡ ✈❛❧✉❡s ❛r❡ t❤❡ s❛♠❡✱ ♦♥❧② t❤❡ ✐♥♣✉ts ❝❤❛♥❣❡✦ ❚✇♦ ❞✉❛❧ ♣❛✐rs ♦❢ ❢♦r♠s ❛r❡ ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

❚❤❡② ❝♦♠❡ ✐♥ ♣❛✐rs✱ ♣r✐♠❛❧ ❛♥❞ ❞✉❛❧✿



■❢



■❢



f ✐s ❛ ♣r✐♠❛❧ 1✲❢♦r♠✳ g

♥♦❞❡ ❢✉♥❝t✐♦♥✱ ❛

0✲❢♦r♠✱

t❤❡♥

✐s ❛ ♣r✐♠❛❧ ❡❞❣❡ ❢✉♥❝t✐♦♥✱ ❛

1✲❢♦r♠✱

t❤❡♥

0✲❢♦r♠✳

❚❤❡s❡ ❛r❡ t❤❡ ❣r❛♣❤s ♦❢ t❤❡ ❞✉❛❧ ❢♦r♠s✿

  f ⋆ [x − h/2, x + h/2] = f (x)   g ⋆ (x) = g [x − h/2, x + h/2]

✐s ❛ ❞✉❛❧ ❡❞❣❡ ❢✉♥❝t✐♦♥✱

✐s ❛ ❞✉❛❧ ♥♦❞❡ ❢✉♥❝t✐♦♥✱

✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞

✶✷✶

■❢ ✇❡ ③♦♦♠ ♦✉t✱ t❤❡ ❢✉♥❝t✐♦♥s ❞✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r ✇✐❧❧ ❧♦♦❦ ✐❞❡♥t✐❝❛❧✦ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s t❤❡ ❣❡♥❡r❛❧ ❛♣♣r♦❛❝❤✳

❉❡✜♥✐t✐♦♥ ✶✳✶✺✳✸✿ ❞✉❛❧ ❢♦r♠s ●✐✈❡♥ ❛ ♥♦❞❡✴❡❞❣❡ ❢✉♥❝t✐♦♥ s✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛♥ ❡❞❣❡✴♥♦❞❡ ❢✉♥❝t✐♦♥✿ s⋆ (a) = s(a⋆ )

❚❤❡♥✱ s⋆ ✐s ❝❛❧❧❡❞ t❤❡ ❞✉❛❧ ❢✉♥❝t✐♦♥ ♦❢ s✳ ❚❤✐s s❡t✉♣ ❛❧❧♦✇s ✉s t♦ ❝♦♥str✉❝t t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ 0✲❢♦r♠✳ ❲❡ ♣❛rt✐t✐♦♥ t❤❡ s❡❣♠❡♥t ✐♥t♦ n − 1 ✐♥t❡r✈❛❧s ❜② ❣✐✈✐♥❣ ♥♦❞❡s t♦ t❤❡ ❡❞❣❡s ♦❢ t❤❡ ❧❛st ❞❡❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ t❤❡ s❛♠❡ ♥❛♠❡s✿ p = c1 , c2 , c3 , ..., cn−1 , cn = q .

❚❤❡♥ t❤❡ ✐♥❝r❡♠❡♥ts ❛r❡✿ ∆ck = ck+1 − ck .

◆♦✇✱ ✇❤❛t ❛r❡ t❤❡ ♥♦❞❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❡❞❣❡s ♦❢ t❤✐s ♥❡✇ ❞❡❝♦♠♣♦s✐t✐♦♥❄ ❚❤❡ ♥♦❞❡s ♦❢ t❤❡ ❧❛st ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❝♦✉rs❡✦ ■♥❞❡❡❞✱ ✇❡ ❤❛✈❡✿ x1 ✐♥ [c1 , c2 ], x2 ✐♥ [c2 , c3 ], ..., xn−1 ✐♥ [cn−1 , cn ] . ∆f

✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ❢✉♥❝t✐♦♥ ♦❢ ❲❡ ❛♣♣❧② t❤❡ s❛♠❡ ❝♦♥str✉❝t✐♦♥s t♦ t❤✐s ❞❡❝♦♠♣♦s✐t✐♦♥ t♦ t❤❡ ❢✉♥❝t✐♦♥ g = ∆x g ✐s ❞❡✜♥❡❞ ❛t t❤❡ ❡❞❣❡s ♦❢ t❤❡ ♥❡✇ ❞❡❝♦♠♣♦s✐t✐♦♥ ❜②✿ ∆g(xk ) = g(ck+1 ) − g(ck )

❢♦r ❡❛❝❤ k = 1, 2, ..., n − 1 ✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢✉♥❝t✐♦♥ ♦❢ g ✐s ❞❡✜♥❡❞ ❛t t❤❡ ❡❞❣❡s ♦❢ t❤❡ ♥❡✇ ❞❡❝♦♠♣♦s✐t✐♦♥ ❜②✿ g(ck+1 ) − g(ck ) ∆g (xk ) = ∆x ck+1 − ck

❢♦r ❡❛❝❤ k = 1, 2, ..., n − 1 ✳

❉❡✜♥✐t✐♦♥ ✶✳✶✺✳✹✿ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡

❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ f ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❞✐❢✲ ❢❡r❡♥❝❡✱ ✐✳❡✳✱ ✐t ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✭❛♥❞ ❞❡♥♦t❡❞✮ ❛s ❢♦❧❧♦✇s✿ ∆2 f (xk ) = ∆f (ck+1 ) − ∆f (ck )

✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞

✶✷✷

❢♦r ❡❛❝❤ k = 1, 2, ..., n − 1 ✳

❉❡✜♥✐t✐♦♥ ✶✳✶✺✳✺✿ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ ❛ ❢✉♥❝t✐♦♥ f ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ ✐✳❡✳✱ ✐t ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ t❤❡ ❞❡❝♦♠✲ ♣♦s✐t✐♦♥ ✭❛♥❞ ❞❡♥♦t❡❞✮ ❛s ❢♦❧❧♦✇s✿ ∆2 f (xk ) = ∆x2

− ∆f (c ) ∆x k ck+1 − ck

∆f (c ) ∆x k+1

❢♦r ❡❛❝❤ k = 1, 2, ..., n − 1 ✳ ◆♦t❡ t❤❛t t❤❡r❡ ❛r❡✿ • n + 1 ✈❛❧✉❡s ♦❢ f ✭❛t t❤❡ ♥♦❞❡s✮✱

∆f ✭❛t t❤❡ ❡❞❣❡s✮✱ ❛♥❞ ∆x ∆2 f ✭❛t t❤❡ ♥♦❞❡s ❡①❝❡♣t a ❛♥❞ b✮✳ • n − 1 ✈❛❧✉❡s ♦❢ ∆x2

• n ✈❛❧✉❡s ♦❢

❲❡ ✇✐❧❧ ♦❢t❡♥ ♦♠✐t t❤❡ s✉❜s❝r✐♣ts ❢♦r t❤❡ s✐♠♣❧✐✜❡❞ ♥♦t❛t✐♦♥✿

❙❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ∆2 f (x) = ∆f (c+∆c)−∆f (c) .

✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞

✶✷✸

❙❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ∆2 f (x) = ∆x2

∆f (c ∆x

+ ∆c) − ∆c

∆f (c) ∆x

.

❊①❛♠♣❧❡ ✶✳✶✺✳✻✿ ❝✉r✈❛t✉r❡ ❆s ✇❡ ❦♥♦✇✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✐s ❝♦♥st❛♥t✳ ❚❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s✱ t❤❡r❡❢♦r❡✱ ③❡r♦✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ❛ ♥♦♥✲③❡r♦ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐♥❞✐❝❛t❡s ❛ ♥♦♥✲❧✐♥❡❛r ❣r❛♣❤✿

❆❜♦✈❡✱ t❤❡ s❧♦♣❡s r❡♠❛✐♥ t❤❡ s❛♠❡✱

2✱

❛t ✜rst❀ t❤❡r❡❢♦r❡✱ t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ③❡r♦✿

∆2 f = 0. ∆x2 ❚❤❡♥✱ t❤❡ s❧♦♣❡ ❝❤❛♥❣❡s t♦

1

❛♥❞ t❤✐s ❝❤❛♥❣❡ ✐s t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✭❛ss✉♠✐♥❣

∆x = 1✮✿

∆2 f = −1 . ∆x2 ❆s ❛♥♦t❤❡r ✇❛② t♦ s❡❡ t❤✐s ✐❞❡❛✱ ✐♠❛❣✐♥❡ ②♦✉rs❡❧❢ ❞r✐✈✐♥❣ ❛❧♦♥❣ ❛ str❛✐❣❤t ♣❛rt ♦❢ t❤❡ r♦❛❞ ❛♥❞ s❡❡✐♥❣ ❛ ♣❛rt✐❝✉❧❛r tr❡❡ ❛❤❡❛❞ ✭♥♦ ❝✉r✈❛t✉r❡✮✱ t❤❡♥✱ ❛s ②♦✉ st❛rt t♦ t✉r♥✱ t❤❡ tr❡❡s st❛rt t♦ ♣❛ss ②♦✉r ✜❡❧❞ ♦❢ ✈✐s✐♦♥ ❢r♦♠ r✐❣❤t t♦ ❧❡❢t ✭❝✉r✈❛t✉r❡✮✿

❋✉rt❤❡r♠♦r❡✱ ❤✐❣❤❡r ✈❛❧✉❡s ♦❢ t❤❡ ❝✉r✈❛t✉r❡ ♦❢ t❤❡ ❣r❛♣❤ ♦❢

y = f (x)✳

❚❤❡ ❤✐❣❤❡r ✈❛❧✉❡ ♦❢ t❤❡ s❡❝♦♥❞

❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♠❡❛♥s ❤✐❣❤❡r ✈❛❧✉❡s ♦❢ t❤❡ ❝✉r✈❛t✉r❡ ♦❢ t❤❡ ❣r❛♣❤ ♦❢

y = f (x)✳

❊①❛♠♣❧❡ ✶✳✶✺✳✼✿ ❢r❡❡ ❢❛❧❧ ❉❡s❝r✐♣t✐♦♥✿ ✏❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❝♦♥st❛♥t✑✳ ❲❡ ❤❛✈❡ ❛ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ♦❢ s❡❝♦♥❞ ♦r❞❡r✿

∆2 p = a(∆t)2 . ❚❤✐s ✐s ❥✉st ❛ ♥❡✇ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ s❛♠❡ ❞✐s❝r❡t❡ ♠♦❞❡❧ ✇❡ ❤❛✈❡ ✉s❡❞ ❜❡❢♦r❡✳ ❚♦ ✐❧❧✉str❛t❡✱ ❧❡t✬s tr② t❤✐s s♣❡❝✐✜❝ ❝❤♦✐❝❡ ♦❢

a=1

❛♥❞

∆t = 1✿

✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞

✶✷✹

❲❡ st❛rt ✇✐t❤ ❛ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥✱ ❞♦ s✉♠♠❛t✐♦♥ t✇✐❝❡✱ ❛♥❞ ❛rr✐✈❡ t♦ ❛ ❢✉♥❝t✐♦♥ t❤❛t s❡❡♠s q✉❛❞r❛t✐❝✳

❊①❛♠♣❧❡ ✶✳✶✺✳✽✿ ♦s❝✐❧❧❛t✐♥❣ s♣r✐♥❣

❉❡s❝r✐♣t✐♦♥✿ ✏❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ ❛♥ ♦❜❥❡❝t ♦♥ ❛ s♣r✐♥❣ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ♥❡❣❛t✐✈❡ ♦❢ t❤❡ ❝✉rr❡♥t ❧♦❝❛t✐♦♥✑✳ ❲❡ ❤❛✈❡ ❛ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ♦❢ s❡❝♦♥❞ ♦r❞❡r ❢♦r t❤❡ ♣♦s✐t✐♦♥

p

♦❢ t❤❡ ❡♥❞ ♦❢ t❤❡ s♣r✐♥❣✿

∆2 p = −kp(∆t)2 . ❚❤✐s ✐s ❥✉st ❛ ♥❡✇ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ s❛♠❡ ❞✐s❝r❡t❡ ♠♦❞❡❧ ✇❡ ❤❛✈❡ ✉s❡❞ ❜❡❢♦r❡✳ ❚♦ ✐❧❧✉str❛t❡✱ ❧❡t✬s tr② t❤✐s s♣❡❝✐✜❝ ❝❤♦✐❝❡ ♦❢ ❛ ♣❡r✐♦❞✐❝

p

✇✐t❤

k=1

❛♥❞

∆t = 1✿

❆❢t❡r t❛❦✐♥❣ t✇♦ ❞✐✛❡r❡♥❝❡s ✭t❤❡ st❛r ♦♣❡r❛t♦rs ❛r❡ ♦♠✐tt❡❞✮✱ ✇❡ ❛rr✐✈❡ t♦

−p✳

❆ ♠❛t❝❤✦

✶✳✶✺✳ ❉✐s❝r❡t❡ ❢♦r♠s✱ ❝♦♥t✐♥✉❡❞

❊①❡r❝✐s❡ ✶✳✶✺✳✾ ■❧❧✉str❛t❡ s✐♠✐❧❛r❧② ♦t❤❡r ♠♦❞❡❧s ❢r♦♠ t❤✐s ❝❤❛♣t❡r✳

✶✷✺

❈❤❛♣t❡r ✷✿ ❱❡❝t♦r ✈❛r✐❛❜❧❡s

❈♦♥t❡♥ts ✷✳✶ ✷✳✷ ✷✳✸ ✷✳✹ ✷✳✺ ✷✳✻ ✷✳✼ ✷✳✽ ✷✳✾

❲❤❡r❡ ♠❛tr✐❝❡s ❝♦♠❡ ❢r♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ ❡✐❣❡♥✈❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇❛s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❡✐❣❡♥✈❛❧✉❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷✻ ✶✸✸ ✶✹✶ ✶✹✾ ✶✻✷ ✶✻✽ ✶✼✾ ✶✽✹ ✶✾✶

✷✳✶✳ ❲❤❡r❡ ♠❛tr✐❝❡s ❝♦♠❡ ❢r♦♠

■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ s❡t ❛s✐❞❡ t❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡s ✕ ♠❡❛s✉r✐♥❣ ❞✐st❛♥❝❡s ❛♥❞ ❛♥❣❧❡s ✕ ❛♥❞ ❝♦♥❝❡♥tr❛t❡ ♦♥ ♣✉r❡ ❛❧❣❡❜r❛✳ ▲❡t✬s r❡❝❛❧❧ t❤❡s❡ ♣r♦❜❧❡♠s ❛❜♦✉t Pr♦❜❧❡♠✱ ❞✐♠❡♥s✐♦♥

$60❄



♠✐①t✉r❡s

1✿ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ t②♣❡ ♦❢ ❝♦✛❡❡ t❤❛t ❝♦sts $3 ♣❡r ♣♦✉♥❞✳ ❍♦✇ ♠✉❝❤ ❞♦ ✇❡ ❣❡t ❢♦r

▲❡t x ❜❡ t❤❡ ✇❡✐❣❤t ♦❢ t❤❡ ❝♦✛❡❡✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ ❙❡t✉♣✿ 3x = 60 . ❙♦❧✉t✐♦♥✿ x =

60 . 3

2✿ ❙✉♣♣♦s❡ t❤❡ ❑❡♥②❛♥ ❝♦✛❡❡ ❝♦sts $2 ♣❡r ♣♦✉♥❞ ❛♥❞ t❤❡ ❈♦❧♦♠❜✐❛♥ ❝♦✛❡❡ ❝♦sts $3 ♣❡r ♣♦✉♥❞✳ ❍♦✇ ♠✉❝❤ ♦❢ ❡❛❝❤ ❞♦ ②♦✉ ♥❡❡❞ t♦ ❤❛✈❡ 6 ♣♦✉♥❞s ♦❢ ❜❧❡♥❞ ✇✐t❤ ❛ t♦t❛❧ ♣r✐❝❡ ♦❢ $14❄

Pr♦❜❧❡♠✱ ❞✐♠❡♥s✐♦♥

▲❡t x ❜❡ t❤❡ ✇❡✐❣❤t ♦❢ t❤❡ ❑❡♥②❛♥ ❝♦✛❡❡ ❛♥❞ ❧❡t y ❜❡ t❤❡ ✇❡✐❣❤t ♦❢ ❈♦❧♦♠❜✐❛♥ ❝♦✛❡❡✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ ❙❡t✉♣✿

x + y =6 2x + 3y = 14

❙♦❧✉t✐♦♥✿ ❋r♦♠ t❤❡ ✜rst ❡q✉❛t✐♦♥✱ ✇❡ ❞❡r✐✈❡✿ y = 6 − x✳ ❚❤❡♥ s✉❜st✐t✉t❡ ✐t ✐♥t♦ t❤❡ s❡❝♦♥❞ ❡q✉❛t✐♦♥✿ 2x + 3(6 − x) = 14✳ ❙♦❧✈❡ t❤❡ ♥❡✇ ❡q✉❛t✐♦♥✿ −x = −4✱ ♦r x = 4✳ ❙✉❜st✐t✉t❡ t❤✐s ❜❛❝❦ ✐♥t♦ t❤❡ ✜rst ❡q✉❛t✐♦♥✿ (4) + y = 6✱ t❤❡♥ y = 2✳ ❇✉t ✐t ✇❛s s♦ ♠✉❝❤ s✐♠♣❧❡r ❢♦r t❤❡ ❢♦r♠❡r ♣r♦❜❧❡♠✦ ■s ✐t ♣♦ss✐❜❧❡ t♦ ♠✐♠✐❝ t❤❡ s❡t✉♣✱ ✐✳❡✳✱ t❤❡ ❡q✉❛t✐♦♥✱ ❛♥❞ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ❢♦r t❤❡ 2✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡❄ ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ ✈❡❝t♦r ❛❧❣❡❜r❛ s✉❣❣❡sts t❤❛t ✐t ♠✐❣❤t ❜❡ ♣♦ss✐❜❧❡✳

✷✳✶✳

❲❤❡r❡ ♠❛tr✐❝❡s ❝♦♠❡ ❢r♦♠

✶✷✼

▲❡t✬s r❡❝❛❧❧ t❤❡ ✇❛②s ✇❡ ❤❛✈❡ ✐♥t❡r♣r❡t❡❞ t❤❡ ♣r♦❜❧❡♠✳ ❋✐rst✿ ♣♦✐♥ts ❛♥❞ ❧✐♥❡s✳ ❲❡ t❤✐♥❦ ♦❢ t❤❡ t✇♦ ❡q✉❛t✐♦♥s ❛s ❡q✉❛t✐♦♥s ❛❜♦✉t t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ♣♦✐♥ts✱

(

(x, y)✱

✐♥ t❤❡ ♣❧❛♥❡✿

x + y = 6, 2x + 3y = 14.

❊✐t❤❡r ❡q✉❛t✐♦♥ ✐s ❛ ❧✐♥❡ ♦♥ t❤❡ ♣❧❛♥❡✳ ❚❤❡ s♦❧✉t✐♦♥

(x, y) = (4, 2)

✐s t❤❡ ♣♦✐♥t ♦❢ t❤❡✐r ✐♥t❡rs❡❝t✐♦♥✿

❙❡❝♦♥❞✿ ✈❡❝t♦rs ❛♥❞ t❤❡✐r ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s✳ ▲❡t✬s ♣✉t t❤❡ ❡q✉❛t✐♦♥s ✐♥ t❤❡s❡ t❛❜❧❡s✿

1·x + 1·y = 6 2 · x + 3 · y = 14

❛❞❞✐t✐♦♥ ♦❢ ✈❡❝t♦rs

❚❤❡ t❛❜❧❡ ✐s s♣❧✐t ❤♦r✐③♦♥t❛❧❧② t♦ r❡✈❡❛❧ t❤❡ ❡q✉❛t✐♦♥s✳ ◆❡①t✱ ✇❡ st❛rt t♦ s♣❧✐t ✈❡rt✐❝❛❧❧② ❛♥❞ r❡❛❧✐③❡ t❤❛t ✇❡ s❡❡ ❛ ❝♦♠♣♦♥❡♥t✇✐s❡



1 · x + 1 · y = 6 2 · x + 3 · y = 14 ❲❡ ❤❛✈❡✿

1 · x

❇✉t

x✬s

❛♥❞

y ✬s

+

2 · x

1 · y

6 =

3 · y

14

❛r❡ r❡♣❡❛t❡❞✦ ❲❡ r❡❛❧✐③❡ t❤❛t ✇❡ s❡❡ ❛ ❝♦♠♣♦♥❡♥t✇✐s❡

1 2

1 · x +

3

s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ✈❡❝t♦rs

6 · y =

14

❱❡❝t♦rs st❛rt t♦ ❛♣♣❡❛r✳ ■♥❞❡❡❞✱ ♦✉r s②st❡♠ ❤❛s ❜❡❡♥ r❡❞✉❝❡❞ t♦ ❛ s✐♥❣❧❡

✈❡❝t♦r

❡q✉❛t✐♦♥✿

      6 1 1 y= x+ 14 3 2 2✳ ❚❤✐s ✐s♥✬t ❛ ❝♦✐♥❝✐❞❡♥❝❡✳ ❚❤❡②         ✇❡✐❣❤t ✭✐♥ ♣♦✉♥❞s✮ 6 1 1 . ❛r❡ , , ❝♦st ✭✐♥ ❞♦❧❧❛rs✮ 14 3 2

❲❡ s❡❡ t❤r❡❡ ✈❡❝t♦rs ♦❢ t❤❡ s❛♠❡ ❞✐♠❡♥s✐♦♥

❚❤❡② ❧✐✈❡ ✐♥ t❤❡ s❛♠❡ s♣❛❝❡

R2

❛r❡ ♦❢ t❤❡ s❛♠❡ ♥❛t✉r❡✿

❛♥❞✱ t❤❡r❡❢♦r❡✱ s✉❜❥❡❝t t♦ t❤❡ ♦♣❡r❛t✐♦♥s ♦❢ ✈❡❝t♦r ❛❧❣❡❜r❛✳



✷✳✶✳

❲❤❡r❡ ♠❛tr✐❝❡s ❝♦♠❡ ❢r♦♠

✶✷✽

❚❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ s②st❡♠ ✐♥ t❤✐s ✐♥t❡r♣r❡t❛t✐♦♥ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❧❣❡❜r❛✐❝ ♠❡❛♥✐♥❣✳ ❲❡ ❝❛♥ t❤✐♥❦ ♦❢ t❤❡

x ❛♥❞ y ✱       6 1 1 . = +y x 14 3 2

t✇♦ ❡q✉❛t✐♦♥s ❛s ❛ s✐♥❣❧❡ ❡q✉❛t✐♦♥ ❛❜♦✉t t❤❡ ❝♦❡✣❝✐❡♥ts✱

♦❢ t❤❡s❡ ✈❡❝t♦rs ✐♥ t❤❡ ♣❧❛♥❡✿

❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s

●❡♦♠❡tr✐❝❛❧❧②✱ ✇❡ ♥❡❡❞ t♦ ✜♥❞ ❛ ✇❛② t♦ str❡t❝❤ t❤❡s❡ t✇♦ ✈❡❝t♦rs s♦ t❤❛t ❛❢t❡r ❛❞❞✐♥❣ t❤❡♠ t❤❡ r❡s✉❧t ✐s t❤❡ ✈❡❝t♦r ♦♥ t❤❡ r✐❣❤t✳ ❲❡ s♣❡❛❦ ♦❢



❚❤❡ s❡t✉♣ ✐s ♦♥ t❤❡ ❧❡❢t ❢♦❧❧♦✇❡❞ ❜② ❛ tr✐❛❧✲❛♥❞✲❡rr♦r ♦♥ t❤❡ r✐❣❤t✿

❙♦✱ t❤❡ ♥❡✇ ♣♦✐♥t ♦❢ ✈✐❡✇ ❤❛s ❝❤❛♥❣❡❞✿ ■♥st❡❛❞ ♦❢ t❤❡

❧♦❝❛t✐♦♥s

✱ ✇❡ ❛r❡ ❛❢t❡r t❤❡

❞✐r❡❝t✐♦♥s



❊①❡r❝✐s❡ ✷✳✶✳✶

❆r❡ t❤❡r❡ ♦t❤❡r ✈❡❝t♦rs ❤❡r❡❄

❚❤✐r❞✿ tr❛♥s❢♦r♠❛t✐♦♥s✳ ■♥✐t✐❛❧❧②✱ ✇❡ ✉s❡ ♣♦✐♥ts✳ ❉✐♠❡♥s✐♦♥



1

♣r♦❜❧❡♠✿

❆ tr❛♥s❢♦r♠❛t✐♦♥

f :R→R

✐s ❣✐✈❡♥ ❜②

f (x) = 30x . •

❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿

f (x) = 60 .

❉✐♠❡♥s✐♦♥



2

♣r♦❜❧❡♠✿

❆ tr❛♥s❢♦r♠❛t✐♦♥

f : R2 → R2

✐s ❣✐✈❡♥ ❜②

F (x, y) = (x + y, 2x + 3y) . •

❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿

F (x, y) = (6, 14) .

◆♦✇✱ ✇❡ ♣r❡❢❡r t♦ ✉s❡ ✈❡❝t♦rs✳ ▲❡t✬s ✜♥❞ ❛❧❧

2✲❞✐♠❡♥s✐♦♥❛❧

✈❡❝t♦rs ✐♥ t❤❡ ❡q✉❛t✐♦♥s✿

1·x + 1·y = 6 2 · x + 3 · y = 14 ❚❤❡ ✜rst ✐s ♦♥ t❤❡ r✐❣❤t❀ ✐t ❝♦♥s✐sts ♦❢ t❤❡ t✇♦ ✏❢r❡❡✑ t❡r♠s ✭❢r❡❡ ♦❢

  6 B= . 14

x✬s

❛♥❞

y ✬s✦✮

♦♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✿

✷✳✶✳

✶✷✾

❲❤❡r❡ ♠❛tr✐❝❡s ❝♦♠❡ ❢r♦♠

❆♥♦t❤❡r ♦♥❡ ✐s ❧❡ss ✈✐s✐❜❧❡❀ ✐t ✐s ♠❛❞❡ ✉♣ ♦❢ t❤❡ t✇♦ ✉♥❦♥♦✇♥s✿   x . X= y

❊✈❡♥ t❤♦✉❣❤ ✐ts ❞✐♠❡♥s✐♦♥ ✐s ❛❧s♦ 2✱ ✐t✬s ♥♦t ♦❢ t❤❡ s❛♠❡ ♥❛t✉r❡ ❛s t❤❡ ♦t❤❡rs✿     # ♦❢ ♣♦✉♥❞s x . ✐s # ♦❢ ♣♦✉♥❞s y

■t ❧✐✈❡s ✐♥ ❛ ❞✐✛❡r❡♥t R2 ✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ s♣❛❝❡s✿ F : R2 → R2 ,

Y = F (X) .

■ts ❢♦r♠✉❧❛ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ t❡r♠s ♦❢ ✈❡❝t♦rs✿     x+y x . = F 2x + 3y y

❖✉r ♣r♦❜❧❡♠ ❜❡❝♦♠❡s ❛ ♣r♦❜❧❡♠ ♦❢ s♦❧✈✐♥❣ ❛♥ ❡q✉❛t✐♦♥ ❢♦r X ✿ F (X) = B . ❲❛r♥✐♥❣✦ ❙✐♥❝❡ ✇❡ ❛r❡♥✬t ❞♦✐♥❣ ❛♥② ❣❡♦♠❡tr② ❜✉t ✇❡ ❛r❡ ❞♦✲ ✐♥❣ ✈❡❝t♦r ❛❧❣❡❜r❛✱ t❤❡ ✈❡❝t♦r ❛♣♣r♦❛❝❤ ✇✐❧❧ ❜❡ ♣r❡✲ ❢❡rr❡❞ t❤r♦✉❣❤♦✉t t❤❡ ❝❤❛♣t❡r✳

❚❤❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ♥❡❡❞❡❞ t♦ ❝♦♠♣✉t❡ F ❛r❡ s♦ s✐♠♣❧❡ t❤❛t t❤❡② ✇✐❧❧ ❜❡ ❡❛s② t♦ ❛❜❜r❡✈✐❛t❡✳ ▲❡t✬s r❡✈✐❡✇ t❤❡ s❡t✉♣✳ ❚❤❡ ♣r♦❜❧❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ n ❤❛s n ✐♥❣r❡❞✐❡♥ts✿ t❤❡ ✉♥❦♥♦✇♥ ♠✉❧t✐♣❧✐❡❞ ❜② ✐s ❡q✉❛❧ t♦

dim 1 dim 2 x X =< x, y > 3 ? 60 B =< 6, 14 >

❲❡ ❤❛✈❡ tr❛♥s✐t✐♦♥❡❞ ❢r♦♠ ♥✉♠❜❡rs t♦ ✈❡❝t♦rs✳ ❇✉t ✇❤❛t ✐s t❤❡ ♦♣❡r❛t✐♦♥ t❤❛t ♠❛❦❡s B ❢r♦♠ X ❄ ◆♦♥❡ ♦❢ t❤❡ ❢❛♠✐❧✐❛r ♦♥❡s✳ ❚❤❡ ❢♦✉r ❝♦❡✣❝✐❡♥ts ♦❢ x, y ❢♦r♠ ❛ t❛❜❧❡✿



 1 1 F = . 2 3

■t ❤❛s t✇♦ r♦✇s ❛♥❞ t✇♦ ❝♦❧✉♠♥s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤✐s ✐s ❛ 2 × 2 ♠❛tr✐①✳

❇♦t❤ X ❛♥❞ B ❛r❡ ❝♦❧✉♠♥✲✈❡❝t♦rs ✐♥ ❞✐♠❡♥s✐♦♥ 2✱ ❛♥❞ ♠❛tr✐① F t✉r♥s X ✐♥t♦ B ✳ ❚❤✐s ✐s ✈❡r② s✐♠✐❧❛r t♦ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ♥✉♠❜❡rs❀ ❛❢t❡r ❛❧❧✱ t❤❡② ❛r❡ ✈❡❝t♦rs ♦❢ ❞✐♠❡♥s✐♦♥ 1✳ ▲❡t✬s ♠❛t❝❤ t❤❡ s❡t✉♣s ♦❢ t❤❡ t✇♦ ♣r♦❜❧❡♠s✿ dim 1 : m · x = b dim 2 : F · X = B

■❢ ✇❡ ❝❛♥ ❥✉st ♠❛❦❡ s❡♥s❡ ♦❢ t❤❡ ♥❡✇ ❛❧❣❡❜r❛✦

❍❡r❡ F X = B ✐s ❛ ♠❛tr✐① ❡q✉❛t✐♦♥✱ ❛♥❞ ✐t✬s s✉♣♣♦s❡❞ t♦ ❝❛♣t✉r❡ t❤❡ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✳ ▲❡t✬s ❝♦♠♣❛r❡ t❤❡ ♦r✐❣✐♥❛❧ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s t♦ F X = B ✿ x +y = 6 ✱ r❡✇r✐tt❡♥ ❛s 2x +3y = 14



     1 1 x 6 · = . 2 3 y 14

✷✳✶✳ ❲❤❡r❡ ♠❛tr✐❝❡s ❝♦♠❡ ❢r♦♠

✶✸✵

❲❡ ❝❛♥ s❡❡ t❤❡s❡ ❡q✉❛t✐♦♥s ♦♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❛s t❤❡s❡ t✇♦ ❞♦t ♣r♦❞✉❝ts✳ ❋✐rst✿

1·x+1·y =6



✱ r❡✇r✐tt❡♥ ❛s

❙❡❝♦♥❞✿

2x + 3y = 14 ❚❤✐s s✉❣❣❡sts ✇❤❛t t❤❡ ♠❡❛♥✐♥❣ ♦❢

X

FX



✱ r❡✇r✐tt❡♥ ❛s

  x 1 1 · = 6. y 

  x 2 3 · = 14 . y 

s❤♦✉❧❞ ❜❡✳ ❲❡ ✏♠✉❧t✐♣❧②✑ ❡✐t❤❡r r♦✇ ✐♥

A✱

❛s ❛ ✈❡❝t♦r✱ ❜② t❤❡ ✈❡❝t♦r

✕ ✈✐❛ t❤❡ ❞♦t ♣r♦❞✉❝t✿

❉❡✜♥✐t✐♦♥ ✷✳✶✳✷✿ ♣r♦❞✉❝t ♦❢ ♠❛tr✐① ❛♥❞ ✈❡❝t♦r ❚❤❡ ♣r♦❞✉❝t

FX

♦❢ ❛

2×2

♠❛tr✐①

F

❛♥❞ ❛

2✲✈❡❝t♦r X ✱

   x a b , X= F = y c d 

✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣

2✲✈❡❝t♦r✿

     ax + by x a b = · FX = cx + dy y c d 

❲❡ ❝❛♥ st✐❧❧ s❡❡ t❤❡s❡ ❞♦t ♣r♦❞✉❝ts ✐♥ t❤❡ r❡s✉❧t✿



     a b x ax + by = · y cx + dy c d

❛♥❞



     x a b ax + by · = . y c d cx + dy

❲❛r♥✐♥❣✦ ❆ ♠❛tr✐① ✐s ♥♦t❤✐♥❣ ❜✉t ❛♥ ❛❜❜r❡✈✐❛t✐♦♥ ♦❢ ❛ tr❛♥s✲ ❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡✿

F X = F (X) . ❍♦✇❡✈❡r✱

♥♦t

❛❧❧

tr❛♥s❢♦r♠❛t✐♦♥s

❝❛♥

❜❡

r❡♣r❡✲

s❡♥t❡❞ ❜② ♠❛tr✐❝❡s✳

❊①❛♠♣❧❡ ✷✳✶✳✸✿ 3 ✈❛r✐❛❜❧❡s ❲❤❛t ✐❢ t❤❡ ❜❧❡♥❞ ✐s t♦ ❝♦♥t❛✐♥ ❛♥♦t❤❡r✱ t❤✐r❞✱ t②♣❡ ♦❢ ❝♦✛❡❡❄ ●✐✈❡♥ t❤r❡❡ ♣r✐❝❡s ♣❡r ♣♦✉♥❞✱ ❤♦✇ ♠✉❝❤ ♦❢ ❡❛❝❤ ❞♦ ②♦✉ ♥❡❡❞ t♦ ❤❛✈❡

x✱ y ✱ ❛♥❞ z ❜❡ t❤❡ ❜❧❡♥❞ ✐s 14✳ ❚❤❡r❡❢♦r❡✱ ▲❡t

6

♣♦✉♥❞s ♦❢ ❜❧❡♥❞ ✇✐t❤ ❛ t♦t❛❧ ♣r✐❝❡ ♦❢

2, 3, 5✱

14❄

✇❡✐❣❤ts ♦❢ t❤❡ t❤r❡❡ t②♣❡s ♦❢ ❝♦✛❡❡✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡♥ t❤❡ t♦t❛❧ ♣r✐❝❡ ♦❢ t❤❡ ✇❡ ❤❛✈❡ ❛ s②st❡♠✿



x + y + z =6 2x + 3y + 5z = 14

❊✐t❤❡r ♦❢ t❤❡s❡ ❡q✉❛t✐♦♥s r❡♣r❡s❡♥ts ❛ ♣❧❛♥❡ ✐♥

R3 ✳

❚❤❡ s♦❧✉t✐♦♥ s❡t t❤❡♥ ❝♦♠❡s ❢r♦♠ t❤❡✐r ✐♥t❡rs❡❝t✐♦♥✿

✷✳✶✳

❲❤❡r❡ ♠❛tr✐❝❡s ❝♦♠❡ ❢r♦♠

✶✸✶

❚❤❡r❡ ❛r❡✱ ♦❢ ❝♦✉rs❡✱ ✐♥✜♥✐t❡❧② ♠❛♥② s♦❧✉t✐♦♥s✳ ❆♥ ❛❞❞✐t✐♦♥❛❧ r❡str✐❝t✐♦♥ ✐♥ t❤❡ ❢♦r♠ ♦❢ ❛♥♦t❤❡r ❧✐♥❡❛r ❡q✉❛t✐♦♥ ♠❛② r❡❞✉❝❡ t❤❡ ♥✉♠❜❡r t♦ ♦♥❡✳✳✳ ♦r ♥♦♥❡✳ ❚❤❡ ✈❛r✐❡t② ♦❢ ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡s ✐s✱ ❜② ❢❛r✱ ❤✐❣❤❡r t❤❛♥ ✐♥ t❤❡

2✲❞✐♠❡♥s✐♦♥❛❧

❝❛s❡❀ t❤❡② ❛r❡ ♥♦t ❞✐s❝✉ss❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✳

❚❤❡ ✈❡❝t♦r ❛❧❣❡❜r❛✱ ❤♦✇❡✈❡r✱ ✐s t❤❡ s❛♠❡✦ ❚❤❡ t❤r❡❡ ✇❡✐❣❤ts ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ ❛ ✈❡❝t♦r✱

< 1, 1, 1 >✱

❛♥❞ t❤❡ ✜rst ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s t❤❡ ❞♦t ♣r♦❞✉❝t✿

< 1, 1, 1 > · < x, y, z >= 6 . ❚❤❡ t❤r❡❡ ♣r✐❝❡s ♣❡r ♣♦✉♥❞ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ ❛ ✈❡❝t♦r✱

< 2, 3, 5 >✱

❛♥❞ t❤❡ ✜rst ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s t❤❡

❞♦t ♣r♦❞✉❝t✿

❋✐♥❛❧❧②✱ ✇❡ ❤❛✈❡ ❛

♠❛tr✐① ❡q✉❛t✐♦♥

< 2, 3, 5 > · < x, y, z >= 14 . ✿



    x 6 1 1 1   . · y = 14 2 3 5 z 

❲✐t❤♦✉t ❤❛r♠✱ ✇❡ ❝❛♥ ♠❛❦❡ t❤❡ ♠❛tr✐① sq✉❛r❡✿

     6 x 1 1 1 2 3 5 · y  = 14 . 0 z 9 0 0  ❊①❛♠♣❧❡ ✷✳✶✳✹✿ s♣r❡❛❞s❤❡❡t

❖♥❡ ❝❛♥ ✉t✐❧✐③❡ ❛ s♣r❡❛❞s❤❡❡t ❛♥❞ ♦t❤❡r s♦❢t✇❛r❡ t♦ t❤✐s ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❢♦r ♠❛tr✐❝❡s ♦❢ ❛♥② ❞✐♠❡♥s✐♦♥s✳ ■♥ ♦r❞❡r t♦ ♠❛❦❡ t❤✐s ✇♦r❦✱ t❤❡ ✈❡❝t♦r

❚❤✐s ✐s t❤❡ ❝♦❞❡ ❢♦r t❤❡ tr❛♥s♣♦s❡ ♦❢

X

❤❛s t♦ ❜❡ ✏tr❛♥s♣♦s❡❞✑ ✭❜♦tt♦♠✮✿

X✿

❂❚❘❆◆❙P❖❙❊✭❘❬✲✺❪❈✿❘❬✲✸❪❈✮

✷✳✶✳ ❲❤❡r❡ ♠❛tr✐❝❡s ❝♦♠❡ ❢r♦♠ ❚❤✐s ✐s t❤❡ ❝♦❞❡ ❢♦r Y ✿

✶✸✷ ❂❙❯▼P❘❖❉❯❈❚✭❘❈✷✿❘❈✹✱❘✽❈❬✲✷❪✿❘✽❈✮

❚❤❡ ✇❤♦❧❡ s②st❡♠ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ t❤❡ ❢♦r♠ ♦❢ ❡①❛❝t❧② t❤❡ s❛♠❡ ♠❛tr✐① ❡q✉❛t✐♦♥✿ FX = B .

❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ❡①❡❝✉t❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛② t♦♦✿   ❝ ♦     ❧ r r ♦ ♦ ✇ ✇ ·  ✉  = rc + ro + ol + ou + wm + wn   ♠ ♥

●❡♥❡r❛❧❧②✱ ✇❡ ❢❛❝❡ ❛ s②st❡♠ ✇✐t❤✿ ✶✳ t❤❡ ♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s m✱ ❛♥❞ ✷✳ t❤❡ ♥✉♠❜❡r ♦❢ ❡q✉❛t✐♦♥s n✳ ❲❡ ✇✐❧❧ ❤❛✈❡ ❛♥ n × m ♠❛tr✐①✿

1 1 2 2  0   n 3

2 3 ... m  0 3 ... 2 6 2 ... 0     ... 1 0 ... 12

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

✳✳✳

❍❡r❡✱ t❤❡ ♥✉♠❜❡r ❛t t❤❡ ij ✲♣♦s✐t✐♦♥ ✐s t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ j t❤ ✈❛r✐❛❜❧❡ ✐♥ t❤❡ it❤ ❡q✉❛t✐♦♥✳ ❘❡❝❛❧❧ ❤♦✇ ❛♥ ✐♥❞❡① ♣♦✐♥ts ❛t ❛ ❧♦❝❛t✐♦♥ ✇✐t❤✐♥ ❛ s❡q✉❡♥❝❡✳ ❙✐♠✐❧❛r❧②✱ ✇❡ ✉s❡ ❞♦✉❜❧❡ ✐♥❞❡① t♦ ♣♦✐♥t ❛t t❤❡ ❝♦rr❡❝t ❧♦❝❛t✐♦♥ ✇✐t❤✐♥ ❛ t❛❜❧❡✳ ❉❡✜♥✐t✐♦♥ ✷✳✶✳✺✿ ❡♥tr✐❡s ♦❢ ♠❛tr✐①

❚❤❡ ij ✲❡♥tr② ✐♥ ❛♥ n × m ♠❛tr✐① A ✐s t❤❡ ♥✉♠❜❡r ❛t t❤❡ it❤ r♦✇ ❛♥❞ j t❤ ❝♦❧✉♠♥✱ ❞❡♥♦t❡❞ ❜② Aij

❢♦r ❡❛❝❤ i = 1, 2, ..., m ❛♥❞ ❡❛❝❤ j = 1, 2, ..., n✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❤❛✈❡ ❢♦r t❤❡ ❛❜♦✈❡ ♠❛tr✐①✿ 1 2 3 i\j A1,1 = 2 A1,2 = 0 A1,3 = 3 1 A 2   2,1 = 0 A2,2 = 6 A2,3 = 2   m Am,1 = 3 Am,2 = 1 Am,3 = 0

✳✳✳

✳✳✳

✳✳✳

✳✳✳

... ... ...

n  A1,n = 2 A2,n = 0     ... ... Am,n = 12

✳✳✳

❲❛r♥✐♥❣✦ ❲❤❛t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❛❜❧❡s ♦❢ ♥✉♠❜❡rs ❛♥❞ ♠❛tr✐❝❡s❄ ❚❤❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ❞✐s❝✉ss❡❞ ❤❡r❡✳

✷✳✷✳

❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡

✶✸✸

✷✳✷✳ ❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡

❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ❛r❡ ♠❛❞❡ ✉♣ ♦❢ t✇♦ r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳

❉❡✜♥✐t✐♦♥ ✷✳✷✳✶✿ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ ❆

tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ ✐s ❛ ❢✉♥❝t✐♦♥ F : ❘2 → ❘2 ,

❣✐✈❡♥ ❜② ❛♥② ♣❛✐r ♦❢ ❢✉♥❝t✐♦♥s f, g ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✿

F (x, y) = (u, v) =



f (x, y), g(x, y)



❲❤❡♥ ❛♣♣r♦♣r✐❛t❡✱ ✇❡ ❝❛♥ ❛❧s♦ ❧♦♦❦ ❛t t❤❡ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts ❛s ✈❡❝t♦rs ✿     u x , 7→ < u, v >= < x, y >= v y

✐♥st❡❛❞ ♦❢ ♣♦✐♥ts✳

▲❡t✬s r❡✈✐❡✇ ❛ ❢❡✇ ❡①❛♠♣❧❡s ♦❢ s✉❝❤ tr❛♥s❢♦r♠❛t✐♦♥s✳

❊①❛♠♣❧❡ ✷✳✷✳✷✿ t✇♦ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❲❡ ♥❡❡❞ t✇♦ r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳ ❈♦♥s✐❞❡r u = f (x, y) = 2x − 3y ✿

f : R2 → R, ♠❡❛♥✐♥❣ f : (x, y) → u = 2x − 3y ❈♦♥s✐❞❡r ❛❧s♦ v = g(x, y) = x + 5y ✿

g : R2 → R, ♠❡❛♥✐♥❣ g : (x, y) → v = x + 5y ▲❡t✬s ❜✉✐❧❞ ❛ ♥❡✇ ❢✉♥❝t✐♦♥ ❢r♦♠ t❤❡s❡ t✇♦✳ ❲❡ t❛❦❡ t❤❡ ✐♥♣✉t t♦ ❜❡ t❤❡ s❛♠❡ ✕ ❛ ♣♦✐♥t ✐♥ t❤❡ ♣❧❛♥❡ ✕ ❛♥❞ ✇❡ ❝♦♠❜✐♥❡ t❤❡ t✇♦ ♦✉t♣✉ts ✐♥t♦ ❛ s✐♥❣❧❡ ♣♦✐♥t (u, v) ✕ ✐♥ ❛♥♦t❤❡r ♣❧❛♥❡✳ ❚❤❡♥ ✇❤❛t ✇❡ ❤❛✈❡ ✐s ❛ s✐♥❣❧❡ ❢✉♥❝t✐♦♥✿ F : R2 → R2 , ♠❡❛♥✐♥❣ F : (x, y) → (u, v) = (2x − 3y, x + 5y)

■♥ s❤♦rt✱ t❤✐s ✐s t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤✐s ❢✉♥❝t✐♦♥✿

F (x, y) = (2x − 3y, x + 5y) . ■♥ t❡r♠s ♦❢ ✈❡❝t♦rs✿

    2x − 3y x 7→ F : x + 5y y

❚❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ♠❛tr✐① ♦❢ F ❛r❡ r❡❛❞ ❢r♦♠ t❤❛t r❡♣r❡s❡♥t❛t✐♦♥✿   2 −3 F = 1 5 ❲❤❛t t❤✐s ❢✉♥❝t✐♦♥ ❞♦❡s t♦ t❤❡ ♣❧❛♥❡ r❡♠❛✐♥s t♦ ❜❡ ❞❡t❡r♠✐♥❡❞✳

✷✳✷✳ ❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡

✶✸✹

❊①❛♠♣❧❡ ✷✳✷✳✸✿ ✈❡rt✐❝❛❧ s❤✐❢t ❚❤❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❜② F (x, y) = (x, y + 3)

✐s ❛ ✈❡rt✐❝❛❧ s❤✐❢t ✿

✉♣ k

(x, y) −−−−−−→ (x, y + k)

❲❡ ✈✐s✉❛❧✐③❡ t❤❡s❡ tr❛♥s❢♦r♠❛t✐♦♥s ❜② ❞r❛✇✐♥❣ s♦♠❡t❤✐♥❣ ♦♥ t❤❡ ♦r✐❣✐♥❛❧ ♣❧❛♥❡ ✭t❤❡ ❞♦♠❛✐♥✮ ❛♥❞ t❤❡♥ s❡❡ ✇❤❛t t❤❛t ❧♦♦❦s ❧✐❦❡ ✐♥ t❤❡ ♥❡✇ ♣❧❛♥❡ ✭t❤❡ ❝♦✲❞♦♠❛✐♥✮✿

Pr❡❞✐❝t❛❜❧②✱ t❤❡ ❢♦r♠✉❧❛✿ F (x, y) = (x + a, y + b) = (x, y)+ < a, b > ,

❣✐✈❡s t❤❡ s❤✐❢t ❜② ✈❡❝t♦r < a, b >✳

❊①❡r❝✐s❡ ✷✳✷✳✹ ❊①♣❧❛✐♥ ✇❤② t❤❡r❡ ✐s ♥♦ ♠❛tr✐①✳

❊①❛♠♣❧❡ ✷✳✷✳✺✿ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧ ✢✐♣ ◆♦✇ t❤❡ ❤♦r✐③♦♥t❛❧ ✢✐♣✳ ❲❡ ❧✐❢t✱ t❤❡♥ ✢✐♣ t❤❡ s❤❡❡t ♦❢ ♣❛♣❡r ✇✐t❤ xy ✲♣❧❛♥❡ ♦♥ ✐t✱ ❛♥❞ ✜♥❛❧❧② ♣❧❛❝❡ ✐t ♦♥ t♦♣ ♦❢ ❛♥♦t❤❡r s✉❝❤ s❤❡❡t s♦ t❤❛t t❤❡ y ✲❛①❡s ❛❧✐❣♥✳ ■❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❣✐✈❡♥ ❜② F (x, y) = (−x, y) ,

t❤❡♥ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❇❡❧♦✇ ✇❡ ✐❧❧✉str❛t❡ t❤❡ ❢❛❝t t❤❛t t❤❡ ♣❛r❛❜♦❧❛✬s ❧❡❢t ❜r❛♥❝❤ ✐s ❛ ♠✐rr♦r ✐♠❛❣❡ ♦❢ ✐ts r✐❣❤t ❜r❛♥❝❤✿

❲❡ ❝❛♥ ❛❧s♦ r❡♣r❡s❡♥t t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ ✈✐❛ ✈❡❝t♦rs✿       (−1)x + 0y −x x = 7→ F : 0x + 1y y y

✷✳✷✳ ❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡

✶✸✺

❚❤❡♥✱ ✇❡ ❤❛✈❡ ✐ts ♠❛tr✐①✿ ■♥❞❡❡❞✱

 −1 0 . F = 0 1 

         −x −1 0 x x x . = = =F F y 0 1 y y y

◆❡①t✱ ❝♦♥s✐❞❡r ✈❡rt✐❝❛❧ ✢✐♣✳ ❲❡ ❧✐❢t✱ t❤❡♥ ✢✐♣ t❤❡ s❤❡❡t ♦❢ ♣❛♣❡r ✇✐t❤ xy ✲♣❧❛♥❡ ♦♥ ✐t✱ ❛♥❞ ✜♥❛❧❧② ♣❧❛❝❡ ✐t ♦♥ t♦♣ ♦❢ ❛♥♦t❤❡r s✉❝❤ s❤❡❡t s♦ t❤❛t t❤❡ x✲❛①❡s ❛❧✐❣♥✳ ■❢ G(x, y) = (x, −y) ,

t❤❡♥ ✇❡ ❤❛✈❡✿

✈❡rt✐❝❛❧ ✢✐♣

(x, y) −−−−−−−−−−→ (x, −y).

❲❡ ❝❛♥ ❛❧s♦ r❡♣r❡s❡♥t t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ ✈✐❛ ❛ ♠❛tr✐①✿  1 0 . G= 0 −1 

■♥❞❡❡❞✱

       x −1 0 x x . = = G −y 0 1 y y

❊①❛♠♣❧❡ ✷✳✷✳✻✿ ❝❡♥tr❛❧ s②♠♠❡tr②

❍♦✇ ❛❜♦✉t t❤❡ ✢✐♣ ❛❜♦✉t t❤❡ ♦r✐❣✐♥ ❄ ❚❤✐s ✐s t❤❡ ❢♦r♠✉❧❛✱ F (x, y) = (−x, −y) ,

♦❢ ✇❤❛t ✐s ❛❧s♦ ❦♥♦✇♥ ❛s t❤❡ ❝❡♥tr❛❧ s②♠♠❡tr②✿

❚❤✐s ✐s ✇❤❛t t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ❞♦❡s t♦ ❛ st❛r✿

✷✳✷✳

❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡

✶✸✻

❲❡ ❝❛♥ ❛❧s♦ r❡♣r❡s❡♥t t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ ✈✐❛ ✈❡❝t♦rs✿

      (−1)x + 0y −x x = 7→ F : 0x + (−1)y y y ❚❤❡♥✱ ✇❡ ❤❛✈❡ ✐ts ♠❛tr✐①✿

 −1 0 . F = 0 −1 

■♥❞❡❡❞✱

         −x x −1 0 x x . = = =F F −y 0 −1 y y y ❊①❛♠♣❧❡ ✷✳✷✳✼✿ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧ str❡t❝❤ ◆♦✇ t❤❡

❤♦r✐③♦♥t❛❧ str❡t❝❤

✳ ❲❡ ❣r❛❜ ❛ r✉❜❜❡r s❤❡❡t ❜② t❤❡ t♦♣ ❛♥❞ t❤❡ ❜♦tt♦♠ ❛♥❞ ♣✉❧❧ t❤❡♠ ❛♣❛rt

✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡

y ✲❛①✐s

❞♦❡s♥✬t ♠♦✈❡✳ ❍❡r❡✱

F (x, y) = (kx, y) .

❙✐♠✐❧❛r❧②✱ t❤❡

❤♦r✐③♦♥t❛❧ str❡t❝❤

✐s ❣✐✈❡♥ ❜②✿

G(x, y) = (x, ky) .

✷✳✷✳

✶✸✼

❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡

❲❡ ❝❛♥ ❛❧s♦ r❡♣r❡s❡♥t t❤❡s❡ tr❛♥s❢♦r♠❛t✐♦♥s ✈✐❛ ♠❛tr✐❝❡s✿     1 0 k 0 . ❛♥❞ G = F = 0 k 0 1 ❊①❛♠♣❧❡ ✷✳✷✳✽✿ r❡✲s❝❛❧✐♥❣

❍♦✇ ❛❜♦✉t t❤❡

✉♥✐❢♦r♠ str❡t❝❤

✭s❛♠❡ ✐♥ ❛❧❧ ❞✐r❡❝t✐♦♥s✮❄ ❚❤✐s ✐s ✐ts ❢♦r♠✉❧❛✿

F (x, y) = (kx, ky) .

❚❤❡ r❡s✉❧t ✐s r❡✲s❝❛❧✐♥❣✳ ❚❤❡ r❡❛s♦♥ ❝♦♠❡s ❢r♦♠ ✇❤❛t ✇❡ ❦♥♦✇ ❢r♦♠ ❣❡♦♠❡tr②✿ ❙✐♠✐❧❛r tr✐❛♥❣❧❡s ❤❛✈❡ t❤❡ s❛♠❡ ❛♥❣❧❡s✳ ❍❡r❡ ✐s ❛♥ ✐❧❧✉str❛t✐♦♥ ✇❤②✿

❲❡ ❝❛♥ ❛❧s♦ r❡♣r❡s❡♥t t❤✐s tr❛♥s❢♦r♠❛t✐♦♥s ✈✐❛ ❛ ♠❛tr✐①✿   k 0 . F = 0 k ❊①❡r❝✐s❡ ✷✳✷✳✾

❋✐♥❞ t❤❡ ♠❛tr✐① ❢♦r ❛ ❞✐s♣r♦♣♦rt✐♦♥❛❧ str❡t❝❤✳ ❏✉st ❛s ❜❡❢♦r❡✱ ✇❡ ♣✉t ❛❧❧ ♥❡✇❧② ✐♥tr♦❞✉❝❡❞ ❢✉♥❝t✐♦♥s ✐♥ ❜r♦❛❞ ❝❛t❡❣♦r✐❡s✳ ❙♦♠❡ ♦❢ t❤❡s❡ ❝❛t❡❣♦r✐❡s ✕ s✉❝❤ ❛s ♠♦♥♦t♦♥✐❝✐t② ✕ ❤❛✈❡ ❜❡❝♦♠❡ ✐rr❡❧❡✈❛♥t✳ ❖t❤❡rs ✕ s✉❝❤ ❛s s②♠♠❡tr② ✕ ❤❛✈❡ ❜❡❝♦♠❡ ❜② ❢❛r ♠♦r❡ ❝♦♠♣❧❡①✳ ❚✇♦ t❤❛t ✇✐❧❧ ❜❡ ♣✉rs✉❡❞ ❛r❡ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✳ ❘❡❝❛❧❧✿

• ❲❡ ❝❛❧❧ ❛ ❢✉♥❝t✐♦♥

• ❲❡ ❝❛❧❧ ❛ ❢✉♥❝t✐♦♥

♦♥❡✲t♦✲♦♥❡ ♦♥t♦

✐❢ t❤❡r❡ ✐s ♥♦ ♠♦r❡ t❤❛t ♦♥❡ ✐♥♣✉t ❢♦r ❡❛❝❤ ♦✉t♣✉t✳

✐❢ t❤❡r❡ ✐s ❛t ❧❡❛st ♦♥❡ ✐♥♣✉t ❢♦r ❡❛❝❤ ♦✉t♣✉t✳

❚❤❡ ❢✉♥❝t✐♦♥s ❛❜♦✈❡ ❛r❡ ❛❧❧ ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✳ ❊①❡r❝✐s❡ ✷✳✷✳✶✵

Pr♦✈❡ t❤❡ ❧❛st st❛t❡♠❡♥t✳

✷✳✷✳

❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡

✶✸✽

❊①❛♠♣❧❡ ✷✳✷✳✶✶✿ ♣r♦❥❡❝t✐♦♥s ❚❤❡ ❢✉♥❝t✐♦♥s t❤❛t ❛r❡ ♥♦t ♦♥❡✲t♦✲♦♥❡ ♦r ♦♥t♦ ❛r❡ t❤❡

♣r♦❥❡❝t✐♦♥s

✳ ❚❤❡r❡ ❛r❡ ❛t ❧❡❛st t✇♦ t②♣❡s✿

❚❤✐s ✐s t❤❡ ✈❡rt✐❝❛❧ ♦♥❡✿

F (x, y) = (x, 0) . ■t ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦♥ t❤❡

x✲❛①✐s✳

■t✬s ❛s ✐❢ t❤❡ s❤❡❡t ♦❢ t❤❡

xy ✲♣❧❛♥❡

✐s r♦❧❧❡❞ ✐♥t♦ ❛ t❤✐♥ s❝r♦❧❧✿

❲❡ ❝❛♥ ❛❧s♦ r❡♣r❡s❡♥t t❤✐s tr❛♥s❢♦r♠❛t✐♦♥s ✈✐❛ ❛ ♠❛tr✐①✿

 1 0 . F = 0 0 

❊①❡r❝✐s❡ ✷✳✷✳✶✷ ❋✐♥❞ t❤❡ ❢♦r♠✉❧❛ ❛♥❞ t❤❡ ♠❛tr✐① ❢♦r t❤❡ ♣r♦ ❥❡❝t✐♦♥ ♦♥ t❤❡

y ✲❛①✐s✳

❊①❡r❝✐s❡ ✷✳✷✳✶✸ ❋✐♥❞ t❤❡ ❢♦r♠✉❧❛ ❛♥❞ t❤❡ ♠❛tr✐① ❢♦r t❤❡ ♣r♦ ❥❡❝t✐♦♥ ♦♥ t❤❡ ❧✐♥❡

❊①❛♠♣❧❡ ✷✳✷✳✶✹✿ ❝♦❧❧❛♣s❡ ❋✐♥❛❧❧②✱ ✇❡ ❤❛✈❡ t❤❡

❝♦❧❧❛♣s❡



y = x✳

✷✳✷✳ ❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡

✶✸✾

■t ✐s ❛ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥✿ F (x, y) = (x0 , y0 ) .

■t✬s ❛s ✐❢ t❤❡ s❤❡❡t ✐s ❝r✉s❤❡❞ ✐♥t♦ ❛ t✐♥② ❜❛❧❧✿

❚❤❡r❡ ✐s ❛ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ✇❤❡♥ t❤✐s ♣♦✐♥t ✐s t❤❡ ♦r✐❣✐♥✿  0 0 . F = 0 0 

❊①❡r❝✐s❡ ✷✳✷✳✶✺ ❙❤♦✇ t❤❛t t❤❡r❡ ✐s ♥♦ ♠❛tr✐① ✉♥❧❡ss x0 = 0, y0 = 0✳ ❚❤❡ ♠❡❛♥✐♥❣ ♦❢ ❡❛❝❤ ♥✉♠❜❡r ✐♥ t❤❡ ♠❛tr✐① ❞❡♣❡♥❞s ♦♥ ✐ts ❧♦❝❛t✐♦♥✿ x y x a b y c d

❚❤✐s ✐s ❛ s♣❡❝✐❛❧ ❝❛s❡ t❤❛t ✇❡ ❤❛✈❡ ❧❡❛r♥❡❞ ❛❜♦✉t s♦ ❢❛r✿

▼❛tr✐① ❉❡❝♦♥str✉❝t✐♦♥ str❡t❝❤❡❞ ♦r ✢✐♣♣❡❞ x → a, 0 ← t❤❡r❡ ✐s ♥♦ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ x, y t❤❡r❡ ✐s ♥♦ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ x, y → 0, d ← str❡t❝❤❡❞ ♦r ✢✐♣♣❡❞ y ❚❤❡r❡ ❛r❡ ♠❛♥② ♠♦r❡ tr❛♥s❢♦r♠❛t✐♦♥s✱ ❤♦✇❡✈❡r✱ t❤❛t ❛r❡♥✬t ❝♦✈❡r❡❞ s♦ ❢❛r ❜❡❝❛✉s❡ t❤❡② ❝❛♥♥♦t ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ tr❛♥s❢♦r♠❛t✐♦♥s✳

❊①❛♠♣❧❡ ✷✳✷✳✶✻✿ ✢✐♣ ❛❜♦✉t ❞✐❛❣♦♥❛❧ ❆ ✢✐♣ ❛❜♦✉t t❤❡ ❧✐♥❡ x = y t❤❛t ❛♣♣❡❛r❡❞ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ✜♥❞✐♥❣ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥✿

❆s ✇❡ ❛❝q✉✐r❡ t❤❡ ✐♥✈❡rs❡ ❜② ✐♥t❡r❝❤❛♥❣✐♥❣ x ❛♥❞ y ✱ ✇❡ ❤❛✈❡ t❤❡ s❛♠❡ ❤❡r❡✿ (x, y) 7→ (y, x) .

❚❤❡ ♠❛tr✐① ✐s✿

 0 1 . F = 1 0 

✷✳✷✳

❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡

✶✹✵

❲❡ s❡❡ ❤❡r❡ s♦♠❡ ♥❡✇ ❢❡❛t✉r❡s✿

▼❛tr✐① ❉❡❝♦♥str✉❝t✐♦♥

x

❞♦❡s♥✬t ❞❡♣❡♥❞s ♦♥

y

❞❡♣❡♥❞s ♦♥

x → 0, 1 ← x ❞❡♣❡♥❞s ♦♥ y x → 1, 0 ← y ❞♦❡s♥✬t ❞❡♣❡♥❞s

♦♥

y

❊①❡r❝✐s❡ ✷✳✷✳✶✼ ❋✐♥❞ t❤❡ ♠❛tr✐① ❢♦r t❤✐s r♦t❛t✐♦♥✿

❊①❛♠♣❧❡ ✷✳✷✳✶✽✿ ❝♦♠♣♦s✐t✐♦♥s ▼♦r❡ ❝♦♠♣❧❡① tr❛♥s❢♦r♠❛t✐♦♥s✱ ❤♦✇❡✈❡r✱ ✇✐❧❧ r❡q✉✐r❡ ❢✉rt❤❡r st✉❞②✳ ❇❡❧♦✇✱ ✇❡ ✈✐s✉❛❧✐③❡ t❤❡ r♦t❛t✐♦♥ ✇✐t❤ ❛ str❡t❝❤ ✇✐t❤ ❛ ❢❛❝t♦r ♦❢

2✿

❊①❛♠♣❧❡ ✷✳✷✳✶✾✿ ❢♦❧❞✐♥❣ ❆♠♦♥❣ ♦t❤❡rs✱ ✇❡ ♠❛② ❝♦♥s✐❞❡r

❢♦❧❞✐♥❣

t❤❡ ♣❧❛♥❡✿

F (x, y) = (|x|, y) .

90✲❞❡❣r❡❡

✷✳✸✳

▲✐♥❡❛r ♦♣❡r❛t♦rs

✶✹✶

❊①❡r❝✐s❡ ✷✳✷✳✷✵

❈♦♥✜r♠ t❤❛t t❤✐s ❢✉♥❝t✐♦♥ ❝❛♥♥♦t ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ ♠❛tr✐①✳ ❖❢ ❝♦✉rs❡✱ ❛♥② ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ Rn ❝❛♥ ❜❡ ✕ ✐♥ ❛ s✐♠✐❧❛r ♠❛♥♥❡r ✕ r♦t❛t❡❞ ✭❛r♦✉♥❞ ✈❛r✐♦✉s ❛①❡s✮✱ str❡t❝❤❡❞ ✭✐♥ ✈❛r✐♦✉s ❞✐r❡❝t✐♦♥s✮✱ ♣r♦❥❡❝t❡❞ ✭♦♥t♦ ✈❛r✐♦✉s ❧✐♥❡s ♦r ♣❧❛♥❡s✮✱ ♦r ❝♦❧❧❛♣s❡❞✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ❤♦✇ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦♥ t❤❡ xy ✲♣❧❛♥❡

F (x, y, z) = (x, y, 0) ✇♦r❦s✿

❚❤✐s ✐s ❤♦✇ ♠❛♣s ❛r❡ ♠❛❞❡ ✭st❡r❡♦❣r❛♣❤✐❝ ♣r♦❥❡❝t✐♦♥✮✿

✷✳✸✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs

❲❡ ❝♦♥s✐❞❡r tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ❛s ❜❡❢♦r❡ ❜✉t ✇❡ t❤✐♥❦ ♦❢ t❤❡ ♣♦✐♥ts ♦♥ t❤❡ ♣❧❛♥❡ ❛s t❤❡ ❡♥❞s ♦❢ 2✲✈❡❝t♦rs✳ ■t ♠❛❦❡s ♥♦ ❞✐✛❡r❡♥❝❡ ✇❤❡♥ t❤❡② ❛r❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ t❤❡✐r ❝♦♠♣♦♥❡♥ts✿

< x, y > 7→ < u, v > ❚❤❡♥✱ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❢✉♥❝t✐♦♥s❄ ■t ✉s❡❞ t♦ ❜❡ t❤❡ ❝❛s❡ t❤❛t t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♦✉t♣✉t ❞❡♣❡♥❞ ♦♥ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ✐♥♣✉t✿ F (x, y) = (u, v) = 2x − 3y, x + 5y) .

✷✳✸✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs

✶✹✷

◆♦✇ ♦✉r ❢✉♥❝t✐♦♥ s❤♦✇s ❤♦✇ t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ♦✉t♣✉t ❞❡♣❡♥❞ ♦♥ t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ✐♥♣✉t✿ F < x, y >



=< u, v >=< 2x − 3y, x + 5y > .

❍❡r❡ < x, y > ✐s t❤❡ ✐♥♣✉t ❛♥❞ < u, v > ✐s t❤❡ ♦✉t♣✉t✿ ✐♥♣✉t

❢✉♥❝t✐♦♥

< x, y > →

❇♦t❤ ❛r❡ ✈❡❝t♦rs❀ ✐t✬s ❛ ✈❡❝t♦r ❢✉♥❝t✐♦♥ ✿

♦✉t♣✉t → < u, v >

F

F : R2 → R2 . ❲❛r♥✐♥❣✦

❲❡ ❝♦✉❧❞ ✐♥t❡r♣r❡t F ❛s ❛ ✈❡❝t♦r ✜❡❧❞✱ ❜✉t ✇❡ ✇♦♥✬t✳ ❊①❛♠♣❧❡ ✷✳✸✳✶✿ r❡✲s❝❛❧✐♥❣

❙♦♠❡ ♦❢ t❤❡ ❜❡♥❡✜ts ♦❢ t❤✐s ♣♦✐♥t ♦❢ ✈✐❡✇ ❛r❡ ✐♠♠❡❞✐❛t❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❡✈❡♥ ✐❢ ✇❡ ❞✐❞♥✬t ❦♥♦✇ t❤❛t t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ❣✐✈❡♥ ❜② F (x, y) = (2x, 2y)

✐s ❛ ✉♥✐❢♦r♠ str❡t❝❤✱ ✇❡ ❝❛♥ ❞✐s❝♦✈❡r t❤❛t ❢❛❝t ✇✐t❤ ♦✉r ❦♥♦✇❧❡❞❣❡ ♦❢ ✈❡❝t♦r ❛❧❣❡❜r❛✳ ❲❡ ✇r✐t❡ t❤✐s ❢✉♥❝t✐♦♥ ✐♥ t❡r♠s ♦❢ ✈❡❝t♦rs ❛♥❞ ❞✐s❝♦✈❡r s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ F (< x, y >) =< 2x, 2y >= 2 < x, y > .

■♥ ❢❛❝t✱ t❤✐s ✐❞❡❛ ✇✐❧❧ ✇♦r❦ ✐♥ ❛♥② ❞✐♠❡♥s✐♦♥✳ ❚❤✐s ✐s ❤♦✇ ②♦✉ str❡t❝❤ t❤❡ s♣❛❝❡ ❜② ❛ ❢❛❝t♦r ♦❢ 2✱ ✉s✐♥❣ t❤❡ ❝♦♠♣♦♥❡♥t✲❢r❡❡ ❛♣♣r♦❛❝❤✿ F : Rn → Rn ❣✐✈❡♥ ❜② F (X) = 2X .

❚❤❡ ❛♣♣r♦❛❝❤ ❣✐✈❡s ✉s ❛ ❜❡tt❡r r❡♣r❡s❡♥t❛t✐♦♥ ✇❤❡♥ t❤❡ ❢✉♥❝t✐♦♥s t❤❛t ♠❛❦❡ ✉♣ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ❤❛♣♣❡♥ t♦ ❜❡ ❧✐♥❡❛r✳ ▼❛tr✐❝❡s r❡♣❧② ♦♥ ✈❡❝t♦r ❛❧❣❡❜r❛✿    x a b =F ·X. · F (X) = f (x, y), g(x, y) = y c d 



❊①❡r❝✐s❡ ✷✳✸✳✷

❉♦❡s ✐t ✇♦r❦ ✇❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ✐s♥✬t ❧✐♥❡❛r❄ ❚r② F (x, y) =



 1 e , ✳ y x

❈♦♥✈❡rs❡❧②✱ ✐❢ ✇❡ ❞♦ ❤❛✈❡ ❛ ♠❛tr✐①✱ ✇❡ ❝❛♥ ❛❧✇❛②s ✉♥❞❡rst❛♥❞ ✐t ❛s ❛ ❢✉♥❝t✐♦♥✱ ❛s ❢♦❧❧♦✇s✿ 

a b c d

    ax + by x =< ax + by, cx + dy > , = cx + dy y

❢♦r s♦♠❡ a, b, c, d ✜①❡❞✳ ❙♦✱ ♠❛tr✐① F ❝♦♥t❛✐♥s ❛❧❧ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❢✉♥❝t✐♦♥ F ✳ ❖♥❡ ❝❛♥ t❤✐♥❦ ♦❢ F ✭❛ t❛❜❧❡✮ ❛s ❛♥ ❛❜❜r❡✈✐❛t✐♦♥ ♦❢ F X ✭❛ ❢♦r♠✉❧❛✮✳ ❲❛r♥✐♥❣✦

❲❡ ✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ ✉s❡ t❤❡ s❛♠❡ ❧❡tt❡r ❢♦r t❤❡ ❢✉♥❝✲ t✐♦♥ ❛♥❞ t❤❡ ♠❛tr✐①✳

❈❧❡❛r❧②✱ ❛ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜② ❛ ♠❛tr✐① ✐s ❛ s♣❡❝✐❛❧ ♦♥❡✳ ❲❤❛t ✐s s♦ s♣❡❝✐❛❧ ❛❜♦✉t ✐t❄

✷✳✸✳

▲✐♥❡❛r ♦♣❡r❛t♦rs

◆♦✇✱ t❤❡ ❞♦♠❛✐♥

R2

✶✹✸

♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s ❛

✈❡❝t♦r s♣❛❝❡✱

❛♥❞ s♦ ✐s ✐ts ❝♦❞♦♠❛✐♥✳ ❍♦✇ ❞♦❡s s✉❝❤ ❛ ❢✉♥❝t✐♦♥

✐♥t❡r❛❝t ✇✐t❤ t❤❡ ❛❧❣❡❜r❛ ♦❢ t❤❡s❡ t✇♦ s♣❛❝❡s❄ ❲❤❛t ❤❛♣♣❡♥s t♦ t❤❡

✈❡❝t♦r ♦♣❡r❛t✐♦♥s

✉♥❞❡r

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛❞❞✐t✐♦♥ ❛♥❞ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❝❛rr✐❡❞ ♦✉t ✐♥ t❤❡ ❞♦♠❛✐♥ s♣❛❝❡ ♦❢

F❄

F✳

❖♥❝❡

F

❤❛s

tr❛♥s❢♦r♠❡❞ t❤❡ ♣❧❛♥❡✱ ✇❤❛t ❞♦ t❤❡s❡ ♦♣❡r❛t✐♦♥s ❧♦♦❦ ❧✐❦❡ ♥♦✇❄ ❊①❛♠♣❧❡ ✷✳✸✳✸✿ ❞✐♠❡♥s✐♦♥

❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜②

1

2✱ f (x) = 2x

✏♣r❡s❡r✈❡s ❛❞❞✐t✐♦♥✑✿

f (x + x′ ) = 2(x + x′ ) = 2x + 2x′ = f (x) + f (x′ ) . ❆❢t❡r ❛❧❧✱ t❤✐s ✐s ❥✉st ❛ str❡t❝❤ ❜② ❛ ❢❛❝t♦r ♦❢

2✳

❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ✐s ❥✉st ❛♥ ❛❜❜r❡✈✐❛t✐♦♥ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐❛❣r❛♠✿

x,x′  ·2 y

+

′ x+ x  ·2 y

−−−−→ +

2x, 2x′ −−−−→ 2x + 2x′ = 2(x + x′ ) ■♥ t❤❡ ❞✐❛❣r❛♠✱ ✇❡ st❛rt ✇✐t❤ ❛ ♣❛✐r ♦❢ ♥✉♠❜❡rs ❛t t❤❡ t♦♣ ❧❡❢t ❛♥❞ t❤❡♥ ✇❡ ♣r♦❝❡❡❞ ✐♥ t✇♦ ✇❛②s✿

• •

❘✐❣❤t✿ ❆❞❞ t❤❡♠✳ ❚❤❡♥ ❞♦✇♥✿ ❆♣♣❧② t❤❡ ❢✉♥❝t✐♦♥ t♦ t❤❡ r❡s✉❧t✳ ❉♦✇♥✿ ❆♣♣❧② t❤❡ ❢✉♥❝t✐♦♥ t♦ t❤❡♠✳ ❚❤❡♥ r✐❣❤t✿ ❆❞❞ t❤❡ r❡s✉❧ts✳

❆ s❤✐❢t ❜②

1✱ f (x) = x + 1✱

❞♦❡s♥✬t ♣r❡s❡r✈❡ ❛❞❞✐t✐♦♥✿

f (x + x′ ) = (x + x′ ) + 1 = x + x′ + 1 6= x + x′ + 2 = (x + 1) + (x′ + 1) = f (x) + f (x′ ) . ❊①❡r❝✐s❡ ✷✳✸✳✹

❲❤❛t ❡✛❡❝t ❞♦❡s t❤❡ ❢✉♥❝t✐♦♥

f :R→R

❣✐✈❡♥ ❜②

f (x) = 2x

❤❛✈❡ ♦♥ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐♥ ✐ts ❞♦♠❛✐♥❄

❊①❛♠♣❧❡ ✷✳✸✳✺✿ ❛❞❞✐t✐♦♥ ✉♥❞❡r r♦t❛t✐♦♥

♣❧❛♥❡ ✐s tr❛♥s❢♦r♠❡❞❄ r♦t❛t❡❞✱ ❛s ✐❢ ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r✱ ✐t ✇✐❧❧ r❡♠❛✐♥ ❛♥ ❛❞❞✐t✐♦♥ ❞✐❛❣r❛♠✿

❲❤❛t ❤❛♣♣❡♥s t♦ ❛♥ ❛❞❞✐t✐♦♥ ❞✐❛❣r❛♠ ✭t❤❡ ♣❛r❛❧❧❡❧♦❣r❛♠ ❝♦♥str✉❝t✐♦♥✮ ✇❤❡♥ t❤❡ ■❢ s✉❝❤ ❛ ❞✐❛❣r❛♠ ✐s

❲❡ s❡❡ t❤❡ ♣❛r❛❧❧❡❧♦❣r❛♠ r✉❧❡ ♦❢ ❛❞❞✐t✐♦♥ ♦♥ t❤❡ ❧❡❢t ❛♥❞ ♦♥ t❤❡ r✐❣❤t✳ ❚❤✐s ✐s t❤❡ ❛❧❣❡❜r❛✿

A, B   r♦t❛t❡❞ y

+

−−−−→ +

A+ B  r♦t❛t❡❞ y

F (A), F (B) −−−−→ F (A) + F (B) = F (A + B) ❲❤❛t ❤❛♣♣❡♥s t♦ ❛♥ ❛❞❞✐t✐♦♥ ❞✐❛❣r❛♠ ✇❤❡♥ t❤❡ ✈❡❝t♦r s♣❛❝❡ ✐s tr❛♥s❢♦r♠❡❞❄ ❲❤❡♥ ✐t ✐s st✐❧❧ ❛♥ ❛❞❞✐t✐♦♥ ❞✐❛❣r❛♠✱ t❤✐s ✐s t❤❡ ❧❛♥❣✉❛❣❡ ✇❡ ✇✐❧❧ ✉s❡✿

✷✳✸✳

▲✐♥❡❛r ♦♣❡r❛t♦rs

✶✹✹

❉❡✜♥✐t✐♦♥ ✷✳✸✳✻✿ ❛❞❞✐t✐♦♥ ✐s ♣r❡s❡r✈❡❞ ❲❡ s❛② t❤❛t

❛❞❞✐t✐♦♥ ✐s ♣r❡s❡r✈❡❞

✉♥❞❡r ❛ ❢✉♥❝t✐♦♥

F : Rn → Rm

✐❢

F (A + B) = F (A) + F (B) ❢♦r ❛♥② ✈❡❝t♦rs

A

❛♥❞

B✳

❊①❛♠♣❧❡ ✷✳✸✳✼✿ str❡t❝❤ ❞✐♠❡♥s✐♦♥ ✷ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ❡①❛♠♣❧❡ ♦❢ ❛ str❡t❝❤ ❜❡❧♦✇ s❤♦✇s t❤❛t t❤❡ tr✐❛♥❣❧❡s ♦❢ t❤❡ ❞✐❛❣r❛♠✱ ✐❢ ♥♦t ✐❞❡♥t✐❝❛❧✱ ❛r❡

s✐♠✐❧❛r

t♦ t❤❡ ♦r✐❣✐♥❛❧✿

■t✬s ❛s ✐❢ ✇❡ ❥✉st st❡♣♣❡❞ ❛✇❛② ❢r♦♠ t❤❡ ♣✐❡❝❡ ♦❢ ♣❛♣❡r t❤❛t ❤❛s t❤❡ ❛❞❞✐t✐♦♥ ❞✐❛❣r❛♠ ♦♥ ✐t ♦r ♣✉t t❤❡ ❞✐❛❣r❛♠ ✉♥❞❡r ❛ ♠❛❣♥✐❢②✐♥❣ ❣❧❛ss✳

❊①❛♠♣❧❡ ✷✳✸✳✽✿ r❡✢❡❝t✐♦♥ ❞✐♠❡♥s✐♦♥ ✷ ❲❡ ❝❛♥ ❛❧s♦ s❡❡ t❤❡ ❛❞❞✐t✐♦♥ ❞✐❛❣r❛♠ ✐♥ t❤❡ ♠✐rr♦r✱ ❛♥❞ ✐t✬s st✐❧❧ ❛♥ ❛❞❞✐t✐♦♥ ❞✐❛❣r❛♠✿

❊①❡r❝✐s❡ ✷✳✸✳✾ ❙❤♦✇ t❤❛t ❛ ❢♦❧❞ ❞♦❡s♥✬t ♣r❡s❡r✈❡ ✈❡❝t♦r ❛❞❞✐t✐♦♥✳ ❙✉❣❣❡st ♦t❤❡r ❡①❛♠♣❧❡s✳

❲❤❛t ❛❜♦✉t t❤❡ ❣❡♥❡r❛❧ ❝❛s❡❄

❚❤❡♦r❡♠ ✷✳✸✳✶✵✿ Pr❡s❡r✈✐♥❣ ❆❞❞✐t✐♦♥ ■❢ ❛ ❢✉♥❝t✐♦♥ ❛❞❞✐t✐♦♥✳

F : Rn → Rm

✐s ❣✐✈❡♥ ❜② ❛ ♠❛tr✐①✱

F (X) = F X ✱

✐t ♣r❡s❡r✈❡s

✷✳✸✳

▲✐♥❡❛r ♦♣❡r❛t♦rs

✶✹✺

Pr♦♦❢✳

❈♦♥s✐❞❡r

F : R 2 → R2

❛♥❞ t✇♦ ✐♥♣✉t ✈❡❝t♦rs✿

  x A= y

❛♥❞

 ′ x B= ′ . y

▲❡t✬s ❝♦♥✜r♠ t❤❡ ❢♦r♠✉❧❛✿

F (A + B) = F (A) + F (B) . ▲❡t✬s ❝♦♠♣❛r❡✿

▲❡❢t✲❤❛♥❞ s✐❞❡✿

     ′  x   x   F   +    y y′        ′  a b   x   x   =    +    c d y y′    ′ a b  x + x  =   ′ c d y+y   ′ ′ a(x + x ) + b(y + y ) =  ′ ′ c(x + x ) + d(y + y )

❘✐❣❤t✲❤❛♥❞ s✐❞❡✿

    ′ x   x F  +F   y y′       ′  a b   x  a b   x  =   +    c d y c d y′     ′ ′ ax + by  ax + by  = +  ′ ′ cx + dy cx + dy   ′ ′ ax + by + ax + by  =  ′ ′ cx + dy + cx + dy

❚❤❡s❡ ❛r❡ t❤❡ s❛♠❡✱ ❛❢t❡r ❢❛❝t♦r✐♥❣✳

❊①❛♠♣❧❡ ✷✳✸✳✶✶✿ str❡t❝❤ ❞✐♠❡♥s✐♦♥ ✷

❍❡r❡ ✐s ❛ ❝♦♥✜r♠❛t✐♦♥ ♦❢ t❤❡ r❡s✉❧t ❢♦r ❛ ❤♦r✐③♦♥t❛❧ str❡t❝❤✿

❲❡ s✐♠♣❧② ❝♦♠♣❛r❡ ✇❤❛t ❤❛♣♣❡♥s ✐♥ t❤❡ ❞♦♠❛✐♥ ✇✐t❤ ✐ts ✏r❡✢❡❝t✐♦♥✑ ✐♥ t❤❡ ❝♦❞♦♠❛✐♥✳

❚❤❡ ❞✐❛❣r❛♠ ♦❢ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ♠✉❝❤ s✐♠♣❧❡r✿

✷✳✸✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs

✶✹✻

■t ✐s ❛ str❡t❝❤ ♦❢ t❤❡ ✈❡❝t♦r❀ r♦t❛t❡❞ ♦r str❡t❝❤❡❞✱ ✐t r❡♠❛✐♥s ❛ str❡t❝❤✳ ❚❤✐s ✐s t❤❡ ❧❛♥❣✉❛❣❡ ✇❡ ✇✐❧❧ ✉s❡✿

❉❡✜♥✐t✐♦♥ ✷✳✸✳✶✷✿ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ♣r❡s❡r✈❡❞ ❲❡ s❛② t❤❛t s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ♣r❡s❡r✈❡❞ ✉♥❞❡r ❛ ❢✉♥❝t✐♦♥ F : Rn → Rm ✐❢ F (kA) = kF (A)

❢♦r ❛♥② ✈❡❝t♦r X ❛♥❞ r❡❛❧ k✳ ❚❤❡ ❢♦r♠✉❧❛s ✐♥ t❤❡ t✇♦ ❞❡✜♥✐t✐♦♥s ❛r❡ ❥✉st ❛❜❜r❡✈✐❛t✐♦♥s ♦❢ t❤❡s❡ ❞✐❛❣r❛♠s✿ A,B  F y

+

−−−−→

A+ B  F y

+

F (A), F (B) −−−−→ F (A) + F (B) = F (A + B)

A   F y

·k

−−−−→ ·k

kA   F y

F (A) −−−−→ kF (A) = F (kA)

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♦r❞❡r ♦❢ t❤❡s❡ ♦♣❡r❛t✐♦♥s ♠❛❦❡s ♥♦ ❞✐✛❡r❡♥❝❡✳

❊①❛♠♣❧❡ ✷✳✸✳✶✸✿ ♠♦t✐♦♥s ❚❤❡ s❛♠❡ ❝♦♥❝❧✉s✐♦♥ ✐s q✉✐❝❦❧② r❡❛❝❤❡❞ ❢♦r t❤❡ ✢✐♣ ❛♥❞ ♦t❤❡r ♠♦t✐♦♥s✿ ❚❤❡ tr✐❛♥❣❧❡s ♦❢ t❤❡ ♥❡✇ ❞✐❛❣r❛♠ ❛r❡ ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ♦r✐❣✐♥❛❧✳ ❲❡ ❝❛♥ ❥✉st ✐♠❛❣✐♥❡ t❤❛t t❤❡ ❛❞❞✐t✐♦♥ ❞✐❛❣r❛♠ ✐s ❞r❛✇♥ ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r ✇✐t❤ ♥♦ ❣r✐❞✱ ✇❤✐❝❤ t❤❡♥ ❤❛s ❜❡❡♥ r♦t❛t❡❞✿

❚❤❡♦r❡♠ ✷✳✸✳✶✹✿ Pr❡s❡r✈✐♥❣ ❙❝❛❧❛r ▼✉❧t✐♣❧✐❝❛t✐♦♥ ■❢ ❛ ❢✉♥❝t✐♦♥

F : Rn → Rn

✐s ❣✐✈❡♥ ❜② ❛ ♠❛tr✐①✱

F (A) = F A✱

✐t ♣r❡s❡r✈❡s s❝❛❧❛r

♠✉❧t✐♣❧✐❝❛t✐♦♥✳

Pr♦♦❢✳ ❈♦♥s✐❞❡r ❛♥ ✐♥♣✉t ✈❡❝t♦r X =< x, y > ❛♥❞ ❛ s❝❛❧❛r k✳ ❚❤❡♥✱ 

a b c d

               a b x ax + by akx + bky kx a b x . =k =k = = k c d y cx + dy ckx + dky ky c d y

◆♦✇✱ t❤✐s ✐s t❤❡ ❣❡♥❡r❛❧ ❝❛s❡✿

✷✳✸✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs

✶✹✼

❉❡✜♥✐t✐♦♥ ✷✳✸✳✶✺✿ ❧✐♥❡❛r ♦♣❡r❛t♦r ❆ ❢✉♥❝t✐♦♥ F : Rn → Rm t❤❛t ♣r❡s❡r✈❡s ❜♦t❤ ❛❞❞✐t✐♦♥ ❛♥❞ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ❝❛❧❧❡❞ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ✭♦r ❛ ❧✐♥❡❛r ♠❛♣✮❀ ✐✳❡✳✱✿ F (U + V ) = F (U ) + F (V ) F (kV ) = kF (V )

❲❛r♥✐♥❣✦ y = ax + b ❤❛s ❜❡❡♥ ❝❛❧❧❡❞ ❛ ✏❧✐♥❡❛r ◆♦✇✱ y = ax ✐s ❝❛❧❧❡❞ ❛ ✏❧✐♥❡❛r ♦♣❡r❛t♦r✑✳

Pr❡✈✐♦✉s❧②✱ ❢✉♥❝t✐♦♥✑✳

❲❡ ❝♦♠❜✐♥❡ t❤❡ t✇♦ ♦♣❡r❛t✐♦♥s t♦❣❡t❤❡r✿

❚❤❡♦r❡♠ ✷✳✸✳✶✻✿ ▲✐♥❡❛r ❖♣❡r❛t♦rs ❛♥❞ ▲✐♥❡❛r ❈♦♠❜✐♥❛t✐♦♥s ❆ ❧✐♥❡❛r ♦♣❡r❛t♦r F : Rn → Rm ♣r❡s❡r✈❡s ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s❀ ✐✳❡✳✱ X = xA + yB =⇒ F (X) = xF (A) + yF (B) ,

❢♦r ❛♥② ✈❡❝t♦rs A ❛♥❞ B ❛♥❞ ❛♥② r❡❛❧ ❝♦❡✣❝✐❡♥ts x ❛♥❞ y ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❞✐❛❣r❛♠ ♦❢ ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ✇✐❧❧ r❡♠❛✐♥ s✉❝❤ ✉♥❞❡r ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✿

❊①❡r❝✐s❡ ✷✳✸✳✶✼ ❉❡s❝r✐❜❡ ✇❤❛t t❤✐s ❧✐♥❡❛r ♦♣❡r❛t♦r ❞♦❡s ❛♥❞ ✜♥❞ ✐ts ♠❛tr✐①✳ ❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ ♦✉r ❛♥❛❧②s✐s✳

❚❤❡♦r❡♠ ✷✳✸✳✶✽✿ ▲✐♥❡❛r ❖♣❡r❛t♦rs ✈s✳ ▼❛tr✐❝❡s • ❚❤❡ ❢✉♥❝t✐♦♥ F : R2 → R2 ❞❡✜♥❡❞ ✈✐❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛ 2 × 2 ♠❛tr✐① F ✱ F (X) = F X ,

✐s ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✳

✷✳✸✳

▲✐♥❡❛r ♦♣❡r❛t♦rs

✶✹✽

• ❈♦♥✈❡rs❡❧②✱ ❡✈❡r② ❧✐♥❡❛r ♦♣❡r❛t♦r F : R2 → R2 ✐s ❞❡✜♥❡❞ ✈✐❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② s♦♠❡ 2 × 2 ♠❛tr✐①✳

Pr♦♦❢✳ ❚❤❡ ✜rst ♣❛rt ♦❢ t❤❡ t❤❡♦r❡♠ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ t✇♦ t❤❡♦r❡♠s ❛❜♦✈❡✳ ❚❤❡ ❝♦♥✈❡rs❡ ✐s ♣r♦✈❡♥ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳

❲❛r♥✐♥❣✦ ▲✐♥❡❛r ♦♣❡r❛t♦rs ❛♥❞ ♠❛tr✐❝❡s ❛r❡♥✬t ✐♥t❡r❝❤❛♥❣❡✲ ❛❜❧❡✱ ❜❡❝❛✉s❡ ♠❛tr✐❝❡s ❡♠❡r❣❡ ♦♥❧② ✇❤❡♥ ❛ ❈❛rt❡✲ s✐❛♥ s②st❡♠ ❤❛s ❜❡❡♥ s♣❡❝✐✜❡❞✳

❈♦r♦❧❧❛r② ✷✳✸✳✶✾✿ ▲✐♥❡❛r ❖♣❡r❛t♦r ❛t 0 ❆ ❧✐♥❡❛r ♦♣❡r❛t♦r t❛❦❡s t❤❡ ③❡r♦ ✈❡❝t♦r t♦ t❤❡ ③❡r♦ ✈❡❝t♦r✿ F (0) = 0 .

❊①❡r❝✐s❡ ✷✳✸✳✷✵ Pr♦✈❡ t❤❡ ❝♦r♦❧❧❛r② ✭❛✮ ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✱ ✭❜✮ ❜② ❡①❛♠✐♥✐♥❣ t❤❡ ♠❛tr✐① ♠✉❧t✐♣❧✐✲ ❝❛t✐♦♥✳ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ✇✐❧❧ ❜❡ ✈✐s✐❜❧❡ ✐♥ t❤❡ ❡①❛♠♣❧❡s ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳ ❚❤❡ ❧❛tt❡r ♣❛rt ♦❢ t❤❡ t❤❡♦r❡♠ ✐s ♠❛t❡r✐❛❧✐③❡❞ ✇❤❡♥ ❛ ♠❛tr✐① ✐s ❢♦✉♥❞ ❢♦r ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ❞❡s❝r✐❜❡❞ ❜② ✇❤❛t ✐t ❞♦❡s✳ ❲❡ st❛rt ✇✐t❤ ❛ ❝♦✉♣❧❡ ♦❢ s✐♠♣❧❡ ❡①❛♠♣❧❡s✳ ▲❡t✬s ♥♦t ❢♦r❣❡t✿

◮ ▲✐♥❡❛r ♦♣❡r❛t♦rs ❛r❡ ❢✉♥❝t✐♦♥s✳ ❚❤❡ t✇♦ s✐♠♣❧❡st ❢✉♥❝t✐♦♥s ✕ ♥♦ ♠❛tt❡r ✇❤❛t t❤❡ ❞♦♠❛✐♥ ❛♥❞ ❝♦❞♦♠❛✐♥ ❛r❡ ✕ ❛r❡ t❤❡ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ✐❞❡♥t✐t② ❢✉♥❝t✐♦♥✳ ❍♦✇❡✈❡r✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛st ❝♦r♦❧❧❛r②✱ t❤❡r❡ ❝❛♥ ❜❡ ♦♥❧② ♦♥❡ ❝♦♥st❛♥t✱ 0✱ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ♦♥❧② ♦♥❡ ❝♦♥st❛♥t ❧✐♥❡❛r ♦♣❡r❛t♦r✳ ❚❤✐s ✐s t❤❡ s✐♠♣❧❡st ❧✐♥❡❛r ♦♣❡r❛t♦r✿

❉❡✜♥✐t✐♦♥ ✷✳✸✳✷✶✿ ③❡r♦ ♦♣❡r❛t♦r ❚❤❡ ❢✉♥❝t✐♦♥ F : Rn → Rm t❤❛t t❛❦❡s ❡✈❡r② ✈❡❝t♦r t♦ t❤❡ ③❡r♦ ✈❡❝t♦r✱

F (0) = 0 , ✐s ❝❛❧❧❡❞ t❤❡

③❡r♦ ♦♣❡r❛t♦r✳

❚❤❡ ♥♦t❛t✐♦♥ ✐s ❛s ❢♦❧❧♦✇s✿

0 : Rn → Rm , ❚❤❡ ♠❛tr✐① ♦❢ t❤❡ ③❡r♦ ♦♣❡r❛t♦r ❝♦♥s✐sts✱ ♦❢ ❝♦✉rs❡✱ ♦❢ ❛❧❧ ③❡r♦s✿



0 0 0= ... 0

0 0 ... 0

0 0 ... 0

... ... ... ...

 0 0  ... 0

0(X) = 0 .

✷✳✹✳

❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs

✶✹✾

❲❡ ❝❛♥ ✇r✐t❡✿

0ij = 0 . ❲❤❛t ❛❜♦✉t t❤❡ ✐❞❡♥t✐t② ❢✉♥❝t✐♦♥❄ ❚❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡ ❞♦♠❛✐♥ ❛♥❞ t❤❡ ❝♦❞♦♠❛✐♥ ♠✉st ❜❡ t❤❡ s❛♠❡✿

❉❡✜♥✐t✐♦♥ ✷✳✸✳✷✷✿ ✐❞❡♥t✐t② ♦♣❡r❛t♦r ❚❤❡ ❢✉♥❝t✐♦♥

F : Rn → Rn

t❤❛t t❛❦❡s ❡✈❡r② ✈❡❝t♦r t♦ ✐ts❡❧❢✱

F (X) = X , ✐s ❝❛❧❧❡❞ t❤❡

✐❞❡♥t✐t② ♦♣❡r❛t♦r✳

❚❤❡ ♥♦t❛t✐♦♥ ✐s ❛s ❢♦❧❧♦✇s✿

I : Rn → R n ,

I(X) = X .

❚❤❡ ♠❛tr✐① ♦❢ t❤❡ ✐❞❡♥t✐t② ♦♣❡r❛t♦r ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿



1 0 I= ... 0 ■t ❤❛s

1✬s

♦♥ t❤❡ ♠❛✐♥ ❞✐❛❣♦♥❛❧ ❛♥❞

0✬s

0 1 ... 0

0 0 ... 0

... ... ... ...

 0 0  ... 1

❡❧s❡✇❤❡r❡✳ ❲❡ ❝❛♥ ✇r✐t❡✿

Iij =

(

1 0

✐❢ ✐❢

i=j, i 6= j .

✷✳✹✳ ❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs

❚❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ❛r❡ ✐❧❧✉str❛t❡❞ ♣r❡✈✐♦✉s❧② ✇✐t❤ ❝✉r✈❡s ♣❧♦tt❡❞ ♦♥ t❤❡ ♦r✐❣✐♥❛❧ ♣❧❛♥❡ ❛♥❞ t❤❡♥ s❡❡♥ tr❛♥s❢♦r♠❡❞✳

❇❡②♦♥❞ ❥✉st s❛②✐♥❣ t❤❛t t❤❡ ❝✉r✈❡ ✇❛s ❞r❛✇♥ ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r ♦r ❛ s❤❡❡t ♦❢

r✉❜❜❡r✱ ✇❤❛t ❡①❛❝t❧② ❤❛♣♣❡♥s t♦ t❤♦s❡ ❝✉r✈❡s❄ ❯s✐♥❣ t❤❡ ❣r❛♣❤s ♦❢ ❢✉♥❝t✐♦♥s t♦ r❡♣r❡s❡♥t t❤❡s❡ ❝✉r✈❡s ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ❢❛✐❧s✳ ❡①❛♠♣❧❡✱ r♦t❛t✐♥❣ s✉❝❤ ❛ ❝✉r✈❡✱

u = g(v)✱

✐♥ t❤❡ ❝♦❞♦♠❛✐♥✿

y = f (x)✱

❋♦r

✐s ❧✐❦❡❧② t♦ ♣r♦❞✉❝❡ ❛ ❝✉r✈❡ t❤❛t ✐s♥✬t t❤❡ ❣r❛♣❤ ♦❢ ❛♥② ❢✉♥❝t✐♦♥✱

✷✳✹✳

❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs

❖✉r ❝❤♦✐❝❡ ✐s✱ t❤❡♥✱ ♣❛r❛♠❡tr✐❝ ❣✐✈❡♥ ❜②✿

❝✉r✈❡s ✿

✶✺✵

P : R → R2 ,

X = P (t) ♦r x = x(t), y = y(t) .

❊①❛♠♣❧❡ ✷✳✹✳✶✿ ♥♦♥✲✉♥✐❢♦r♠ r❡✲s❝❛❧❡

▲❡t✬s ❝♦♥s✐❞❡r t❤✐s tr❛♥s❢♦r♠❛t✐♦♥✿ ❍❡r❡✱ t❤✐s ❢✉♥❝t✐♦♥ ✐s ❣✐✈❡♥ ❜② t❤❡ ♠❛tr✐①✿



u = 2x v = 4y



2 0 F = 0 4



❲❡ ❝❛♥ s❡❡ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❧✐♥❡s t❤r♦✉❣❤ 0✿

❚❤❡② ❛❧❧ r❡♠❛✐♥ str❛✐❣❤t ❡✈❡♥ t❤♦✉❣❤ s♦♠❡ ♦❢ t❤❡♠ r♦t❛t❡✳ ▲❡t✬s t❛❦❡ ❛ ❝❧♦s❡r ❧♦♦❦ ❛t t❤❡ str❛✐❣❤t ❧✐♥❡s t❤❛t ♣❛ss t❤r♦✉❣❤ t❤❡ ♦r✐❣✐♥✳ ❚❤❡ ❡q✉❛t✐♦♥ ♦❢ s✉❝❤ ❛ ❧✐♥❡ ✐s ✈❡r② s✐♠♣❧❡✿ P (t) = tV ,

✇❤❡r❡ V 6= 0 ✐s s♦♠❡ ✜①❡❞ ✈❡❝t♦r✳ ❆s V ✐s tr❛♥s❢♦r♠❡❞ ❜② F ✱ t❤❡♥ s♦ ❛r❡ ❛❧❧ ♦❢ ✐ts ♠✉❧t✐♣❧❡s✿

■t✬s ❛♥♦t❤❡r str❛✐❣❤t ❧✐♥❡✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs ❞♦♥✬t ❜❡♥❞✦ ❲❡ ❝♦♥✜r♠ t❤❡ ♦❜s❡r✈❛t✐♦♥✿ ❚❤❡♦r❡♠ ✷✳✹✳✷✿ ■♠❛❣❡s ♦❢ ▲✐♥❡s ❚❤❡ ✐♠❛❣❡ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡ t❤r♦✉❣❤ t❤❡ ♦r✐❣✐♥ ✉♥❞❡r ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ✇✐❧❧ ♣r♦❞✉❝❡ ❛♥♦t❤❡r str❛✐❣❤t ❧✐♥❡ t❤♦r♦✉❣❤ t❤❡ ♦r✐❣✐♥✳

✷✳✹✳

❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs

✶✺✶

Pr♦♦❢✳

❙✉♣♣♦s❡

F

✐s s✉❝❤ ❛♥ ♦♣❡r❛t♦r ❛♥❞

V

P (t) = tV ❚❤✐s ✐s ❛ ❧✐♥❡ ✇✐t❤

F (V )

✐s t❤❡ ❞✐r❡❝t✐♦♥ ✈❡❝t♦r ♦❢ t❤❡ ❧✐♥❡✳ ❚❤❡♥✿

F

−−−−→ (F ◦ P )(t) = F (tV ) = tF (V )

❛s t❤❡ ❞✐r❡❝t✐♦♥ ✈❡❝t♦r✳

■♥ ❣❡♥❡r❛❧✱ ✇❡ ✇✐t♥❡ss t❤❡ ❢♦❧❧♦✇✐♥❣✿ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

F

−−−−→

♣❛r❛♠❡tr✐❝ ❝✉r✈❡

▲❡t✬s ❝♦♥s✐❞❡r t✇♦ ♠♦r❡ ❜❛s✐❝ t②♣❡s ♦❢ ❝✉r✈❡s✳ ❚❤❡ ❣r❛♣❤ ♦❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s ❝❛❧❧❡❞ ❛ ✏♣❛r❛❜♦❧❛✑✳ ❚❤✐s ✐s ✇❤❛t ✇❡ ❝❛r❡ ❛❜♦✉t✿

t❤❡ ♣❛r❛❜♦❧❛ ♦❢ f (x) = x2 ✈✐❛ ❛ ✈❡rt✐❝❛❧ str❡t❝❤ ♦r s❤r✐♥❦✳ ■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t ❡✈❡r② q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ❤❛s ✐ts ✈❡rt❡① ❢♦r♠ ✿ ◮

❆♥② ♣❛r❛❜♦❧❛ ❝❛♥ ❜❡ ❛❝q✉✐r❡❞ ❢r♦♠

f (x) = a(x − h)2 + k .

❚❤❡r❡ ❛r❡ ♠❛♥② ❝✐r❝❧❡s ♦♥ t❤❡ ♣❧❛♥❡✳ ❚❤✐s ✐s ✇❤❛t ✇❡ ❝❛r❡ ❛❜♦✉t✿



❆♥② ❝✐r❝❧❡ ❝❛♥ ❜❡ ❛❝q✉✐r❡❞ ❢r♦♠

t❤❡

❝✐r❝❧❡ ♦❢

■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t ❡✈❡r② ❝✐r❝❧❡ ❤❛s ✐ts

x2 + y 2 = 1

✈✐❛ ❛ ✉♥✐❢♦r♠ str❡t❝❤ ♦r s❤r✐♥❦✳

❝❡♥t❡r❡❞ ❢♦r♠ ✿

(x − h)2 + (y − k)2 = r2 .

❊①❡r❝✐s❡ ✷✳✹✳✸

❊①♣❧❛✐♥ ❤♦✇ t❤❡ ❣r❛♣❤ ♦❢ ❛♥② ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥s✳ ❙❛♠❡ ❢♦r t❤❡ ❧♦❣❛r✐t❤♠s✳

ax

❝❛♥ ❜❡ ❛❝q✉✐r❡❞ ❢r♦♠ t❤❡ ♥❛t✉r❛❧ ♦♥❡

ex

✈✐❛

✷✳✹✳

❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs

✶✺✷

❲❡ t❛❦❡ ❛❞✈❛♥t❛❣❡ ♦❢ ♦✉r ❦♥♦✇❧❡❞❣❡ ♦❢ ✈❡❝t♦r ❛❧❣❡❜r❛ t♦ s✉♠♠❛r✐③❡ t❤❡s❡ t❤r❡❡ ❝❛s❡s✿ ✏t❡♠♣❧❛t❡✑

/

❧✐♥❡✿

y=x ⌣

♣❛r❛❜♦❧❛✿

y = x2 ◦

❝✐r❝❧❡✿

x2 + y 2 = 1

r❡❧❛t✐♦♥

♣❛r❛♠❡tr✐❝

y − 3 = 2(x − 1)

(x, y) = (1, 3) + t < 1, 2 >

✈❡rt✐❝❛❧ str❡t❝❤ ❜②

2,

s❤✐❢t ✉♣ ❜②

< 1, 3 >

y − 3 = 2(x − 1)2

(x, y) = (1, 3)+ < t, 2t2 >

✈❡rt✐❝❛❧ str❡t❝❤ ❜②

2,

s❤✐❢t ✉♣ ❜②

< 1, 3 >

(y − 3)2 + (x − 1)2 = 22 ✉♥✐❢♦r♠ str❡t❝❤ ❜②

2,

(x, y) = (1, 3) + 2 < cos t, sin t >

s❤✐❢t ✉♣ ❜②

< 1, 3 >

❲❡ ✇✐❧❧ ♥❡❡❞ t♦ ✈✐s✉❛❧✐③❡ ❡①❛♠♣❧❡s ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ♦♥ t❤❡ ♣❧❛♥❡✿

F : R2 → R2 . ❲❡ ✐❧❧✉str❛t❡ t❤❡♠ ❜② ❝r❡❛t✐♥❣ ♠❛r❦s ♦♥ t❤❡ ♦r✐❣✐♥❛❧ ♣❧❛♥❡ ❛♥❞ t❤❡♥ s❡❡ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡♠ ❛s t❤❡② ❛♣♣❡❛r ♦♥ t❤❡ ♥❡✇ ♣❧❛♥❡✳ ❊❛❝❤ ♦❢ t❤❡s❡ ♠❛r❦s ✇✐❧❧ ❜❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐♥ t❤❡ ❞♦♠❛✐♥✿

P : R → R2 , X = P (t) . ❲❡ t❤❡♥ ♣❧♦t ✐ts ✐♠❛❣❡ ✐♥ t❤❡ ❝♦❞♦♠❛✐♥ ✉♥❞❡r t❤❡ tr❛♥s❢♦r♠❛t✐♦♥

❚♦ ❜❡ ♣r❡❝✐s❡✱ ✇❡ ♣❧♦t t❤❡ ✐♠❛❣❡ ♦❢ t❤❡

❝♦♠♣♦s✐t✐♦♥

Y = F (X)✿

♦❢ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✿

F ◦ P : R → R2 , Y = F (P (t)) . ■t ✐s ❛❧s♦ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

F

R2

−−−−→

↑P

րF ◦P

R2

R ❲❡ ✇✐❧❧ ♣❧♦t ♠❛♥② ♣❛✐rs ♦❢ ❝✉r✈❡s ❡❛❝❤ t✐♠❡✿

♦❧❞ ❝✉r✈❡✿

R2 ↑P R

R2 −→

♥❡✇ ❝✉r✈❡✿

R

րF ◦P

❲❡ ❝❛♥ ✉s❡ t❤✐s s❡t✉♣ ✐♥ t✇♦ ✇❛②s✿ ✶✳ ❲❡ ❝❛♥ st✉❞② ❛ ❝✉r✈❡ ❜② ❛♣♣❧②✐♥❣ ✈❛r✐♦✉s tr❛♥s❢♦r♠❛t✐♦♥s t♦ t❤❡ ♣❧❛♥❡✳ ✷✳ ❲❡ ❝❛♥ st✉❞② ❛ tr❛♥s❢♦r♠❛t✐♦♥ ❜② ❛♣♣❧②✐♥❣ ✐t t♦ ✈❛r✐♦✉s ❝✉r✈❡s✳ ❲❡ ❞✐❞ t❤❡ ❢♦r♠❡r ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ s❡❝t✐♦♥✳ ◆♦✇ t❤❡ ❧❛tt❡r✳

✷✳✹✳

❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs

✶✺✸

❊①❛♠♣❧❡ ✷✳✹✳✹✿ str❡t❝❤

❙♦✱ ❛♣♣❧②✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥s t♦ ❝✉r✈❡s ✇✐❧❧ ❣✐✈❡ ✉s ♥❡✇ ❝✉r✈❡s✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ st❛rt ✇✐t❤ t❤❡ ❝✐r❝❧❡s✿ P (t) = r < cos t, sin t >, r > 0 .

❚❤❡♥✱ ✉s✐♥❣ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② 2 ♦♥ ❛❧❧ ✈❡❝t♦rs ♠❡❛♥s t❤❡♥ ❞✐s❝♦✈❡r t❤❛t t❤❡ ✐♠❛❣❡ ♦❢ t❤❡ ❝✉r✈❡ ✐s ❣✐✈❡♥ ❜②✿

str❡t❝❤✐♥❣ r❛❞✐❛❧❧② t❤❡ ✇❤♦❧❡ s♣❛❝❡✳ ❲❡

Q(t) = 2P (t) = 2r < cos t, sin t > .

■t ✐s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s 2r✿

❲❤❛t ❝✉r✈❡s ❞♦ ✇❡ ❝❤♦♦s❡❄ ❋❛♠✐❧✐❛r ♦♥❡s✿ str❛✐❣❤t ❧✐♥❡s ❛♥❞ ❝✐r❝❧❡s✳ ❍♦✇ ♠❛♥②❄ ❚❤❡ ✇❤♦❧❡ ❣r✐❞✳ ❲❡ ❤❛✈❡ t✇♦ ♣♦ss✐❜✐❧✐t✐❡s✿ ❈❛rt❡s✐❛♥ ❣r✐❞✿ P♦❧❛r ❣r✐❞✿

❛ r❡❝t❛♥❣✉❧❛r ❣r✐❞ ♦❢ ❧✐♥❡s

❛ ❣r✐❞ ♦❢ ❝✐r❝❧❡s ❛♥❞ r❛❞✐✐

❚❤❡ ❈❛rt❡s✐❛♥ ❣r✐❞ ✐s ❝r❡❛t❡❞ ✇✐t❤ t❤❡s❡ t✇♦ t②♣❡s ♦❢ ❧✐♥❡s✿

❚❤❡s❡ ❧✐♥❡s ❛r❡ ❞❡✜♥❡❞ ♣❛r❛♠❡tr✐❝❛❧❧②✿ ✶✳ ❤♦r✐③♦♥t❛❧✿ x = t, y = k, k = ... − 3, −2, −1, 0, 1, 2, 3, ... ✷✳ ✈❡rt✐❝❛❧✿ x = k, y = t, k = ... − 3, −2, −1, 0, 1, 2, 3, ...

❚❤❡ ♣♦❧❛r ❣r✐❞ ✐s ❝r❡❛t❡❞ ✇✐t❤ t❤❡s❡ t✇♦ t②♣❡s ♦❢ ❧✐♥❡s✿

✷✳✹✳

❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs

✶✺✹

❍❡r❡ t❤❡② ❛r❡ ❞❡✜♥❡❞ ♣❛r❛♠❡tr✐❝❛❧❧②✿ ✶✳ r❛②s✿ x = at, y = bt, a, b r❡❛❧❀ ❛♥❞ ✷✳ ❝✐r❝❧❡s✿ x = r cos t, y = r sin t, r r❡❛❧✳ ❲❡ ✇✐❧❧ ❛♣♣❧② t❤❡ ❈❛rt❡s✐❛♥ ♦r t❤❡ ♣♦❧❛r ❛s ♥❡❡❞❡❞✳ ❊①❛♠♣❧❡ ✷✳✹✳✺✿ ❝♦❧❧❛♣s❡ ♦♥ ❛①✐s

▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ✈❡r② s✐♠♣❧❡ ❢✉♥❝t✐♦♥✿ 

u = 2x v =0

      x 2 0 u . · = ✱ r❡✲✇r✐tt❡♥✿ y 0 0 v

❇❡❧♦✇✱ ♦♥❡ ❝❛♥ s❡❡ ❤♦✇ t❤✐s ❢✉♥❝t✐♦♥ ❝♦❧❧❛♣s❡s t❤❡ ✇❤♦❧❡ ♣❧❛♥❡ t♦ t❤❡ x✲❛①✐s✿

❚❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s t♦ ❡✈❡r② ❝✐r❝❧❡✿

■♥ t❤❡ ♠❡❛♥t✐♠❡✱ t❤❡ x✲❛①✐s ✐s str❡t❝❤❡❞ ❜② ❛ ❢❛❝t♦r ♦❢ 2✳ ❲❡ ❝❛♥ s❡❡ ❜♦t❤ ✐♥ t❤❡ ♠❛tr✐①✿ str❡t❝❤ ♦❢ x → 2, 0 ← y ❞♦❡s♥✬t ❞❡♣❡♥❞s ♦♥ x → 0, 0 ←



❝♦❧❧❛♣s❡

❇❡❝❛✉s❡ ♦❢ t❤❡ ❝♦❧❧❛♣s❡ ♦❢ t❤❡ y ✲❛①✐s✱ t❤❡ ❢✉♥❝t✐♦♥ ✐s ♥❡✐t❤❡r ♦♥❡✲t♦✲♦♥❡ ♥♦r ♦♥t♦✳ ❊①❛♠♣❧❡ ✷✳✹✳✻✿ str❡t❝❤✲s❤r✐♥❦ ❛❧♦♥❣ ❛①❡s

▲❡t✬s r❡✈✐s✐t t❤✐s ❧✐♥❡❛r ♦♣❡r❛t♦r✿ ❲❡ ❧♦♦❦ ❛t t❤❡ ❝✐r❝❧❡s ✜rst✿

 2 0 . F = 0 4 

✷✳✹✳

❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs

✶✺✺

❚❤❡ ❝✐r❝❧❡s ❤❛✈❡ ❜❡❝♦♠❡ ❡❧❧✐♣s❡s✦ ❲❡ ❝❛♥ s❡❡ ✇❤❛t ❤❛♣♣❡♥s ✐♥ t❤❡ ♠❛tr✐①✿ str❡t❝❤ ♦❢

y

❞♦❡s♥✬t ❞❡♣❡♥❞s ♦♥

x → 2, 0 ← x ❞♦❡s♥✬t ❞❡♣❡♥❞s x → 0, 4 ← str❡t❝❤ ♦❢ y

♦♥

y

❚❤❡ ❛①❡s st❛② ♣✉t✳ ❲❤❛t ❤❛♣♣❡♥s t♦ t❤❡ r❡st ♦❢ t❤❡ ♣❧❛♥❡❄ ▲❡t✬s ❧♦♦❦ ❛t t❤❡ ❧✐♥❡s ♥♦✇✿

❙✐♥❝❡ t❤❡ str❡t❝❤✐♥❣ ✐s ♥♦♥✲✉♥✐❢♦r♠✱ t❤❡ ✈❡❝t♦rs t✉r♥✳ ❍♦✇❡✈❡r✱ s✐♥❝❡ t❤❡ ❜❛s✐s ✈❡❝t♦rs

e1

❛♥❞

e2

❞♦♥✬t

t✉r♥✱ t❤✐s ✐s ♥♦t ❛ r♦t❛t✐♦♥ ❜✉t r❛t❤❡r ❛ ✏❢❛♥♥✐♥❣ ♦✉t✑ ♦❢ t❤❡ ✈❡❝t♦rs✳ ❚❤❡✐r s❧♦♣❡s ❤❛✈❡ ✐♥❝r❡❛s❡❞✳ ❲❡ ❛❧s♦ ❞✐s❝♦✈❡r t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✳

❊①❛♠♣❧❡ ✷✳✹✳✼✿ str❡t❝❤✲s❤r✐♥❦ ❛❧♦♥❣ ❛①❡s ❆ s❧✐❣❤t❧② ❞✐✛❡r❡♥t ❢✉♥❝t✐♦♥ ✐s✿



u = −x v = 4y

■t ✐s s✐♠♣❧❡ ❜❡❝❛✉s❡ t❤❡ t✇♦ ✈❛r✐❛❜❧❡s ❛r❡ ❢✉❧❧② s❡♣❛r❛t❡❞✳ ❏✉st t❤❡ ❝✐r❝❧❡s✿

❚❤❡ s❧✐❣❤t ❝❤❛♥❣❡ t♦ t❤❡ ❢✉♥❝t✐♦♥ ♣r♦❞✉❝❡s ❛ s✐♠✐❧❛r ❜✉t ❞✐✛❡r❡♥t ♣❛tt❡r♥✿ ❲❡ s❡❡ t❤❡ r❡✈❡rs❛❧ ♦❢ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❡❧❧✐♣s❡ ❛r♦✉♥❞ t❤❡ ♦r✐❣✐♥✳ ❲❡ ✇❛② t❤❛t

t❤❡ ♦r✐❡♥t❛t✐♦♥ ❤❛s ❝❤❛♥❣❡❞✳

❚❤❡ ♠❛tr✐① ♦❢

F

✷✳✹✳

❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs

✐s st✐❧❧ ❞✐❛❣♦♥❛❧✿ ❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✳

✶✺✻

 −1 0 . F = 0 4 

❊①❛♠♣❧❡ ✷✳✹✳✽✿ ❡①♣❡r✐♠❡♥t

▲❡t✬s ❝♦♥s✐❞❡r ❛ ♠♦r❡ ❣❡♥❡r❛❧ ❢✉♥❝t✐♦♥✿ 

u = x +2y v = 2x +4y

 1 2 . =⇒ F = 2 4 

■t ✐s ❤❛r❞ t♦ t❡❧❧ ✇❤❛t ✐t ❞♦❡s✱ ❥✉❞❣✐♥❣ ❜② ✐ts ♠❛tr✐①✳ ❲❡ ❡①♣❡r✐♠❡♥t ✿

■t ❛♣♣❡❛rs t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s str❡t❝❤✐♥❣ t❤❡ ♣❧❛♥❡ ✐♥ ♦♥❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ❝♦❧❧❛♣s✐♥❣ ✐♥ ❛♥♦t❤❡r✳ ❚❤❛t✬s ✇❤② t❤❡r❡ ✐s ❛ ✇❤♦❧❡ ❧✐♥❡ ♦❢ ♣♦✐♥ts X ✇✐t❤ F X = 0✳ ❚♦ ✜♥❞ ✐t✱ ✇❡ s♦❧✈❡ t❤✐s ❡q✉❛t✐♦♥✿ 

x +2y = 0 2x +4y = 0

=⇒ x = −2y .

❚❤❡ ✈❡❝t♦r < 1, 2 > ✐s✱ ✐♥ ❢❛❝t✱ ✈✐s✐❜❧❡ ✐♥ t❤❡ ♠❛tr✐①✳ ❇❡❝❛✉s❡ ♦❢ t❤❡ ❝♦❧❧❛♣s❡ ♦❢ t❤❡ ❣r❡❡♥ ❧✐♥❡ t♦ t❤❡ ♦r✐❣✐♥✱ t❤❡ ❢✉♥❝t✐♦♥ ✐s ♥❡✐t❤❡r ♦♥❡✲t♦✲♦♥❡ ♥♦r ♦♥t♦✳ ❊①❡r❝✐s❡ ✷✳✹✳✾

❙❤♦✇ ✇❤❡r❡ ❡❛❝❤ ♣♦✐♥t ❣♦❡s✳ ❊①❛♠♣❧❡ ✷✳✹✳✶✵✿ ❡①♣❡r✐♠❡♥t

❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛tr✐① F ✿

 −1 −2 . F = 1 −4 

❆❣❛✐♥✱ ✐t✬s t♦♦ ❝♦♠♣❧❡① t♦ r❡✈❡❛❧ ✇❤❛t ✐t ❞♦❡s✱ ❛♥❞ ✇❡ ❤❛✈❡ t♦ ❡①♣❡r✐♠❡♥t✿

✷✳✹✳

❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs

✶✺✼

■t ❧♦♦❦s ❧✐❦❡ ❛ ♥♦♥✲✉♥✐❢♦r♠ str❡t❝❤ ❛❧♦♥❣ ❞✐❛❣♦♥❛❧ ❞✐r❡❝t✐♦♥s✳

❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞

♦♥t♦✳

❊①❡r❝✐s❡ ✷✳✹✳✶✶

F✿   1 2 . F = 3 2

❉❡s❝r✐❜❡ ✇❤❛t ✐s ❤❛♣♣❡♥✐♥❣ ✉♥❞❡r t❤✐s ♦♣❡r❛t♦r

❊①❛♠♣❧❡ ✷✳✹✳✶✷✿ s❦❡✇✐♥❣✲s❤❡❛r✐♥❣ ❈♦♥s✐❞❡r t❤✐s ♠❛tr✐①✿

❇❡❧♦✇✱ ✇❡ r❡♣❧❛❝❡ ❝✐r❝❧❡s ✇✐t❤

❡❧❧✐♣s❡s

 1 1 . F = 0 1 

❛♥❞ s❡❡ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡♠ ✉♥❞❡r s✉❝❤ ❛ ❢✉♥❝t✐♦♥✿

❚❤❡r❡ s❡❡♠s t♦ ❜❡ ♥♦ str❡t❝❤ ❛❧♦♥❣ t❤❡

x✲❛①✐s✳

❚❤❡r❡ ✐s st✐❧❧ ❛♥❣✉❧❛r str❡t❝❤✲s❤r✐♥❦ ❜✉t t❤✐s t✐♠❡ ✐t ✐s

❜❡t✇❡❡♥ t❤❡ t✇♦ ❡♥❞s ♦❢ t❤❡ s❛♠❡ ❧✐♥❡✳

❚♦ s❡❡ ♠♦r❡ ❝❧❡❛r❧②✱ ❝♦♥s✐❞❡r t❤❡ ❈❛rt❡s✐❛♥ ❣r✐❞✳ ❚❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s t♦ ❛ sq✉❛r❡✿

✷✳✹✳

❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs

✶✺✽

❚❤❡ ♣❧❛♥❡ ✐s s❦❡✇❡❞✱ ❧✐❦❡ ❛ ❞❡❝❦ ♦❢ ❝❛r❞s✿

❙✉❝❤ ❛ s❦❡✇✐♥❣ ❝❛♥ ❜❡ ❝❛rr✐❡❞ ♦✉t ✇✐t❤ ❛♥② ✐♠❛❣❡✲❡❞✐t✐♥❣ s♦❢t✇❛r❡✳ ❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✳ ❊①❛♠♣❧❡ ✷✳✹✳✶✸✿ r♦t❛t✐♦♥

π/2

❈♦♥s✐❞❡r ❛ r♦t❛t✐♦♥ t❤r♦✉❣❤ 90 ❞❡❣r❡❡s✿ (x, y) ❜❡❝♦♠❡s (−y, x)✳ ❲❡ ❤❛✈❡✿ 

      x 0 1 u . · = ✱ r❡✲✇r✐tt❡♥✿ y −1 0 v

u = −y v =x

❚❤❡ ❡①♣❡r✐♠❡♥t ❝♦♥✜r♠s ✇❤❛t ✇❡ ❦♥♦✇✿

❲❡✬✈❡ ❤❛❞ ♠❛♥② ❡①❛♠♣❧❡s✱ ❜✉t ❤♦✇ ❞♦ ✇❡ ❜✉✐❧❞ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ❢r♦♠ ❛ ❞❡s❝r✐♣t✐♦♥❄ ❚❤❡ s♦❧✉t✐♦♥ r❡❧✐❡s ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s✐♠♣❧❡ ♦❜s❡r✈❛t✐♦♥✿ ❚❤❡♦r❡♠ ✷✳✹✳✶✹✿ ❈♦❧✉♠♥s ❛r❡ ❱❛❧✉❡s ♦❢ ❇❛s✐s ❱❡❝t♦rs ❚❤❡ t✇♦ ❝♦❧✉♠♥s ♦❢ t❤❡ ♠❛tr✐① ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ❛r❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ t✇♦ ❜❛s✐s ✈❡❝t♦rs ✉♥❞❡r t❤✐s ♦♣❡r❛t♦r✿



a b F = c d



    a 1 = =⇒ F c 0

❛♥❞

    b 0 . = F d 1

✷✳✹✳

❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs

✶✺✾

❊①❡r❝✐s❡ ✷✳✹✳✶✺

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❚❤❡ ❝♦♥✈❡rs❡ ✐s ❥✉st ❛s ✐♠♣♦rt❛♥t✿ ❚❤❡♦r❡♠ ✷✳✹✳✶✻✿ ❱❛❧✉❡s ♦❢ ❇❛s✐s ❱❡❝t♦rs ❆r❡ ❈♦❧✉♠♥s ❚❤❡ ♠❛tr✐① ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ✐s ❢✉❧❧② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ t✇♦ ❜❛s✐s ✈❡❝t♦rs ✉♥❞❡r t❤✐s ♦♣❡r❛t♦r✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ♠❡r❣❡ t❤❡ t✇♦ ❝♦❧✉♠♥✲✈❡❝t♦rs ✐♥t♦ ❛ ♠❛tr✐①✿     1 a 7→ F : , 0 c

      0 b a b 7→ F : ▼❡r❣❡✿ F = . 1 d c d

❊①❛♠♣❧❡ ✷✳✹✳✶✼✿ ♠❛tr✐❝❡s ❢r♦♠ ✈❛❧✉❡s

❚❤✐s ✐s t❤❡ ③❡r♦ ♦♣❡r❛t♦r✿

❚❤✐s ✐s t❤❡ ✐❞❡♥t✐t②✿

    0 1 , 7→ 0 0

    0 0 7→ 0 1

 0 0 . ▼❡r❣❡✿ 0 = 0 0

    1 1 , 7→ 0 0

    0 0 7→ 1 1

 1 0 . ▼❡r❣❡✿ I = 0 1

    2 1 , 7→ 0 0

    0 0 7→ 1 1

 2 0 . ▼❡r❣❡✿ Sx = 0 1

    0 0 7→ 3 1

 1 0 . ▼❡r❣❡✿ Sy = 0 3

    0 0 7→ 1 1

 −1 0 . ▼❡r❣❡✿ Fx = 0 1

    0 0 7→ −1 1

 1 0 . ▼❡r❣❡✿ Fy = 0 −1

❚❤✐s ✐s t❤❡ ❤♦r✐③♦♥t❛❧ str❡t❝❤ ❜② 2✿







❆♥❞ t❤✐s ✐s t❤❡ ✈❡rt✐❝❛❧ str❡t❝❤ ❜② 3✿     1 1 , 7→ 0 0



❚❤✐s ✐s t❤❡ ❤♦r✐③♦♥t❛❧ ✢✐♣✿     −1 1 , 7→ 0 0



❆♥❞ t❤✐s ✐s t❤❡ ✈❡rt✐❝❛❧ ✢✐♣✿     1 1 , 7→ 0 0



❚❤✐s ✐s t❤❡ ✢✐♣ ❛❜♦✉t t❤❡ ❞✐❛❣♦♥❛❧✿     1 0 7→ , 0 1

    0 1 7→ 1 0

 0 1 . ▼❡r❣❡✿ Fd = 1 0 

❲❛r♥✐♥❣✦ ■ts ♠❛tr✐① ✐s ❥✉st ❛♥ ❛❜❜r❡✈✐❛t❡❞ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✳

✷✳✹✳

❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs

✶✻✵

❊①❡r❝✐s❡ ✷✳✹✳✶✽

❙✉♣♣♦s❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r A✿ • ❧❡❛✈❡s t❤❡ x✲❛①✐s ✐♥t❛❝t✱ ❛♥❞ • str❡t❝❤❡s t❤❡ y ✲❛①✐s ❜② ❛ ❢❛❝t♦r ♦❢ 2✳ ❋✐♥❞ t❤❡ ♠❛tr✐① ♦❢ A✳ ❊①❡r❝✐s❡ ✷✳✹✳✶✾

❙✉♣♣♦s❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r A✿ • r♦t❛t❡s t❤❡ x✲❛①✐s 45 ❞❡❣r❡❡s ❝❧♦❝❦✇✐s❡✱ ❛♥❞ • ✢✐♣s t❤❡ y ✲❛①✐s✳ ❋✐♥❞ t❤❡ ♠❛tr✐① ♦❢ A✳ ❊①❡r❝✐s❡ ✷✳✹✳✷✵

❙✉♣♣♦s❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r A✿ • ❧❡❛✈❡s t❤❡ x✲❛①✐s ✐♥t❛❝t✱ ❛♥❞ • str❡t❝❤❡s t❤❡ ❞✐❛❣♦♥❛❧ y = x ❜② ❛ ❢❛❝t♦r ♦❢ 2✳ ❋✐♥❞ t❤❡ ♠❛tr✐① ♦❢ A✳ ❊①❡r❝✐s❡ ✷✳✹✳✷✶

▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ ❧✐♥❡❛r ♦♣❡r❛t♦r ❛♥❞ ✜♥❞ ✐ts ♠❛tr✐①✳ ❘❡♣❡❛t✳ ▲❡t✬s ❛♣♣❧② t❤✐s r❡s✉❧t t♦ s♦♠❡ tr❛♥s❢♦r♠❛t✐♦♥s ✇❡ ❤❛✈❡ ❜❡❡♥ ✐♥t❡r❡st❡❞ ✐♥✳ ❚❤❡♦r❡♠ ✷✳✹✳✷✷✿ ▼❛tr✐① ♦❢ ❘♦t❛t✐♦♥ ❚❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r ♦❢ r♦t❛t✐♦♥ t❤r♦✉❣❤ ❛♥ ❛♥❣❧❡ ♠❛tr✐①✿



cos α − sin α R= sin α cos α

Pr♦♦❢✳

❲❡ ♦♥❧② ♥❡❡❞ t♦ s❡❡ ✇❤❡r❡ t❤❡ ❜❛s✐s ✈❡❝t♦rs < 1, 0 > ❛♥❞ < 0, 1 > ❣♦✳

❚❤❡ ✜rst ♦♥❡ ✐s s✐♠♣❧❡✿

    cos α 1 . = R sin α 0

❚❤❡ s❡❝♦♥❞ ♦♥❡ ✢✐♣s t❤❡ s✐❣♥ ♦❢ t❤❡ x✲❝♦♠♣♦♥❡♥t✿

    − sin α 0 . = R cos α 1



α

✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣

✷✳✹✳

❊①❛♠✐♥✐♥❣ ❛♥❞ ❜✉✐❧❞✐♥❣ ❧✐♥❡❛r ♦♣❡r❛t♦rs

❊①❛♠♣❧❡ ✷✳✹✳✷✸✿ r♦t❛t✐♦♥

✶✻✶

π/4

❈♦♥s✐❞❡r ❛ r♦t❛t✐♦♥ t❤r♦✉❣❤ 45 ❞❡❣r❡❡s✿ (

u = cos π4 x − sin π4 y

✱ r❡✲✇r✐tt❡♥✿

v = sin π4 x + cos π4 y

" # u v

=

"

cos π4 − sin π4 sin π4

cos π4

# " # x · . y

❲❡ ♣❧♦t ❡❧❧✐♣s❡s ✐♥st❡❛❞ ♦❢ ❝✐r❝❧❡s t♦ ♠❛❦❡ ✐t ❡❛s✐❡r t♦ s❡❡ ✇❤❛t ✐s ❤❛♣♣❡♥✐♥❣✿

❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✳ ❊①❛♠♣❧❡ ✷✳✹✳✷✹✿ r♦t❛t✐♦♥ ✇✐t❤ str❡t❝❤✲s❤r✐♥❦

▲❡t✬s ❝♦♥s✐❞❡r ❛ ♠♦r❡ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥✿ 

❍❡r❡✱ t❤❡ ♠❛tr✐① ♦❢ F ✐s ♥♦t ❞✐❛❣♦♥❛❧✿ ❚❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s t♦ ♦✉r ❡❧❧✐♣s❡s✿

u = 3x −13y . v = 5x +y  3 −13 . F = 5 1 

❲❡ s❡❡♠ t♦ ❤❛✈❡ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ str❡t❝❤✐♥❣✱ ✢✐♣♣✐♥❣✱ ❛♥❞ r♦t❛t✐♥❣✳✳✳ ❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❜♦t❤ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♦♥t♦✳ ❋♦r ❧✐♥❡❛r ♦♣❡r❛t♦rs✱ t❤❡r❡ ✐s ❛♥ ❡❛s② ✇❛② t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥ ✇❡ ❤❛✈❡ ❜❡❡♥ ❛s❦✐♥❣✿ ❲❤❡♥ ✐s ✐t ♦♥❡✲t♦✲♦♥❡❄

✷✳✺✳

❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①

✶✻✷

❚❤❡♦r❡♠ ✷✳✹✳✷✺✿ ❖♥❡✲t♦✲♦♥❡ ▲✐♥❡❛r ❖♣❡r❛t♦r ❆ ❧✐♥❡❛r ♦♣❡r❛t♦r

F

✐s ♦♥❡✲t♦✲♦♥❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❡q✉❛t✐♦♥

F (X) = 0

❤❛s ♦♥❧②

t❤❡ ③❡r♦ s♦❧✉t✐♦♥✳

Pr♦♦❢✳

❙✉♣♣♦s❡ t❤❡r❡ ❛r❡ t✇♦ ❞✐st✐♥❝t s♦❧✉t✐♦♥s X 6= Y ✳ ❚❤❡♥✱ ✇❡ ❝♦♥❝❧✉❞❡✿ F (X) = F (Y ) =⇒ F (X) − F (Y ) = 0 =⇒ F (X − Y ) = 0 .

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ ❢♦✉♥❞ s✉❝❤ ❛ Z = X − Y 6= 0 t❤❛t F (Z) = 0 = F (0)✳ ❊①❡r❝✐s❡ ✷✳✹✳✷✻

Pr♦✈❡ t❤❡ r❡st ♦❢ t❤❡ t❤❡♦r❡♠✳ ■♥ ♦t❤❡r ✇♦r❞s✱ F ✐s ♦♥❡✲t♦✲♦♥❡ ✇❤❡♥ F (X) = 0 =⇒ X = 0 .

❙♦✱ t♦ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r ❛ ♠✐①t✉r❡ ♣r♦❜❧❡♠ ❤❛s ❛ s✐♥❣❧❡ s♦❧✉t✐♦♥✱ ✇❡ ❝❤♦♦s❡✱ ✐♥ ❛ t✇✐st✱ t♦ r❡♣❧❛❝❡ ✐t ✇✐t❤ ❛ ♠✐①t✉r❡ ♣r♦❜❧❡♠ t❤❛t r❡q✉✐r❡s t♦ ♣r♦❞✉❝❡ ③❡r♦s ✐♥ ❛❧❧ ❡q✉❛t✐♦♥s✳ ❚❤❡♥ ✇❡ ❛s❦ ✐❢ t❤✐s ♣r♦❜❧❡♠ ❤❛s ❛ ♥♦♥✲③❡r♦ s♦❧✉t✐♦♥✳

✷✳✺✳ ❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①

❊①❛♠♣❧❡ ✷✳✺✳✶✿ ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ♥♦t

❈♦♥s✐❞❡r t❤✐s ♠❛tr✐①✿

 1 1 . A= 0 1 

■s t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r ♦♥❡✲t♦✲♦♥❡❄ ❊✈❡r② ♦♥❡ ♦❢ ♦✉r ❡❧❧✐♣s❡s ✐♥ t❤❡ ❞♦♠❛✐♥ ❤❛s ❜❡❡♥ str❡t❝❤❡❞ ❛♥❞ ♠❛②❜❡ r♦t❛t❡❞ ❜✉t t❤❡② st✐❧❧ ❝♦✈❡r t❤❡ t❤❡ ✇❤♦❧❡ ♣❧❛♥❡ ✐♥ t❤❡ ❝♦❞♦♠❛✐♥✿

■t ✐s ♦♥❡✲t♦✲♦♥❡✳ ❚❤✐s ♦♥❡ ✐s ❞✐✛❡r❡♥t✿

 1 2 . A= 2 4

❲❡ ❥✉st ✇❛t❝❤ ✇❤❡r❡ t❤❡ t✇♦ ❜❛s✐s ✈❡❝t♦rs ❣♦✿



✷✳✺✳

❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①

✶✻✸

❲❤❛t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❢r♦♠ t❤❡ ❢♦r♠❡r ❝❛s❡❄ ❚❤❡② ❣♦ t♦ ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✿ ❚❤❡✐r ✈❛❧✉❡s ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✈❡❝t♦r < 1, 2 >✳ ❲❡ ❝❛♥ s❡❡ t❤❛t ✐♥ t❤❡ ♠❛tr✐① t❤❛t t❤❡ s❡❝♦♥❞ ❝♦❧✉♠♥ ✐s t✇✐❝❡ t❤❡ ✜rst✿     2

2 1 . = 4 2

■♥ ❢❛❝t✱ ❛ ✇❤♦❧❡ ❧✐♥❡ ♦❢ ✈❡❝t♦rs ❣♦❡s t♦ 0❀ ✐t✬s ♥♦t ♦♥❡✲t♦✲♦♥❡✦

❉❡✜♥✐t✐♦♥ ✷✳✺✳✷✿ s✐♥❣✉❧❛r ♠❛tr✐① ❆ 2 × 2 ♠❛tr✐① A ✐s ❝❛❧❧❡❞ s✐♥❣✉❧❛r ✇❤❡♥ ✐ts t✇♦ ❝♦❧✉♠♥s ❛r❡ ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✳ ❙♦✱ ❢♦r ❛ s✐♥❣✉❧❛r ♠❛tr✐①

 a b , A= c d 

t❤❡r❡ ✐s s✉❝❤ ❛♥ x t❤❛t✿

    b a . =x d c

▲❡t✬s ❡①❛♠✐♥❡ t❤✐s ✐❞❡❛✿ ❯♥❞❡r ✇❤❛t ❝✐r❝✉♠st❛♥❝❡s ✐s ❛ ♠❛tr✐① s✐♥❣✉❧❛r❄ ❲❡ ❜r❡❛❦ t❤❡ ✈❡❝t♦r ❡q✉❛t✐♦♥ ❛❜♦✈❡ ✐♥t♦ t✇♦ s❝❛❧❛r ❡q✉❛t✐♦♥s✿ a = xb, c = xd .

■♥st❡❛❞ ♦❢ s♦❧✈✐♥❣ t❤❡♠ ❢♦r x✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡r❡ ✐s s✉❝❤ ❛♥ x✳ ❲❡ ♠✉❧t✐♣❧② t❤❡ ✜rst ❜② d✱ ❛♥❞ t❤❡ s❡❝♦♥❞ ❜② b✱ ❛♥❞ t❤❡♥ s✉❜tr❛❝t t♦ ❡❧✐♠✐♥❛t❡ x✿ ad = xbd cb = xbd ad − bc = 0

❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ✐❢ s✉❝❤ ❛♥ x ❡①✐sts✱ t❤❡♥

ad − bc = 0 .

■♥ t❤✐s ❡①♣r❡ss✐♦♥✱ t❤❡ t❡r♠s ♦❢ t❤❡ ♠❛tr✐① ❛r❡ ❝r♦ss✲♠✉❧t✐♣❧✐❡❞ ❛♥❞ s✉❜tr❛❝t❡❞✿ 

a b c d



→ ad − bc .

❚❤✐s ♥✉♠❜❡r ✐s ❛♥ ✐♠♣♦rt❛♥t ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ t❤❡ ♠❛tr✐①✿

❉❡✜♥✐t✐♦♥ ✷✳✺✳✸✿ ❞❡t❡r♠✐♥❛♥t

❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ 2 × 2 ♠❛tr✐① A ✐s ❞❡✜♥❡❞ ❛♥❞ ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿  a b = ad − bc det A = det c d 

❲❤❛t ❞♦❡s t❤❡ ❞❡t❡r♠✐♥❛♥t ❞❡t❡r♠✐♥❡❄

✷✳✺✳

❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①

✶✻✹

❚❤❡♦r❡♠ ✷✳✺✳✹✿ ❙✐♥❣✉❧❛r ▼❛tr✐① ❛♥❞ ❉❡t❡r♠✐♥❛♥t ❆

2×2

♠❛tr✐①

A

✐s s✐♥❣✉❧❛r ✐❢ ❛♥❞ ♦♥❧② ✐❢

det A = 0✳

Pr♦♦❢✳

✭⇒✮ ❙✉♣♣♦s❡ A ✐s s✐♥❣✉❧❛r✱ t❤❡♥      a = xb b a =⇒ =x =⇒ det A = ad − bc = (xb)d − b(xd) = 0 . c = xd d c

✭⇐✮ ❙✉♣♣♦s❡ ad − bc = 0✱ t❤❡♥ ❧❡t✬s ✜♥❞ x✱ t❤❡ ♠✉❧t✐♣❧❡✳ a • ❈❛s❡ ✶✿ ❆ss✉♠❡ b 6= 0✱ t❤❡♥ ❝❤♦♦s❡ x = ✳ ❚❤❡♥ b a = a, xb = b b a ad bc xd = d = = = c. b b b ❙♦     a b . = x c d • ❈❛s❡ ✷✿ ❆ss✉♠❡ a 6= 0 ✳✳✳ ❊①❡r❝✐s❡ ✷✳✺✳✺

❋✐♥✐s❤ t❤❡ ♣r♦♦❢✳ ❲❡ ♠❛❦❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♦❜s❡r✈❛t✐♦♥ ❛❜♦✉t t❤❡ ❞❡t❡r♠✐♥❛♥t✱ ✇❤✐❝❤ ✇✐❧❧ ❛❧s♦ r❡❛♣♣❡❛r ✐♥ t❤❡ ❝❛s❡ ♦❢ n × n ♠❛tr✐❝❡s✿ ◮ ❚❤❡ ❞❡t❡r♠✐♥❛♥t ✐s ❛♥ ❛❧t❡r♥❛t✐♥❣ s✉♠ ♦❢ t❡r♠s✱ ❡❛❝❤ ♦❢ ✇❤✐❝❤ ✐s t❤❡ ♣r♦❞✉❝t ♦❢ n ♦❢ t❤❡

♠❛tr✐①✬s ❡♥tr✐❡s✱ ❡①❛❝t❧② ♦♥❡ ❢r♦♠ ❡❛❝❤ r♦✇ ❛♥❞ ❡①❛❝t❧② ♦♥❡ ❢r♦♠ ❡❛❝❤ ❝♦❧✉♠♥✳

▲❡t✬s ❝♦♥s✐❞❡r ❛ s♣❡❝✐❛❧ ♠❛tr✐① ❡q✉❛t✐♦♥✱ ✇✐t❤ t❤❡ ③❡r♦ r✐❣❤t✲❤❛♥❞ s✐❞❡✿ F (X) = 0 .

■t ✐s ❝❛❧❧❡❞ ❛ ❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥✳ ❲❡ ❦♥♦✇ t❤❛t t❤❡r❡ ✐s ❛❧✇❛②s ❛t ❧❡❛st ♦♥❡ s♦❧✉t✐♦♥✱ t❤❡ ③❡r♦ ✈❡❝t♦r✦ ❚❤❡ q✉❡st✐♦♥ ✐s t❤❡♥ ❜❡❝♦♠❡s ◮ ❆r❡ t❤❡r❡ ❛♥② ♥♦♥✲③❡r♦ s♦❧✉t✐♦♥s❄

❲❡ ❦♥♦✇ t❤❛t t❤❡ ❛♥s✇❡r ♠❛② ❜❡ ♣r♦✈✐❞❡❞ ❜② ❛ ♠♦r❡ ❜❛s✐❝ q✉❡st✐♦♥✿ ◮ ■s t❤❡ ❢✉♥❝t✐♦♥ ♦♥❡✲t♦✲♦♥❡❄

▲❡t✬s st❛rt ✇✐t❤ ❞✐♠❡♥s✐♦♥ 1✿ ❚❤✐s ✐s s✐♠♣❧❡✿

f (x) = mx, s♦❧✈❡ f (x) = 0 . mx = 0 =⇒ x = 0 ✳✳✳ ✉♥❧❡ss m = 0.

❆❧❧ ❢✉♥❝t✐♦♥s ❡①❝❡♣t t❤❡ ❝♦♥st❛♥t ③❡r♦ ❢✉♥❝t✐♦♥ ❛r❡ ♦♥❡✲t♦✲♦♥❡ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❝❛♥ ♦♥❧② ♣r♦❞✉❝❡ ❛ s✐♥❣❧❡ s♦❧✉t✐♦♥ ❢♦r ♦✉r ❡q✉❛t✐♦♥✿

✷✳✺✳

❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①

✶✻✺

■s t❤❡r❡ ❛ s✐♠✐❧❛r ❞❡❝✐s✐✈❡ ❝♦♥❞✐t✐♦♥ ✐♥ t❤❡ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡❄ ❲❡ ❞♦ ❤❛✈❡ t❤❡ ③❡r♦ ♦♣❡r❛t♦r ✭♠❛tr✐①✮✿

F (X) = 0

❢♦r ❛❧❧

X.

■t ❝♦❧❧❛♣s❡s t❤❡ ✇❤♦❧❡ ♣❧❛♥❡ t♦ t❤❡ ♦r✐❣✐♥✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ❛r❡ ♦t❤❡r✱ ❧❡ss ❡①tr❡♠❡ ❝♦❧❧❛♣s❡s✱ ♣r♦❥❡❝t✐♦♥s ♦♥t♦ ❧✐♥❡s✿

❚❤❡♥ ❛ ✇❤♦❧❡ ❧✐♥❡ ✐s t❛❦❡♥ t♦

0✳

❚❤❡② ❛r❡ ❛❧s♦ ♥♦t ♦♥❡✲t♦✲♦♥❡✦

❲❡ ❞❡♣❧♦② s✐♠♣❧❡ ❛❧❣❡❜r❛ ✐♥ ♦r❞❡r t♦ r❡s♦❧✈❡ t❤✐s ✐ss✉❡✳

❚❤❡♦r❡♠ ✷✳✺✳✻✿ ◆♦♥✲③❡r♦ ❙♦❧✉t✐♦♥s ❙✉♣♣♦s❡

A

t❤❡ ♠❛tr✐①

2 × 2 ♠❛tr✐①✳ ❚❤❡♥✱ det A 6= 0 ✐❢ ❡q✉❛t✐♦♥ AX = 0 ❝♦♥s✐sts ♦❢ ♦♥❧② 0✳ ✐s ❛

❛♥❞ ♦♥❧② ✐❢ t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢

Pr♦♦❢✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ❩❡r♦ ❋❛❝t♦r Pr♦♣❡rt②✿ ❚❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♥✉♠❜❡rs ✐s ③❡r♦ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✐t❤❡r ♦♥❡ ♦❢ t❤❡♠ ✭♦r ❜♦t❤✮ ✐s ③❡r♦❀ ✐✳❡✳✱

a = 0 ❖❘ b = 0 ⇐⇒ ab = 0 . ▲❡t✬s s♦❧✈❡ t❤❡ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿



ax + by = 0, cx + dy = 0.

(1) (2)

❋r♦♠ ✭✶✮✱ ✇❡ ❞❡r✐✈❡✿

y = −ax/b,

♣r♦✈✐❞❡❞

b 6= 0. (3)

❙✉❜st✐t✉t❡ t❤✐s ✐♥t♦ ✭✷✮✿

cx + d(−ax/b) = 0 . ❚❤❡♥

x(c − da/b) = 0 ,

✷✳✺✳

❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①

✶✻✻

♦r✱ ❛❧t❡r♥❛t✐✈❡❧②✱

x(cb − da) = 0, ✇❤❡♥ b 6= 0 .

❖♥❡ ♣♦ss✐❜✐❧✐t② ✐s x = 0❀ ✐t ❢♦❧❧♦✇s ❢r♦♠ ✭✸✮ t❤❛t y = 0 t♦♦✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡ t✇♦ ❝❛s❡s ❢♦r b 6= 0✿ • ❈❛s❡ ✶✿ x = 0, y = 0✱ ♦r • ❈❛s❡ ✷✿ ad − bc = 0✳ ❈❛s❡ ✶ ❞♦❡s♥✬t ✐♥t❡r❡st ✉s✳ ■♥ ❝❛s❡ ✷✱ x ✐s ❛r❜✐tr❛r② ❛♥❞ t❤❡r❡ ♠❛② ❜❡ ♥♦♥✲③❡r♦ s♦❧✉t✐♦♥s✳ ◆♦✇✱ ✇❡ ❛♣♣❧② t❤✐s ❛♥❛❧②s✐s t♦ y ✐♥ ✭✶✮ ✐♥st❡❛❞ ♦❢ x❀ ✇❡ ❤❛✈❡ ❢♦r a 6= 0✿ • ❈❛s❡ ✶✿ x = 0, y = 0✱ ♦r • ❈❛s❡ ✷✿ ad − bc = 0✳ ❚❤❡ r❡s✉❧t ✐s t❤❡ s❛♠❡✦ ❋✉rt❤❡r♠♦r❡✱ ✐❢ ✇❡ ❛♣♣❧② t❤✐s ❛♥❛❧②s✐s ❢♦r x ❛♥❞ y ✐♥ ✭✷✮ ✐♥st❡❛❞ ♦❢ ✭✶✮✱ ✇❡ ❤❛✈❡ t❤❡ s❛♠❡ t✇♦ ❝❛s❡s✳ ❚❤✉s✱ ✇❤❡♥❡✈❡r ♦♥❡ ♦❢ t❤❡ ❢♦✉r ❝♦❡✣❝✐❡♥ts✱ a, b, c, d✱ ✐s ♥♦♥✲③❡r♦✱ ✇❡ ❤❛✈❡ t❤❡s❡ ❝❛s❡s✿ • ❈❛s❡ ✶✿ x = 0, y = 0✱ ♦r • ❈❛s❡ ✷✿ ad − bc = 0✳ ❇✉t ✇❤❡♥ a = b = c = d = 0✱ ❈❛s❡ ✷ ✐s s❛t✐s✜❡❞✳✳✳ ❛♥❞ ✇❡ ❝❛♥ ❤❛✈❡ ❛♥② ✈❛❧✉❡s ❢♦r x ❛♥❞ y ✦ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❛♥❛❧②s✐s ❛❜♦✈❡✿ det A 6= 0 =⇒ x = y = 0 .

❚❤❡ ❝♦♥✈❡rs❡ ✐s ❛❧s♦ tr✉❡✳ ■♥❞❡❡❞✱ ❧❡t✬s ❝♦♥s✐❞❡r ♦✉r s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ❛❣❛✐♥✿ 

ax + by = 0, cx + dy = 0.

(1) (2)

❲❡ ♠✉❧t✐♣❧② ✭✶✮ ❜② c ❛♥❞ ✭✷✮ ❜② a✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿  cax    acx (ca − ac)x    0·x

+ + + −

cby ady (cb − ad)y det A · y

=0 =0 =0 =0

c · (1) a · (2)

❚❤❡ t❤✐r❞ ❡q✉❛t✐♦♥ ✐s t❤❡ r❡s✉❧t ♦❢ s✉❜tr❛❝t✐♦♥ ♦❢ t❤❡ ✜rst t✇♦✳ ❚❤❡ ✇❤♦❧❡ ❡q✉❛t✐♦♥ ✐s ③❡r♦ ✇❤❡♥ det A = 0✦ ❚❤✐s ♠❡❛♥s t❤❛t ❡q✉❛t✐♦♥s ✭✶✮ ❛♥❞ ✭✷✮ r❡♣r❡s❡♥t t✇♦ ✐❞❡♥t✐❝❛❧ ❧✐♥❡s ♦♥ t❤❡ ♣❧❛♥❡✳ ■t ❢♦❧❧♦✇s t❤❛t t❤❡ ♦r✐❣✐♥❛❧ s②st❡♠ ❤❛s ✐♥✜♥✐t❡❧② ♠❛♥② s♦❧✉t✐♦♥s✳ ❊①❡r❝✐s❡ ✷✳✺✳✼

❲❤❛t ✐❢ a = 0 ♦r c = 0❄ ❊①❛♠♣❧❡ ✷✳✺✳✽✿ ❝♦♠♣✉t✐♥❣ ❞❡t❡r♠✐♥❛♥ts

❚❤❡ ✢✐♣ ♦✈❡r t❤❡ y ✲❛①✐s✿

❚❤❡ str❡t❝❤✿

 −1 0 = (−1) · 1 − 0 · 0 = −1 . det 0 1 

 λ 0 = λ · µ − 0 · 0 = λ · µ. det 0 µ 

❚❤❡ r♦t❛t✐♦♥✿  cos α − sin α = cos α · cos α − (− sin α) · sin α = cos2 α + sin2 α = 1 , det sin α cos α 

❜② t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠✳

✷✳✺✳

✶✻✼

❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ♠❛tr✐①

❚❤❡s❡ ❤❛✈❡ ♥♦♥✲③❡r♦ ❞❡t❡r♠✐♥❛♥ts✳ ▼❡❛♥✇❤✐❧❡✱ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦♥ t❤❡ x✲❛①✐s ❤❛s ❛ ③❡r♦ ❞❡t❡r♠✐♥❛♥t✿  λ 0 = λ · 0 − 0 · 0 = 0. det 0 0 

❆s ②♦✉ ❝❛♥ s❡❡✱ ✇❡ ❝❛♥ ❞❡r✐✈❡ s♦♠❡ ♠♦r❡ ✐♥❢♦r♠❛t✐♦♥ ❢r♦♠ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❞❡t❡r♠✐♥❛♥t t❤❛♥ ❥✉st t❤❛t ✐t✬s ♦♥❡✲t♦✲♦♥❡✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s t❤❡ ❝♦♥tr❛✲♣♦s✐t✐✈❡ ❢♦r♠ ♦❢ t❤❡ t❤❡♦r❡♠✿ ❈♦r♦❧❧❛r② ✷✳✺✳✾✿ ◆♦♥✲③❡r♦ ❙♦❧✉t✐♦♥s

A ✐s ❛ 2 × 2 det A = 0✳

❙✉♣♣♦s❡ ♦♥❧② ✐❢

X 6= 0

♠❛tr✐①✳ ❚❤❡♥✱ t❤❡r❡ ✐s s✉❝❤ ❛♥

t❤❛t

AX = 0

✐❢ ❛♥❞

❙✐♥❝❡ A(0) = 0✱ t❤✐s ✐♥❞✐❝❛t❡s t❤❛t A ✐s♥✬t ♦♥❡✲t♦✲♦♥❡✳ ❚❤❡r❡ ✐s ♠♦r❡✿ ❈♦r♦❧❧❛r② ✷✳✺✳✶✵✿ ❇✐❥❡❝t✐♦♥s ❛♥❞ ❉❡t❡r♠✐♥❛♥ts ❙✉♣♣♦s❡

A

✐s ❛

2×2

♠❛tr✐①✳ ■t ✐s ❛ ❜✐❥❡❝t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢

det A 6= 0✳

❊①❡r❝✐s❡ ✷✳✺✳✶✶

Pr♦✈❡ t❤❡ r❡st ♦❢ t❤❡ t❤❡♦r❡♠✳ ❙♦✱ ❛ ③❡r♦ ❞❡t❡r♠✐♥❛♥t ✐♥❞✐❝❛t❡s t❤❛t s♦♠❡ ♥♦♥✲③❡r♦ ✈❡❝t♦r X ✐s t❛❦❡♥ t♦ 0 ❜② A✳ ■t ❢♦❧❧♦✇s t❤❛t ❛❧❧ t❤❡ ♠✉❧t✐♣❧❡s✱ kX ✱ ♦❢ X ❛r❡ ❛❧s♦ t❛❦❡♥ t♦ 0✿ A(kX) = kA(X) = k0 = 0 .

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✇❤♦❧❡ ❧✐♥❡ ✐s ❝♦❧❧❛♣s❡❞ t♦ 0✳ ❚❤❡♦r❡♠ ✷✳✺✳✶✷✿ ▲✐♥❡ ❈♦❧❧❛♣s❡s ■❢ ❛ ✈❡❝t♦r ✐s t❛❦❡♥ t♦ ③❡r♦ ❜② ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✱ t❤❡♥ t❤❡ ✇❤♦❧❡ ❧✐♥❡ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤✐s ✈❡❝t♦r ❢r♦♠ t❤❡ ♦r✐❣✐♥ ✐s t❛❦❡♥ t♦ ③❡r♦✿

 A(X) = 0 =⇒ A {Y : Y = kX, k

r❡❛❧

}



= 0.

❲❡ ❝❛♥ ♣❧❛❝❡ ❞✐✛❡r❡♥t ❝♦♦r❞✐♥❛t❡ s②st❡♠s ♦♥ t❤❡ s❛♠❡ ♣❧❛♥❡✳ ❚❤❡ ♦r✐❣✐♥ ✐s t❤❡ s❛♠❡✱ ❜✉t t❤❡ ✉♥✐ts ❛♥❞ t❤❡ ❛♥❣❧❡s ♦❢ t❤❡ ❛①❡s ♠❛② ❜❡ ❞✐✛❡r❡♥t✿

❲❡ ❦♥♦✇ t❤❛t t❤❡ ✈❡❝t♦r ❛❧❣❡❜r❛ r❡♠❛✐♥s t❤❡ s❛♠❡✳ ❍♦✇❡✈❡r✱ ❛ ❝♦♠♣♦♥❡♥t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ❞♦❡s ❞❡♣❡♥❞ ♦♥ ♦✉r ❝❤♦✐❝❡ ♦❢ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠✳ ❚❤❡r❡❢♦r❡✱ ❛ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ❞❡♣❡♥❞s ♦♥ ♦✉r ❝❤♦✐❝❡ ♦❢ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ t♦♦✳ ❘❡♠❛r❦❛❜❧②✱ t❤✐s ✐s♥✬t tr✉❡ ❢♦r t❤❡ ❞❡t❡r♠✐♥❛♥t✦ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t ❢❛❝t ✐s ❛❝❝❡♣t❡❞ ✇✐t❤♦✉t ♣r♦♦❢✿

✷✳✻✳

✶✻✽

■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs

❈♦r♦❧❧❛r② ✷✳✺✳✶✸✿ ❉❡t❡r♠✐♥❛♥t ■s ■♥tr✐♥s✐❝ ❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r r❡♠❛✐♥s t❤❡ s❛♠❡ ✐♥ ❛♥② ❈❛rt❡s✐❛♥ ❝♦♦r❞✐✲ ♥❛t❡ s②st❡♠✳

❚❤❡ ❞❡t❡r♠✐♥❛♥t ✇✐❧❧ t❡❧❧ ✉s ❛ ❧♦t ❛❜♦✉t t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r✿ • det A < 0 ✐♥❞✐❝❛t❡s t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ✢✐♣✳

• | det A| = 1 ✐♥❞✐❝❛t❡s t❤❛t t❤✐s ✐s ❛ ♠♦t✐♦♥✳

• det A = 0 ✐♥❞✐❝❛t❡s t❤❡ ❝♦❧❧❛♣s❡ ♦r t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ♣r♦❥❡❝t✐♦♥✳

❇✉t ❤♦✇ ❞♦ ✇❡ ❞❡t❡❝t str❡t❝❤❡s ♦r r♦t❛t✐♦♥s❄

✷✳✻✳ ■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs

❆♥ ❡❛s② ♦❜s❡r✈❛t✐♦♥ ❛❜♦✉t r♦t❛t✐♦♥ ✐s t❤❛t ✐❢ ❥✉st ♦♥❡ ✈❡❝t♦r ✐s♥✬t r♦t❛t❡❞✱ t❤❡r❡ ✐s ♥♦ r♦t❛t✐♦♥✦ ■❢ ❛ ✈❡❝t♦r ✐s♥✬t r♦t❛t❡❞✱ ✇❤❛t ❝❛♥ ♣♦ss✐❜❧② ❤❛♣♣❡♥ t♦ ✐t ✉♥❞❡r ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r❄ ❆ str❡t❝❤ ✭✇✐t❤ ❛ ♣♦ss✐❜❧❡ ✢✐♣✮✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐t✬s s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ V 7→ λV ,

❢♦r s♦♠❡ r❡❛❧ λ✳ ❊①❛♠♣❧❡ ✷✳✻✳✶✿ r❡✲s❝❛❧✐♥❣

❈♦♥s✐❞❡r ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ❣✐✈❡♥ ❜② t❤❡ ♠❛tr✐①✿  2 0 . A= 0 3 

❲❤❛t ❡①❛❝t❧② ❞♦❡s t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ ❞♦ t♦ ✐t❄ ❚♦ ❛♥s✇❡r✱ ❥✉st ❝♦♥s✐❞❡r ✇❤❡r❡ A t❛❦❡s t❤❡ st❛♥❞❛r❞ ❜❛s✐s ✈❡❝t♦rs✿ " # " # A : e1 =

A : e2 =

1 0

" # 0 1

7→

7→

2 0

" # 0 3

= 2e1

= 3e2

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❤❛t ❤❛♣♣❡♥s t♦ ❡✐t❤❡r ✐s ❛ ✭❞✐✛❡r❡♥t✮ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ A(e1 ) = 2e1 ❛♥❞ A(e2 ) = 3e2

❋✉rt❤❡r♠♦r❡✱ t❤❡ ❡♥t✐r❡t② ♦❢ ❡❛❝❤ ♦❢ t❤❡ ❛①❡s ✐s str❡t❝❤❡❞ t❤✐s ✇❛②✳ ❙♦✱ ✇❡ ❝❛♥ s❛② t❤❛t A str❡t❝❤❡s t❤❡ ♣❧❛♥❡ ❤♦r✐③♦♥t❛❧❧② ❜② ❛ ❢❛❝t♦r ♦❢ 2 ❛♥❞ ✈❡rt✐❝❛❧❧② ❜② 3✱ ✐♥ ❡✐t❤❡r ♦r❞❡r✿

✷✳✻✳ ■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs

✶✻✾

❊✈❡♥ t❤♦✉❣❤ ✇❡ s♣❡❛❦ ♦❢ str❡t❝❤✐♥❣ t❤❡ ♣❧❛♥❡✱ t❤✐s ✐s ♥♦t t♦ s❛② t❤❛t ❛❧❧ ✈❡❝t♦rs ❛r❡ str❡t❝❤❡❞✳ ■♥❞❡❡❞✱

< 1, 1 >       1 2 1 , 6= λ = A 1 3 1

♦t❤❡r ✈❡❝t♦rs ♠❛② ❜❡ r♦t❛t❡❞❀ ❢♦r ❡①❛♠♣❧❡✱ t❤❡ ✈❛❧✉❡s ♦❢

❢♦r ❛♥② r❡❛❧

✐s♥✬t ✐ts ♠✉❧t✐♣❧❡✿

λ✳

❊①❡r❝✐s❡ ✷✳✻✳✷ ❆♥❛❧②③❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ✇✐t❤ ❛ ❞✐❛❣♦♥❛❧ ♠❛tr✐①✿

 h 0 . A= 0 v 

❊①❛♠♣❧❡ ✷✳✻✳✸✿ str❡t❝❤ ❛❧♦♥❣ ♦t❤❡r ❛①❡s ❲❤❛t ✐❢ t❤❡ ♦♣❡r❛t♦r str❡t❝❤❡s ❛❧♦♥❣ ♦t❤❡r ❧✐♥❡s❄ ❍❡r❡ ✇❡ s✐♠♣❧② r♦t❛t❡ t❤❡ ♣✐❝t✉r❡ ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡ t❤r♦✉❣❤

45

❞❡❣r❡❡s t♦ ♠❛❦❡ t❤✐s ♣♦✐♥t✿

❚❤❡ ❝✐r❝❧❡ ✐s str❡t❝❤❡❞✱ ❜✉t ✐♥ ✇❤❛t ❞✐r❡❝t✐♦♥ ♦r ❞✐r❡❝t✐♦♥s❄ ■s t❤❡r❡ ❛ r♦t❛t✐♦♥ t♦♦❄ ■t ✐s ❤❛r❞ t♦ t❡❧❧ ✇✐t❤♦✉t ♣r✐♦r ❦♥♦✇❧❡❞❣❡✳ ❲❡ ❛❧s♦ ❤❛✈❡ s❡❡♥ t❤✐s✿

❚❤❡ ♣❧❛♥❡ ✐s ✈✐s✐❜❧② str❡t❝❤❡❞✱ ❜✉t ✐♥ ✇❤❛t ❞✐r❡❝t✐♦♥ ♦r ❞✐r❡❝t✐♦♥s❄ ■t ✐s ❤❛r❞ t♦ t❡❧❧ ❜❡❝❛✉s❡ t❤❡ r❡s✉❧t s✐♠♣❧② ❧♦♦❦s s❦❡✇❡❞✳

■t ♠✐❣❤t ❜❡ t②♣✐❝❛❧ t❤❡♥ t❤❛t ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ❖♥ s✉❝❤ ❛ ✈❡❝t♦r

V✱ A

A

r♦t❛t❡s s♦♠❡ ✈❡❝t♦rs✱ ❜✉t

A

❛❧s♦ str❡t❝❤❡s ♦t❤❡r ✈❡❝t♦rs✳

❛❝ts ❛s ❛ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✿

A(V ) = λV , ❢♦r s♦♠❡ ♥✉♠❜❡r

λ✳

❋♦r ❡①❛♠♣❧❡✱ ✇❡ s❡❡ ❞✐s♣r♦♣♦rt✐♦♥❛❧ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧ str❡t❝❤✐♥❣✿

✷✳✻✳ ■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs

✶✼✵

❚❤✐s ✐❞❡❛ ❜r✐♥❣ ✉s t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t ❝♦♥❝❡♣t✿

❉❡✜♥✐t✐♦♥ ✷✳✻✳✹✿ ❡✐❣❡♥✈❛❧✉❡ ●✐✈❡♥ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ♦❢

A

✐❢ ✐t s❛t✐s✜❡s✿

A : R2 → R 2 ✱

❛ ✭r❡❛❧✮ ♥✉♠❜❡r

λ

✐s ❝❛❧❧❡❞ ❛♥ ❡✐❣❡♥✈❛❧✉❡

A(V ) = λV ❢♦r s♦♠❡ ♥♦♥✲③❡r♦ ✈❡❝t♦r s♣♦♥❞✐♥❣ t♦

V

✐♥

R2 ✳

❚❤❡♥✱

V

✐s ❝❛❧❧❡❞ ❛♥ ❡✐❣❡♥✈❡❝t♦r ♦❢

A

❝♦rr❡✲

λ✳

❲❛r♥✐♥❣✦ V =0 A(0) = 0✳ ❱❡❝t♦r

✐s ❡①❝❧✉❞❡❞ ❜❡❝❛✉s❡ ✇❡ ❛❧✇❛②s ❤❛✈❡

◆♦t❡ t❤❛t ✏❡✐❣❡♥✑ ♠❡❛♥s ✏❝❤❛r❛❝t❡r✐st✐❝✑ ✐♥ ●❡r♠❛♥✳ ◆♦✇✱ ❤♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡s❡❄

❊①❛♠♣❧❡ ✷✳✻✳✺✿ ✐❞❡♥t✐t② ♦♣❡r❛t♦r ■❢ t❤✐s ✐s t❤❡ ✐❞❡♥t✐t② ♠❛tr✐①✱

A = I✱

t❤❡ ❡q✉❛t✐♦♥ ✐s ❡❛s② t♦ s♦❧✈❡✿

λV = AV = IV = V. ❙♦✱

λ = 1✳

❚❤✐s ✐s t❤❡ ♦♥❧② ❡✐❣❡♥✈❛❧✉❡✳ ❲❤❛t ❛r❡ ✐ts ❡✐❣❡♥✈❡❝t♦rs❄ ❆❧❧ ✈❡❝t♦rs ❜✉t

0✳

■♥❞❡❡❞✱ ♥♦ ✈❡❝t♦r

✐s r♦t❛t❡❞✦

❊①❡r❝✐s❡ ✷✳✻✳✻ ❲❤❛t ❛❜♦✉t ❛ str❡t❝❤ ❜② ❛ ❢❛❝t♦r ♦❢

k❄

❊①❛♠♣❧❡ ✷✳✻✳✼✿ ❞✐❛❣♦♥❛❧ ♠❛tr✐① ▲❡t✬s r❡✈✐s✐t t❤✐s ❞✐❛❣♦♥❛❧ ♠❛tr✐①✿

❚❤❡♥ ♦✉r ✈❡❝t♦r ❡q✉❛t✐♦♥

AV = λV

 2 0 . A= 0 3 

❜❡❝♦♠❡s✿



2 0 0 3

    x x . =λ y y

▲❡t✬s r❡✇r✐t❡✿



     2x λx 2x = λx ❆◆❉ x(2 − λ) = 0, = =⇒ =⇒ 3y λy 3y = λy y(3 − λ) = 0.

(1) (2)

❚❤❡ t✇♦ ❡q✉❛t✐♦♥s ♠✉st ❜❡ s❛t✐s✜❡❞ s✐♠✉❧t❛♥❡♦✉s❧②✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ❩❡r♦ ❋❛❝t♦r ❘✉❧❡ ❛❣❛✐♥✳ ◆♦✇✱ ✇❡ ❤❛✈❡

V =< x, y >6= 0✱

▲❡t✬s ✉s❡ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥s t♦ ❝♦♥s✐❞❡r t❤❡s❡ t✇♦ ❝❛s❡s✿

s♦ ❡✐t❤❡r

x 6= 0

♦r

y 6= 0✳

✷✳✻✳

✶✼✶

■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs

• ❈❛s❡ ✶✿ x 6= 0✱ t❤❡♥ ❢r♦♠ ✭✶✮✱ ✇❡ ❤❛✈❡✿ 2 − λ = 0 =⇒ λ = 2✳ • ❈❛s❡ ✷✿ y = 6 0✱ t❤❡♥ ❢r♦♠ ✭✷✮✱ ✇❡ ❤❛✈❡✿ 3 − λ = 0 =⇒ λ = 3✳

❚❤❡s❡ ❛r❡ t❤❡ ♦♥❧② t✇♦ ♣♦ss✐❜✐❧✐t✐❡s✳ ❲❡ ❤❛✈❡ ❢♦✉♥❞ t❤❡ ❡✐❣❡♥✈❛❧✉❡s✦ ❚❤❡ s❡❝♦♥❞ ♣❛rt ✐s t♦ ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✳ ■❢ λ = 2✱ t❤❡♥ y = 0 ❢r♦♠ ✭✷✮✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡✐❣❡♥✈❡❝t♦rs ❛r❡✿       x , x 6= 0 , 0

A

x x . =2 0 0

■❢ λ = 3✱ t❤❡♥ x = 0 ❢r♦♠ ✭✶✮✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡✐❣❡♥✈❡❝t♦rs ❛r❡✿     0 0 . =3 A y y

  0 , y 6= 0 , y

❚❤❡s❡ t✇♦ s❡ts ❛r❡ ❛❧♠♦st ❡q✉❛❧ t♦ t❤❡ t✇♦ ❛①❡s✦ ■❢ ✇❡ ❛♣♣❡♥❞ 0 t♦ t❤❡s❡ s❡ts ♦❢ ❡✐❣❡♥✈❡❝t♦rs✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❋♦r λ = 2✱ t❤❡ s❡t ✐s t❤❡ x✲❛①✐s✿    x : x r❡❛❧ 0

❆♥❞ ❢♦r λ = 3✱ t❤❡ s❡t ✐s t❤❡ y ✲❛①✐s✿

   0 : y r❡❛❧ y

❚❤❡ s♦❧✉t✐♦♥ ✐s t✐♠❡✲❝♦♥s✉♠✐♥❣✱ ❜✉t t❤❡r❡ ✇✐❧❧ ❜❡ ❛ s❤♦rt✲❝✉t ❧❛t❡r✳ ❲❡ ❤❛✈❡ ❝♦♥✜r♠❡❞ t❤❡ ❢❛❝t t❤❛t✱ ❜❡❝❛✉s❡ ♦❢ t❤❡ ♥♦♥✲✉♥✐❢♦r♠ r❡✲s❝❛❧✐♥❣✱ ❛❧❧ ✈❡❝t♦rs ❛r❡ r♦t❛t❡❞ ❡①❝❡♣t ❢♦r t❤❡ ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ ♦♥❡s✳ ❊①❡r❝✐s❡ ✷✳✻✳✽

❆♥❛❧②③❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ✇✐t❤ ❛ ❞✐❛❣♦♥❛❧ ♠❛tr✐①✿  h 0 . A= 0 v 

❋r♦♠ t❤❡ ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ ❣✉❡ss ❛ ♣❛tt❡r♥✳ ❚❤❡♦r❡♠ ✷✳✻✳✾✿ ▼✉❧t✐♣❧❡s ♦❢ ❊✐❣❡♥✈❡❝t♦rs

❆♥② ♥♦♥✲③❡r♦ ♠✉❧t✐♣❧❡ ♦❢ ❛♥ ❡✐❣❡♥✈❡❝t♦r ✐s ❛❧s♦ ❛♥ ❡✐❣❡♥✈❡❝t♦r ✕ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s❛♠❡ ❡✐❣❡♥✈❛❧✉❡✳ Pr♦♦❢✳

❙✉♣♣♦s❡ V ✐s ❛♥ ❡✐❣❡♥✈❡❝t♦r ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r A ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❡✐❣❡♥✈❛❧✉❡ λ✿ AV = λV .

■❢ W = kV ✱ t❤❡♥ AW = A(kV ) = kAV = kλV = λ(kV ) = λW

❙✉❜st✐t✉t❡✳ ❯s❡ t❤❡ ❢❛❝t t❤❛t ✐t ♣r❡s❡r✈❡s s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ❯s❡ t❤❡ ❢❛❝t t❤❛t t❤✐s ✐s ❛♥ ❡✐❣❡♥✈❡❝t♦r ♦❢ λ. ❘❡❛rr❛♥❣❡✳ ❙✉❜st✐t✉t❡ ❜❛❝❦✳

❚❤❡ ✇❤♦❧❡ ❧✐♥❡ ✐s ♠❛❞❡ ✉♣ ♦❢ ❡✐❣❡♥✈❡❝t♦rs✳ ■t✬s ❛ ❝♦♣② ♦❢ R✦ ▼♦r❡ ❣❡♥❡r❛❧ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

✷✳✻✳

✶✼✷

■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs

❉❡✜♥✐t✐♦♥ ✷✳✻✳✶✵✿ ❡✐❣❡♥s♣❛❝❡ ❋♦r ❛♥ ❡✐❣❡♥✈❛❧✉❡ λ ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r A✱ t❤❡ ❡✐❣❡♥s♣❛❝❡ ♦❢ A ❝♦rr❡s♣♦♥❞✐♥❣ t♦ λ ✐s ❞❡✜♥❡❞ ❛♥❞ ❞❡♥♦t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ E(λ) = {V : A(V ) = λV } .

■t✬s ❛❧❧ ❡✐❣❡♥✈❡❝t♦rs ♦❢ λ ♣❧✉s 0✳ ❲❡ ✐♥❝❧✉❞❡ ✐t ✐♥ ♦r❞❡r t♦ ♠❛❦❡ t❤✐s s❡t ✐♥t♦ ❛ s♣❛❝❡✱ ❛ ✈❡❝t♦r s♣❛❝❡✳ ❋r♦♠ t❤❡ ❡①❛♠♣❧❡s ❛❜♦✈❡✱ ✇❡ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳

❊①❛♠♣❧❡ ✷✳✻✳✶✶✿ ✐❞❡♥t✐t② ♠❛tr✐① ❋♦r t❤❡ ✐❞❡♥t✐t② ♠❛tr✐①✱ ✇❡ ❤❛✈❡✿

E(1) = R2 .

❊①❛♠♣❧❡ ✷✳✻✳✶✷✿ ❞✐❛❣♦♥❛❧ ♠❛tr✐① ❋♦r

 2 0 , A= 0 3 

✇❡ ❤❛✈❡✿

• E(2) ✐s t❤❡ x✲❛①✐s✳ • E(3) ✐s t❤❡ y ✲❛①✐s✳ ❚✇♦ ❝♦♣✐❡s ♦❢ R✦

❊①❛♠♣❧❡ ✷✳✻✳✶✸✿ r♦t❛t✐♦♥ ❆ r♦t❛t✐♦♥ ❞♦❡s♥✬t str❡t❝❤ ❛♥② ✈❡❝t♦rs✳ ❚❤❡r❡❢♦r❡✱ t❤❡r❡ ❛r❡ ♥♦ ✭r❡❛❧✮ ❡✐❣❡♥✈❛❧✉❡s✳ ❚❤❡r❡❢♦r❡✱ t❤❡r❡ ❛r❡ ♥♦ ❡✐❣❡♥✈❡❝t♦rs ❛♥❞ ♥♦ ❡✐❣❡♥s♣❛❝❡s✳

❊①❛♠♣❧❡ ✷✳✻✳✶✹✿ ③❡r♦ ♠❛tr✐① ❋♦r t❤❡ ③❡r♦ ♠❛tr✐①✱ A = 0✱ ✇❡ ❤❛✈❡✿ AV = λV, ♦r 0 = λV .

❚❤❡r❡❢♦r❡✱ λ = 0 s✐♥❝❡ V 6= 0✳ ❋✉rt❤❡r♠♦r❡✿ E(0) = R2 .

❊①❛♠♣❧❡ ✷✳✻✳✶✺✿ ♣r♦❥❡❝t✐♦♥ ❈♦♥s✐❞❡r ♥♦✇ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦♥ t❤❡ x✲❛①✐s✱  1 0 . P = 0 0 

❚❤❡♥ ♦✉r ♠❛tr✐① ❡q✉❛t✐♦♥ ✐s s♦❧✈❡❞ ❛s ❢♦❧❧♦✇s✿ 

1 0 0 0

         x = λx ❆◆❉ λx x x x =⇒ = =⇒ =λ 0 = λy λy 0 y y

❙♦✱ t❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ❝❛s❡s ❛r❡✿

λ = 0 ❛♥❞ λ = 1 .

■t ❛♣♣❡❛rs t❤❛t t❤❡ ♦♣❡r❛t♦r ✐s ♣r♦❥❡❝t✐♥❣ ✐♥ ♦♥❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ❞♦✐♥❣ ♥♦t❤✐♥❣ ✐♥ ❛♥♦t❤❡r✳

✷✳✻✳

✶✼✸

■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs

◆❡①t✱ ✐♥ ♦r❞❡r t♦ ✜♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡✐❣❡♥✈❡❝t♦rs✱ ✇❡ ♥♦✇ ❣♦ ❜❛❝❦ t♦ t❤❡ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ❢♦r x ❛♥❞ y ✳ ❲❡ ❝♦♥s✐❞❡r t❤❡s❡ t✇♦ ❝❛s❡s✳ ❋✐rst✿ ❈❛s❡ ✶✿ λ = 0 =⇒

(

x = 0 · x ❆◆❉ 0=0·y

=⇒

(

=⇒

(

x = 0 ❆◆❉ y ❛♥②

   0 =⇒ E(0) = : y r❡❛❧ y

x ❛♥② ❆◆❉ y=0

   x =⇒ E(1) = : x r❡❛❧ 0

❚❤✐s ✐s t❤❡ y ✲❛①✐s✳ ❙❡❝♦♥❞✿ ❈❛s❡ ✷✿ λ = 1 =⇒

(

x = 1 · x ❆◆❉ 0=1·y

❚❤✐s ✐s t❤❡ x✲❛①✐s✳ ❚②♣✐❝❛❧❧②✱ ✇❡ ❤❛✈❡ t✇♦ ❡✐❣❡♥✈❡❝t♦rs t❤❛t ❛r❡♥✬t ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✿ • A(V1 ) = λ1 V1 ❛♥❞ • A(V2 ) = λ2 V2 ✱

❢♦r s♦♠❡ ♥✉♠❜❡rs λ1 6= λ2 ✳ ❊①❛♠♣❧❡ ✷✳✻✳✶✻✿ str❡t❝❤ ❛❧♦♥❣ ♦t❤❡r ❛①❡s

▲❡t✬s r❡✈✐s✐t t❤❡ str❡t❝❤ ❛❧♦♥❣ s♣❡❝✐❛❧ ❧✐♥❡s✳ ■t ♠✐❣❤t ❧♦♦❦ ❧✐❦❡ t❤✐s✿

❍♦✇❡✈❡r✱ ❤♦✇ ✇♦✉❧❞ ✇❡ ❡✈❡♥ ✜♥❞ t❤❡s❡ s♣❡❝✐❛❧ ❞✐r❡❝t✐♦♥s❄ ❇❡❧♦✇✱ ✇❡ tr② t❤❡ ❜❛s✐s ✈❡❝t♦rs✱ ❧♦♦❦ ❛t ✇❤❡r❡ t❤❡② ❣♦ ✭❢r♦♠ t❤❡ ♠❛tr✐① ✐ts❡❧❢✮✱ ❛♥❞ s❡❡ t❤❛t t❤❡② ❤❛✈❡ r♦t❛t❡❞✿

❚❤❡② ❛r❡ ♥♦t ❡✐❣❡♥✈❡❝t♦rs✦ ❚❤❡ ❡✐❣❡♥✈❡❝t♦rs ❜❡❧♦✇ ♠❛② ❜❡ ❢♦✉♥❞ ❜② tr✐❛❧ ❛♥❞ ❡rr♦r ♦r ❜② t❤❡ ♠❡t❤♦❞ ♣r❡s❡♥t❡❞ ❜❡❧♦✇❀ t❤❡②✱ ✐♥❞❡❡❞✱ ❞♦♥✬t r♦t❛t❡✿

✷✳✻✳

✶✼✹

■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs

❊①❛♠♣❧❡ ✷✳✻✳✶✼✿ ♥♦ ❡✐❣❡♥✈❡❝t♦rs

❈❛♥ ✇❡ ❞❡r✐✈❡ ✇❤❛t ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ❞♦❡s ❢r♦♠ ✐ts ♠❛tr✐① ♦♥❧②✱ ✇✐t❤♦✉t ✈✐s✉❛❧✐③❛t✐♦♥❄ ❙✉♣♣♦s❡ A ✐s ❣✐✈❡♥✿   0 −1 . 1 0

A=

❚❤❡♥✱ t♦ ✜♥❞ t❤❡ ❡✐❣❡♥✈❛❧✉❡s✱ ✇❡ ❝♦♥s✐❞❡r t❤✐s s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ AV = λV =⇒



0 −1 1 0

    x x . =λ y y

❲❡ s♦❧✈❡ ✐t ❛s ❢♦❧❧♦✇s✿ =⇒



−y = λx ❆◆❉ =⇒ x = λy



−xy = λx2 ❆◆❉ xy = λy 2

=⇒ λx2 = −λy 2 =⇒ x2 = −y 2 ❖❘ λ = 0 .

❆ ❞✐r❡❝t ❡①❛♠✐♥❛t✐♦♥ r❡✈❡❛❧s t❤❛t λ = 0 ✐s ♥♦t ❛♥ ❡✐❣❡♥✈❛❧✉❡✿ AV = 0 · V =⇒



0 −1 1 0

  x = 0 =⇒ x = 0, y = 0 . y

■♥ t❤❡ ♠❡❛♥t✐♠❡✱ t❤❡ ❡q✉❛t✐♦♥ x2 = −y 2 ✐s ✐♠♣♦ss✐❜❧❡ ✉♥❧❡ss ❜♦t❤ x ❛♥❞ y ❛r❡ ③❡r♦s✱ ✇❤✐❝❤ ✐s ♥♦t ❛❧❧♦✇❡❞✳ ❚❤❡r❡ s❡❡♠s t♦ ❜❡ ♥♦ ❡✐❣❡♥✈❛❧✉❡s✱ ❝❡rt❛✐♥❧② ♥♦t r❡❛❧ ♦♥❡s✳✳✳ ❚❤✐s ♠❡❛♥s t❤❛t ❡✈❡r② ✈❡❝t♦r ✐s r♦t❛t❡❞✳ ▼❛②❜❡ t❤✐s ✐s ❛ r♦t❛t✐♦♥❄ ❨❡s✱ ✇❡ r❡❝♦❣♥✐③❡ t❤❡ ♠❛tr✐① ♦❢ t❤❡ 90✲❞❡❣r❡❡ r♦t❛t✐♦♥✳ ❊①❡r❝✐s❡ ✷✳✻✳✶✽

❍♦✇ ❞♦❡s t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ A t❡❧❧ ②♦✉ ✇❤❡t❤❡r 0 ✐s ❛♥ ❡✐❣❡♥✈❛❧✉❡❄ ❊①❡r❝✐s❡ ✷✳✻✳✶✾

❙❤♦✇ t❤❛t ❛ ③❡r♦ ❡✐❣❡♥✈❛❧✉❡ ✐♠♣❧✐❡s ❛ ❝♦❧❧❛♣s❡✳ ❊①❛♠♣❧❡ ✷✳✻✳✷✵✿ ♥❡❡❞ ❢♦r ❤♦♠♦❣❡♥❡♦✉s s②st❡♠

▲❡t✬s r❡✈✐s✐t t❤✐s ❧✐♥❡❛r ♦♣❡r❛t♦r✿

 −1 −2 . A= 1 −4 

✷✳✻✳

✶✼✺

■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs

❖✉r ✈❡❝t♦r ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s✿



−1 −2 1 −4

    x x . =λ y y

❲❡ r❡✇r✐t❡✱ ❛❣❛✐♥✱ ❛s ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ 

    (−1 − λ)x − 2y = 0 ❆◆❉ λx −x − 2y =⇒ = x + (−4 − λ)y = 0 λy x − 4y

❚❤✐s ✐s ❛♥♦t❤❡r s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s t♦ ❜❡ s♦❧✈❡❞✱ ❛❣❛✐♥✳ ■t ✐s ♠♦r❡ ❝♦♠♣❧❡① t❤❛♥ t❤❡ ♦♥❡s ✇❡ s❛✇ ❛❜♦✈❡ ❛♥❞ ♥♦♥❡ ♦❢ t❤❡ s❤♦rt❝✉ts ❛r❡ ❛✈❛✐❧❛❜❧❡✳✳✳ ❚❤❡ s②st❡♠ ❝♦rr❡s♣♦♥❞s t♦ ❛ ❤♦♠♦❣❡♥❡♦✉s ✈❡❝t♦r ❡q✉❛t✐♦♥✿     

❲❤❛t ❞♦ ✇❡ ❦♥♦✇ ❛❜♦✉t t❤♦s❡❄

−1 − λ −2 1 −4 − λ

0 x = . y 0

▲❡t✬s r❡✈✐❡✇✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r A ❛♥❞ ✇❡ ♥❡❡❞ t♦ ✜♥❞ ✐ts ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs✳ ▲❡t✬s✱ ❢♦r ♥♦✇✱ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ❢♦r♠❡r✳ ❙✉♣♣♦s❡ λ ✐s ❛♥ ❡✐❣❡♥✈❛❧✉❡ ♦❢ A✳ ❚❤✐s ♠❡❛♥s t❤❛t λ ✐s ❛ r❡❛❧ ♥✉♠❜❡r ❛♥❞ t❤❡r❡ ✐s s♦♠❡ ♥♦♥✲③❡r♦ ✈❡❝t♦r V t❤❛t s❛t✐s✜❡s✿ AV = λV

▲❡t✬s ❞♦ s♦♠❡ ✈❡❝t♦r ❛❧❣❡❜r❛✿ AV = λV =⇒ AV − λV = 0 .

❲❡ ✇❛♥t t♦ t✉r♥ t❤✐s ❡q✉❛t✐♦♥ ♦❢ ✈❡❝t♦rs ❛♥❞ ♠❛tr✐❝❡s ✐♥t♦ ♦♥❡ ❡♥t✐r❡❧② ♦❢ ♠❛tr✐❝❡s✳ ❲❡ ❝❛♥ t❛❦❡ t❤✐s ❡q✉❛t✐♦♥ ♦♥❡ st❡♣ ❢✉rt❤❡r ❜② ♦❜s❡r✈✐♥❣ t❤❛t λV = λIV ,

✇❤❡r❡ I ✐s t❤❡ ✐❞❡♥t✐t② ♠❛tr✐①✳ ❚❤❡ ❧✐♥❡❛r✐t② ♦❢ t❤❡s❡ ♦♣❡r❛t♦rs ❛❧❧♦✇s t♦ ❢❛❝t♦r V ♦✉t ♦❢ ♦✉r ❡q✉❛t✐♦♥✳ ■t t❛❦❡s ❛ ♥❡✇ ❢♦r♠✿ (A − λI)V = 0

❚❤❡ ❡q✉❛t✐♦♥ ❝❤❛r❛❝t❡r✐③❡s ❛♥ ❡✐❣❡♥✈❡❝t♦r ❛♥❞ ✐ts ❡✐❣❡♥✈❛❧✉❡s ✐♥ ❛ s♣❛❝❡ ♦❢ ❛♥② ❞✐♠❡♥s✐♦♥✳ ❊①❛♠♣❧❡ ✷✳✻✳✷✶✿ ❞✐♠❡♥s✐♦♥

2

■♥ t❤❡ R2 ❝❛s❡✱ ✇❡ ♠❛❦❡ t❤✐s s♣❡❝✐✜❝ ✇❤❡♥ ♦✉r ❧✐♥❡❛r ♦♣❡r❛t♦r A ✐s s♣❡❝✐✜❝✳ ❙✉♣♣♦s❡ ✐t ✐s ❣✐✈❡♥ ❜② ❛ ♠❛tr✐①✿   A=

a b . c d

❲❡ ❝❛rr② ♦✉t t❤❡s❡ ❝♦♠♣✉t❛t✐♦♥s✿ 

a b AV = c d

        λx x ax + by x . = ❛♥❞ λ = λy y cx + dy y

❚❤❡s❡ t✇♦ ✈❡❝t♦rs ❛r❡ s✉♣♣♦s❡❞ t♦ ❜❡ ❡q✉❛❧✱ s♦ ✇❡ ❤❛✈❡ ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✱ ✇❤✐❝❤ ✐s t❤❡♥ tr❛♥s❢♦r♠❡❞ ✐♥t♦ ❛ ❤♦♠♦❣❡♥❡♦✉s ❢♦r♠✿ 

ax + by = λx ❆◆❉ cx + dy = λy.

❚❤❡ ♠❛tr✐① ♦❢ t❤✐s s②st❡♠ ✐s✿

⇐⇒



(a − λ)x + by = 0 ❆◆❉ cx + (d − λ)y = 0.

 a−λ b . G= c d−λ 

✷✳✻✳

✶✼✻

■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs

❋♦❧❧♦✇✐♥❣ t❤❡s❡ ❝♦♠♣✉t❛t✐♦♥s✱ ✇❡ r❡❝♦❣♥✐③❡ s♦♠❡ ♠❛tr✐① ❛❧❣❡❜r❛✿     1 0 a b . −λ G= 0 1 c d ❲❡ ❣♦ ❜❛❝❦ t♦ ♦✉r ❝♦♠♣❛❝t r❡♣r❡s❡♥t❛t✐♦♥✿

❚❤❡♦r❡♠ ✷✳✻✳✷✷✿ ❊✐❣❡♥✈❛❧✉❡s ❛♥❞ ❊✐❣❡♥✈❡❝t♦rs ❙✉♣♣♦s❡

A

❡✐❣❡♥✈❡❝t♦r

✐s ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✳

V

♦❢

A

❚❤❡♥ ❡✈❡r② ♣❛✐r ♦❢ ❛♥ ❡✐❣❡♥✈❛❧✉❡

λ

❛♥❞ ✐ts

s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛tr✐① ❡q✉❛t✐♦♥✿

GV = 0,

✇❤❡r❡

G = A − λI

◆♦✇✱ t❤❡ q✉❡st✐♦♥ ❛❜♦✉t t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ t❤❡ ♠❛tr✐① A ❜❡❝♦♠❡s ♦♥❡ ❛❜♦✉t t❤❡ ♠❛tr✐① G✿

◮ ❯♥❞❡r ✇❤❛t ❝✐r❝✉♠st❛♥❝❡s ❞♦❡s t❤❡ s②st❡♠ GV = 0 ❤❛✈❡ ❛ ♥♦♥✲③❡r♦ s♦❧✉t✐♦♥❄ ❲❡ ❦♥♦✇ t❤❡ ❛♥s✇❡r ❢r♦♠ t❤❡ ❧❛st s❡❝t✐♦♥✿

◮ ❚❤❡ s②st❡♠ GV = 0 ❤❛s ❛ ♥♦♥✲③❡r♦ s♦❧✉t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ det G = 0✳ ❲❡ ❤❛✈❡ ♣r♦✈❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t✿

❚❤❡♦r❡♠ ✷✳✻✳✷✸✿ ❊✐❣❡♥✈❛❧✉❡s ❛s ❘♦♦ts ❙✉♣♣♦s❡

A

✐s ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r

R2 ✳

❚❤❡♥ ❡✈❡r② ❡✐❣❡♥✈❛❧✉❡

λ

♦❢

A

✐s ❛ s♦❧✉t✐♦♥

t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥✿

det(A − λI) = 0 ■♥ ❝♦♥tr❛st t♦ t❤❡ ♠❛tr✐① ❡q✉❛t✐♦♥ ✐♥ t❤❡ ❧❛st t❤❡♦r❡♠✱ t❤✐s s✐♠♣❧❡ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ t❤❡♥✱ ♣♦ss✐❜❧②✱ t❤❡✐r ❡✐❣❡♥✈❡❝t♦rs✳

❛❧❣❡❜r❛✐❝

❡q✉❛t✐♦♥ ❛❧❧♦✇s t♦ ❞✐s❝♦✈❡r

❲❡ ❝♦❞✐❢② t❤✐s ✐❞❡❛ ❜❡❧♦✇✿

❉❡✜♥✐t✐♦♥ ✷✳✻✳✷✹✿ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ❚❤❡

❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧

♦❢ ❛ 2 × 2 ♠❛tr✐① A ✐s ❞❡✜♥❡❞ t♦ ❜❡✿

χA (λ) = det(A − λI) ▼❡❛♥✇❤✐❧❡✱ t❤❡ ❡q✉❛t✐♦♥ χA (λ) = 0 ✐s ❝❛❧❧❡❞ t❤❡



❝❤❛r❛❝t❡r✐st✐❝ ❡q✉❛t✐♦♥

❚❤✐s ✐s t❤❡ ❝♦♥✈❡♥✐❡♥t ❢♦r♠ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✇❡ ❛r❡ t♦ s♦❧✈❡ ❢♦r ❞✐♠❡♥s✐♦♥ n = 2✿   a−λ b = (a − λ)(d − λ) − bc = 0 . χA (λ) = det c d−λ ■t✬s

q✉❛❞r❛t✐❝



❲❡ ❞♦♥✬t ❦♥♦✇ ✇❤❛t ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ❞♦❡s ❜✉t ✕ ❡✈❡♥ ✇✐t❤♦✉t t❤❡ ❡✐❣❡♥✈❡❝t♦rs ✕ ✇❡ ❝❛♥ t❡❧❧ ❛ ❧♦t ❢r♦♠ ✐ts ❡✐❣❡♥✈❛❧✉❡s✳ ❲❡ r❡❞✐s❝♦✈❡r s♦♠❡ ♦❢ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❢❛♠✐❧✐❛r ♦♣❡r❛t♦rs ❜❡❧♦✇✳

✷✳✻✳ ■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs

✶✼✼

❊①❛♠♣❧❡ ✷✳✻✳✷✺✿ r❡✲s❝❛❧✐♥❣❄

❈♦♥s✐❞❡r ❛❣❛✐♥✿ ❚❤❡♥ ✇❡ s♦❧✈❡✿ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

 2 0 . A= 0 3 

 2−λ 0 = (2 − λ)(3 − λ) = 0 . χA (λ) = det 0 3−λ 

λ = 2, 3 .

❲❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r str❡t❝❤❡s t❤❡ ♣❧❛♥❡ ❜② t❤❡s❡ ❢❛❝t♦rs ✐♥ t✇♦ ❞✐✛❡r❡♥t ❞✐r❡❝t✐♦♥s✳ ❲❤❛t ❛r❡ t❤♦s❡ ❞✐r❡❝t✐♦♥s❄ ❲❡ ❝❛♥✬t t❡❧❧ ✇✐t❤♦✉t ✜♥❞✐♥❣ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✳ ❊①❛♠♣❧❡ ✷✳✻✳✷✻✿ ♣r♦❥❡❝t✐♦♥❄

■❢

 1 0 , A= 0 0 

t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❡q✉❛t✐♦♥ ✐s✿

 1−λ 0 = (1 − λ)(−λ) = 0 . χA (λ) = det 0 −λ 

❚❤❡♥✱ λ = 1, 0 .

❲❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r ❞♦❡s ♥♦t❤✐♥❣ ✐♥ ♦♥❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ❝♦❧❧❛♣s❡s ✐♥ ❛♥♦t❤❡r✳ ❚❤❛t✬s ❛ ♣r♦❥❡❝t✐♦♥✦ ❲❤❛t ❛r❡ t❤♦s❡ ❞✐r❡❝t✐♦♥s❄ ❲❡ ❞♦♥✬t ❦♥♦✇ ✇✐t❤♦✉t t❤❡ ❡✐❣❡♥✈❡❝t♦rs✳ ❊①❛♠♣❧❡ ✷✳✻✳✷✼✿ r♦t❛t✐♦♥❄

❈♦♥s✐❞❡r ❚❤❡♥✱

 0 −1 . A= 1 0 

 −λ −1 = λ2 + 1 = 0 . χA (λ) = det 1 −λ 

◆♦ r❡❛❧ s♦❧✉t✐♦♥s✦ ❙♦✱ ♥♦ ♥♦♥✲③❡r♦ ✈❡❝t♦r ✐s t❛❦❡♥ ❜② A t♦ ✐ts ♦✇♥ ♠✉❧t✐♣❧❡✳ ▼❛②❜❡ t❤✐s ✐s ❛ r♦t❛t✐♦♥✳✳✳ ❚❤❡s❡ t❤r❡❡ ❡①❛♠♣❧❡s s✉❣❣❡st ❛ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ♦❢ t❤❡ ♣❧❛♥❡✳ ❇✉t ✜rst ❛ q✉✐❝❦ r❡✈✐❡✇ ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s✳ ❈♦♥s✐❞❡r ♦♥❡✿

f (x) = x2 + px + q .

❚❤❡ ◗✉❛❞r❛t✐❝ ❋♦r♠✉❧❛ t❤❡♥ ♣r♦✈✐❞❡s t❤❡ x✲✐♥t❡r❝❡♣ts ♦❢ t❤✐s ❢✉♥❝t✐♦♥✿ p x=− ± 2

p p2 − 4q . 2

❖❢ ❝♦✉rs❡✱ t❤❡ x✲✐♥t❡r❝❡♣ts ❛r❡ t❤❡ r❡❛❧ s♦❧✉t✐♦♥s ♦❢ t❤✐s ❡q✉❛t✐♦♥ ❛♥❞ t❤❛t ✐s ✇❤② t❤❡ r❡s✉❧t ♦♥❧② ♠❛❦❡s s❡♥s❡ ✇❤❡♥ t❤❡ ❞✐s❝r✐♠✐♥❛♥t ♦❢ t❤❡ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✱ D = p2 − 4q ,

✐s ♥♦♥✲♥❡❣❛t✐✈❡✳

✷✳✻✳

✶✼✽

■t✬s ❛ str❡t❝❤✿ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs

■♥❝r❡❛s✐♥❣ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢r❡❡ t❡r♠ q ♠❛❦❡s t❤❡ ❣r❛♣❤ ♦❢ y = f (x) s❤✐❢t ✉♣✇❛r❞ ❛♥❞✱ ❡✈❡♥t✉❛❧❧②✱ ♣❛ss t❤❡ x✲❛①✐s ❡♥t✐r❡❧②✳ ❲❡ ❝❛♥ ♦❜s❡r✈❡ ❤♦✇ ✐ts t✇♦ x✲✐♥t❡r❝❡♣ts st❛rt t♦ ❣❡t ❝❧♦s❡r t♦ ❡❛❝❤ ♦t❤❡r✱ t❤❡♥ ♠❡r❣❡✱ ❛♥❞ ✜♥❛❧❧② ❞✐s❛♣♣❡❛r✿

❚❤✐s ♣r♦❝❡ss ✐s ❡①♣❧❛✐♥❡❞ ❜② ✇❤❛t ✐s ❤❛♣♣❡♥✐♥❣✱ ✇✐t❤ t❤❡ ❣r♦✇t❤ ♦❢ q ✱ t♦ t❤❡ r♦♦ts ❣✐✈❡♥ ❜② t❤❡ ◗✉❛❞r❛t✐❝ ❋♦r♠✉❧❛ ✿ √ D p x1,2 = − ± . 2 2 ❚❤❡r❡ ❛r❡ t❤r❡❡ st❛t❡s✿ √ D ✶✳ ❙t❛rt✐♥❣ ✇✐t❤ ❛ ♣♦s✐t✐✈❡ ✈❛❧✉❡✱ D ❞❡❝r❡❛s❡s✱ ❛♥❞ ❞❡❝r❡❛s❡s✳ 2 √ D = 0✳ ✷✳ ❚❤❡♥ D ❜❡❝♦♠❡s 0 ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ 2 ✸✳ ❚❤❡♥ D ❜❡❝♦♠❡s ♥❡❣❛t✐✈❡✱ ❛♥❞ t❤❡r❡ ❛r❡ ♥♦ r❡❛❧ r♦♦ts ✭❝♦♠♣❧❡① r♦♦ts ❛r❡ ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✮✳ ❙♦✱ ✇❡ ❤❛✈❡✿

• ❚❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ t❤❡ r❡❛❧ r♦♦ts ♦❢ t❤❡ ✭q✉❛❞r❛t✐❝✮ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ χA ✳

• ❚❤❡r❡❢♦r❡✱ t❤❡ ♥✉♠❜❡r ♦❢ ❡✐❣❡♥✈❛❧✉❡s ✐s ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ 2✱ ❝♦✉♥t✐♥❣ t❤❡✐r ♠✉❧t✐♣❧✐❝✐t✐❡s✳

▲❡t✬s tr② t♦ ❡①♣❛♥❞ t❤❡ ❝❤❛r❛❝t❡rst✐❝ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ s❡❡ ✐❢ ♣❛tt❡r♥s ❡♠❡r❣❡✿   b  a − λ χ(λ) = det   c d−λ

= (a − λ)(d − λ) − bc

= ad − aλ − λd + λ2 − bc = λ2 − (a + d)λ + (ad − bc) = λ2 − tr A λ + det A . ❚❤❡ t❡r♠ ✐♥ t❤❡ ♠✐❞❞❧❡ ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿

❉❡✜♥✐t✐♦♥ ✷✳✻✳✷✽✿ tr❛❝❡ ♦❢ ♠❛tr✐① ❚❤❡

tr❛❝❡

♦❢ ❛ ♠❛tr✐① A ✐s t❤❡ s✉♠ ♦❢ ✐ts ❞✐❛❣♦♥❛❧ ❡❧❡♠❡♥ts✳ ■t ✐s ❞❡♥♦t❡❞ ❜②✿

 a b =a+d tr A = tr c d 

❙♦✱ t❤❡ tr❛❝❡ ❛♣♣❡❛rs ✕ ❛❧♦♥❣ ✇✐t❤ t❤❡ ❞❡t❡r♠✐♥❛♥t ✕ ✐♥ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✿

✷✳✼✳ ❚❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ ❡✐❣❡♥✈❡❝t♦rs

✶✼✾

❚❤❡♦r❡♠ ✷✳✻✳✷✾✿ ❈❤❛r❛❝t❡r✐st✐❝ P♦❧②♥♦♠✐❛❧ ❚❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ♦❢ ♠❛tr✐①

A

t❛❦❡s t❤✐s ❢♦r♠✿

χA (λ) = λ2 − tr A · λ + det A

■t ✐s ❦♥♦✇♥ t❤❛t ♥♦t ♦♥❧② t❤❡ ❞❡t❡r♠✐♥❛♥t ❜✉t ❛❧s♦ t❤❡ tr❛❝❡ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ♦✉r ❝❤♦✐❝❡ ♦❢ ❛ ❈❛rt❡s✐❛♥ s②st❡♠✳ ❚❤❡r❡❢♦r❡✱ s♦ ✐s t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✳ ❚❤❡ ❞✐s❝r✐♠✐♥❛♥t ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ❝❛♥ ♥♦✇ ❜❡ ✉s❡❞ t♦ t❡❧❧ ✇❤❛t t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❞♦❡s✿ D = (tr A)2 − 4 det A .

❚❤❡♦r❡♠ ✷✳✻✳✸✵✿ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r ❖♣❡r❛t♦rs ❙✉♣♣♦s❡

A ✐s ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ❣✐✈❡♥ ❜② ❛ 2 × 2 ♠❛tr✐① ❛♥❞ D

✐s t❤❡ ❞✐s❝r✐♠✐♥❛♥t

♦❢ ✐ts ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✳ ❚❤❡♥ ✇❡ ❤❛✈❡ t❤r❡❡ ❝❛s❡s✿ ✶✳

D > 0✳

❚❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ❞✐st✐♥❝t✿ ❖♣❡r❛t♦r

A

♥♦♥✲✉♥✐❢♦r♠❧② r❡✲s❝❛❧❡s

t❤❡ ♣❧❛♥❡ ✐♥ t❤❡ ❞✐st✐♥❝t ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡✐❣❡♥✈❡❝t♦rs✳ ✷✳

D = 0✳

❚❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ❡q✉❛❧✿ ❖♣❡r❛t♦r

A

✉♥✐❢♦r♠❧② r❡✲s❝❛❧❡s t❤❡

♣❧❛♥❡ ✐♥ ❛❧❧ ❞✐r❡❝t✐♦♥s ✉♥❧❡ss t❤❡ ❡✐❣❡♥✈❡❝t♦rs ❛r❡ ❛❧❧ ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✳ ✸✳

D < 0✳

❚❤❡r❡ ❛r❡ ♥♦ ❡✐❣❡♥✈❛❧✉❡s✿ ❖♣❡r❛t♦r

A

r♦t❛t❡s t❤❡ ♣❧❛♥❡ ✭✇✐t❤ ❛

♣♦ss✐❜❧❡ r❡✲s❝❛❧✐♥❣✮✳

Pr♦♦❢✳ ❋♦r P❛rt ✶✱ ✇❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t t❤❡ ❞✐r❡❝t✐♦♥s ❛r❡ ❞✐st✐♥❝t ❜❡❝❛✉s❡ ✐❢ t✇♦ ❡✐❣❡♥✈❡❝t♦rs ❛r❡ ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✱ t❤❡♥ t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ ❡✐❣❡♥✈❛❧✉❡✳ ■♥❞❡❡❞✿ A(V ) = λV =⇒ A(kV ) = kA(V ) = kλV = λ(kV ) .

P❛rts ✷ ❛♥❞ ✸ ❛r❡ ❛❞❞r❡ss❡❞ ❧❛t❡r ✐♥ t❤❡ ❝❤❛♣t❡r✳

✷✳✼✳ ❚❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ ❡✐❣❡♥✈❡❝t♦rs ❲❡ ❤❛✈❡ s❤♦✇♥ ❤♦✇ ♦♥❡ ❝❛♥ ✈✐s✉❛❧✐③❡ t❤❡ ✇❛② ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r tr❛♥s❢♦r♠s t❤❡ ♣❧❛♥❡✿ ❜② ❡①❛♠✐♥✐♥❣ ✇❤❛t ❤❛♣♣❡♥s t♦ ✈❛r✐♦✉s ❝✉r✈❡s ✐♥ t❤❡ ❞♦♠❛✐♥✳ ❇② ♠❛♣♣✐♥❣ t❤❡s❡ ❝✉r✈❡s✱ ♦♥❡ ❝❛♥ ❞✐s❝♦✈❡r str❡t❝❤✐♥❣✱ s❤r✐♥❦✐♥❣ ✐♥ ✈❛r✐♦✉s ❞✐r❡❝t✐♦♥s✱ r♦t❛t✐♦♥s✱ ❡t❝✳ ■♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ ✇❡ ❛❧s♦ s❛✇ ❤♦✇ ♦♥❡ ❝❛♥ ✉♥❞❡rst❛♥❞ t❤❡ ✇❛② ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r tr❛♥s❢♦r♠s t❤❡ ♣❧❛♥❡✿ ❜② ❡①❛♠✐♥✐♥❣ ✐ts ❡✐❣❡♥✈❛❧✉❡s✳ ❚❤❡ ♠❡t❤♦❞ ✐s ❡♥t✐r❡❧② ❛❧❣❡❜r❛✐❝ r❛t❤❡r t❤❛♥ ❡①♣❡r✐♠❡♥t❛❧✳ ❲❡ s✐♠♣❧② ✜♥❞ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ ♣✉r❡ str❡t❝❤ ❢♦r F ✿ F V = λV

❚❤❡ ✈✐s✉❛❧✐③❛t✐♦♥s ❛r❡ ♣r♦❞✉❝❡❞ ❜② ❛ s♣r❡❛❞s❤❡❡t✳ ❚❤❡ s♣r❡❛❞s❤❡❡t ❛❧s♦ ❝♦♠♣✉t❡s t❤❡ ❡✐❣❡♥✈❡❝t♦r ❛♥❞ ✐ts ❡✐❣❡♥✈❛❧✉❡s❀ t❤❡② ❛r❡ s❤♦✇♥ ❛❜♦✈❡ t❤❡ ❣r❛♣❤s✳❚❤❡ s♣r❡❛❞s❤❡❡t ❛❧s♦ s❤♦✇s ❡✐❣❡♥s♣❛❝❡s ❛s t✇♦ ✭♦r ♦♥❡✱ ♦r ♥♦♥❡✮ str❛✐❣❤t ❧✐♥❡s❀ t❤❡② r❡♠❛✐♥ ✐♥ ♣❧❛❝❡ ✉♥❞❡r t❤❡ tr❛♥s❢♦r♠❛t✐♦♥✳ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❧❡❛r♥ ❤♦✇ t♦ ♣r❡❞✐❝t t❤❡ ♦✉t❝♦♠❡ ❜② ❡①❛♠✐♥✐♥❣ ♦♥❧② ✐ts ♠❛tr✐①✳

✷✳✼✳

❚❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ ❡✐❣❡♥✈❡❝t♦rs

✶✽✵

❇❡❧♦✇ ✐s ❛ ❢❛♠✐❧✐❛r ❢❛❝t t❤❛t ✇✐❧❧ t❛❦❡ ✉s ❞♦✇♥ t❤❛t r♦❛❞✿

❚❤❡♦r❡♠ ✷✳✼✳✶✿ Pr❡✐♠❛❣❡s ♦❢ ❩❡r♦

V 6= 0 ✉♥❞❡r ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r F kV ✳

■❢ t❤❡ ✐♠❛❣❡ ♦❢ ✐ts ♠✉❧t✐♣❧❡s

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✇❤♦❧❡ ❧✐♥❡ ✇✐t❤

V

✐s ③❡r♦✱ t❤❡♥ s♦ ✐s t❤❛t ♦❢ ❛♥② ♦❢

❛s ✐ts ❞✐r❡❝t✐♦♥ ✈❡❝t♦r ✐s ❝♦❧❧❛♣s❡❞ t♦

0

❜②

F✿

❊①❛♠♣❧❡ ✷✳✼✳✷✿ ❝♦❧❧❛♣s❡ ♦♥ ❛①✐s ❲❡ st❛rt ✇✐t❤ ❛ ❢❛♠✐❧✐❛r ❡①❛♠♣❧❡✿



u = 2x v =0

✐s r❡✲✇r✐tt❡♥ ❛s

     2 0 x u . = 0 0 y v

❊✈❡♥ ✇✐t❤♦✉t t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❡q✉❛t✐♦♥✱ ✇❡ ❝❛♥ ❣✉❡ss t❤❡ ❡✐❣❡♥✈❛❧✉❡✲❡✐❣❡♥✈❡❝t♦r ♣❛✐rs✿



2 0 0 0

    1 1 , =2 0 0



2 0 0 0

    0 0 . =0 1 1

❇❡❧♦✇✱ ♦♥❡ ❝❛♥ s❡❡ ❤♦✇ t❤✐s ♦♣❡r❛t♦r ♣r♦ ❥❡❝ts t❤❡ ✇❤♦❧❡ ♣❧❛♥❡ t♦ t❤❡

❚❤❡ ♦♣❡r❛t♦r ❝♦❧❧❛♣s❡s t❤❡

y ✲❛①✐s

t♦

0✱

✇❤✐❧❡ t❤❡

x✲❛①✐s

x✲❛①✐s✿

✐s str❡t❝❤❡❞ ❜② ❛ ❢❛❝t♦r ♦❢

2✳

❚❤❡ st❛♥❞❛r❞

❜❛s✐s ✈❡❝t♦rs ❤❛♣♣❡♥ t♦ ❜❡ ❡✐❣❡♥✈❡❝t♦rs✦ ❚❤❛t✬s t❤❡ r❡❛s♦♥ ✇❤② t❤❡ ♠❛tr✐① ✐s s♦ s✐♠♣❧❡✳

✷✳✼✳

❚❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ ❡✐❣❡♥✈❡❝t♦rs

✶✽✶

❊①❛♠♣❧❡ ✷✳✼✳✸✿ str❡t❝❤✲s❤r✐♥❦ ❛❧♦♥❣ ❛①❡s ▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ❧✐♥❡❛r ♦♣❡r❛t♦r ❛♥❞ ✐ts ♠❛tr✐①



u = 2x v = 4y

❛♥❞

 2 0 . F = 0 4 

❖♥❝❡ ❛❣❛✐♥✱ ✇❡ ❞♦♥✬t ♥❡❡❞ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❡q✉❛t✐♦♥ t♦ s✉❣❣❡st t❤❡ ❡✐❣❡♥✈❛❧✉❡s ✭❛♥❞ t❤❡♥ ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✮✿



2 0 0 0

    1 1 , =2 0 0



2 0 0 0

    0 0 . =4 1 1

❆s ✐t t✉r♥s ♦✉t✱ ✇❡ ♦♥❧② ♥❡❡❞ t♦ tr❛❝❦ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❜❛s✐s ✈❡❝t♦rs✱ ❛♥❞ t❤❡ r❡st ♦❢ t❤❡ ✈❛❧✉❡s ❛r❡ s❡❡♥ ❛s ❛

❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥

♦❢ t❤❡s❡✿

❚❤❡ r❡st ♦❢ t❤❡ ✈❡❝t♦rs t✉r♥ ♥♦♥✲✉♥✐❢♦r♠❧②❀ ✐✳❡✳✱ t❤❡② ✏❢❛♥ ♦✉t✑✿

X =< x, y >✿            0 1 0 2 2 0 x = 2xe1 + 4ye2 . + 4y = 2x +y =x FX = 1 0 4 0 0 4 y

❚❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s t♦ ❛♥ ❛r❜✐tr❛r② ✈❡❝t♦r

❚❤❡ ❧❛st ❡①♣r❡ss✐♦♥ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ t✇♦

st❛♥❞❛r❞

❜❛s✐s ✈❡❝t♦rs✳ ❚❤❡ ♠✐❞❞❧❡✱

❤♦✇❡✈❡r✱ ✐s ❛❧s♦ ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ❜✉t ✇✐t❤ r❡s♣❡❝t t♦ t❤❡s❡ t✇♦ ✈❡❝t♦rs✿

  2 V1 = 0

❛♥❞

  0 . V2 = 4

❚❤❡② ❝❛♥ ❜❡ t❤♦✉❣❤t ♦❢ ❛s ❢♦r♠✐♥❣ ❛ ✏♥♦♥✲st❛♥❞❛r❞✑ ❜❛s✐s✳ ❚❤♦✉❣❤ ♥♦t ✉♥✐t ✈❡❝t♦rs ❛s t❤❡ st❛♥❞❛r❞ ♦♥❡s✱ t❤❡② ❛r❡ st✐❧❧ ❛❧✐❣♥❡❞ ✇✐t❤ t❤❡ ❛①❡s✳ ◆♦✇✱ ✇❤❛t ✐s t❤❡ ♣♦✐♥t❄ ❊✈❡r② ✈❡❝t♦r ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s

✷✳✼✳ ❚❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ ❡✐❣❡♥✈❡❝t♦rs

✶✽✷

❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦✿

y x V1 + V2 . 2 4 ❋✉rt❤❡r♠♦r❡✱ t♦ ❦♥♦✇ ✇❤❡r❡ ❛♥② X ❣♦❡s ✉♥❞❡r F ✱ ✇❡ ♥❡❡❞ ♦♥❧② t♦ ❦♥♦✇ ✇❤❡r❡ t❤❡s❡ t✇♦ ❣♦✿ ■t✬s ♣✉r❡ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✦ ❲❡ ✇✐❧❧ s❡❡ t❤❛t ❛♥② ♣❛✐r ♦❢ ❡✐❣❡♥✈❡❝t♦rs ✕ ✇❤❡♥ ♥♦t ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r ✕ ✇♦✉❧❞ ❞♦✳ < x, y >=

❊①❛♠♣❧❡ ✷✳✼✳✹✿ str❡t❝❤✲s❤r✐♥❦ ❛❧♦♥❣ ❛①❡s

❆ s❧✐❣❤t❧② ❞✐✛❡r❡♥t ♦♣❡r❛t♦r ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿  u = −x v = 4y

 −1 0 . ❛♥❞ F = 0 4 

■t ✐s st✐❧❧ s✐♠♣❧❡ ❜❡❝❛✉s❡ t❤❡ t✇♦ ✈❛r✐❛❜❧❡s r❡♠❛✐♥ ❢✉❧❧② s❡♣❛r❛t❡❞✳ ❆s ❛ r❡s✉❧t✱ t❤❡ t✇♦ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ❛①❡s ❝❛♥ ❜❡ t❤♦✉❣❤t ♦❢ ❛s tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ✇❤♦❧❡ ♣❧❛♥❡✿

❚❤❡ ♥❡❣❛t✐✈❡ s✐❣♥ ♣r♦❞✉❝❡s ❛ ❞✐✛❡r❡♥t ♣❛tt❡r♥✿ ❲❡ s❡❡ t❤❡ r❡✈❡rs❛❧ ♦❢ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❡❧❧✐♣s❡ ❛r♦✉♥❞ t❤❡ ♦r✐❣✐♥✳ ❆❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ ❤❛✈❡ ❛s ❜❡❢♦r❡✿            −1 0 x −1 0 1 0 FX = =x +y = −x = −xe1 + 4ye2 . + 4y 1 0 4 y 0 4 0 ❖♥❝❡ ❛❣❛✐♥✱ t❤❡ ❧❛st ❡①♣r❡ss✐♦♥ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ t✇♦ st❛♥❞❛r❞ ❜❛s✐s ✈❡❝t♦rs✱ ✇❤✐❧❡ t❤❡ ♠✐❞❞❧❡ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ❜✉t ✇✐t❤ r❡s♣❡❝t t♦ ❛♥♦t❤❡r ❜❛s✐s ♠❛❞❡ ✉♣ ♦❢ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✿     0 −1 . ❛♥❞ V2 = V1 = 4 0 ❲❤❛t ✐❢ t❤❡ ♠❛tr✐① ✐s♥✬t ❞✐❛❣♦♥❛❧❄ ❊①❡r❝✐s❡ ✷✳✼✳✺

❙❤♦✇ t❤❛t t❤❡ st❛♥❞❛r❞ ❜❛s✐s ✈❡❝t♦rs e1 , e2 ❛r❡ ❡✐❣❡♥✈❡❝t♦rs ♦❢ ❞✐❛❣♦♥❛❧ ♠❛tr✐❝❡s✳ ■♥st❡❛❞ ♦❢ t❤❡ st❛♥❞❛r❞ ❜❛s✐s ✈❡❝t♦rs✱ ✇❡ ✇✐❧❧ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ❡✐❣❡♥✈❡❝t♦rs ♦❢ t❤❡ ♦♣❡r❛t♦r ✐♥ ♦r❞❡r t♦ ✉♥❞❡rst❛♥❞ ✇❤❛t t❤❡ ♦♣❡r❛t♦r ❞♦❡s✳ ❚♦ ✐❧❧✉str❛t❡✱ ❥✉st ✐♠❛❣✐♥❡ t❤❛t t❤❡ ♣✐❝t✉r❡ ♦♥ t❤❡ ❧❡❢t ❤❛s ❜❡❡♥ s❦❡✇❡❞✱ r❡s✉❧t✐♥❣ ✐♥ t❤❡ ✐♠❛❣❡ ♦❢ t❤❡ r✐❣❤t✿

✷✳✼✳ ❚❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ ❡✐❣❡♥✈❡❝t♦rs

✶✽✸

❚❤❡ ❡✐❣❡♥✈❡❝t♦rs ♦❢ t❤❡ ♠❛tr✐① ✇✐❧❧ s❡r✈❡ ❛s ❛♥ ❛❧t❡r♥❛t✐✈❡ ❜❛s✐s✳ ❊①❛♠♣❧❡ ✷✳✼✳✻✿ ❝♦❧❧❛♣s❡

▲❡t✬s ❝♦♥s✐❞❡r ❛ ♠♦r❡ ❣❡♥❡r❛❧ ❧✐♥❡❛r ♦♣❡r❛t♦r✿ 

u = x + 2y v = 2x + 4y

 1 2 . =⇒ F = 2 4 

■t ❛♣♣❡❛rs t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ❤❛s ❛ str❡t❝❤✐♥❣ ✐♥ ♦♥❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ❛ ❝♦❧❧❛♣s❡ ✐♥ ❛♥♦t❤❡r✳ ❲❤❛t ❛r❡ t❤♦s❡ ❞✐r❡❝t✐♦♥s❄ ▲✐♥❡❛r ❛❧❣❡❜r❛ ❣✐✈❡s t❤❡ ❛♥s✇❡r✳ ❊✈❡♥ ✇✐t❤♦✉t ❧♦♦❦✐♥❣ ❢♦r ❡✐❣❡♥✈❡❝t♦rs✱ ✇❡ ❦♥♦✇ t❤❛t ✇❡ ❝❛♥ ✉s❡ t❤❡ ❢❛❝t t❤❛t t❤❡ ❞❡t❡r♠✐♥❛♥t ✐s ③❡r♦ ✿  1 2 = 1 · 4 − 2 · 2 = 0. det F = det 2 4 

■t✬s ♥♦t ♦♥❡✲t♦✲♦♥❡ ❛♥❞✱ ✐♥ ❢❛❝t✱ t❤❡r❡ ✐s ❛ ✇❤♦❧❡ ❧✐♥❡ ♦❢ ♣♦✐♥ts X ✇✐t❤ F X = 0✳ ❚♦ ✜♥❞ ✐t✱ ✇❡ s♦❧✈❡ t❤✐s ❡q✉❛t✐♦♥ ❜② s♦❧✈✐♥❣ t❤✐s s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✿ 

x + 2y = 0 2x + 4y = 0

=⇒ x = −2y .

❚❤❡ t✇♦ ❡q✉❛t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t ❛♥❞ r❡♣r❡s❡♥t t❤❡ s❛♠❡ ❧✐♥❡✳ ❲❡ ❤❛✈❡✱ ✐♥❞✐r❡❝t❧②✱ ❢♦✉♥❞ t❤❡ ❡✐❣❡♥s♣❛❝❡ ❛♥❞✱ ♦❢ ❝♦✉rs❡✱ t❤❡ ❡✐❣❡♥✈❡❝t♦rs ♦❢ t❤❡ ③❡r♦ ❡✐❣❡♥✈❛❧✉❡ λ1 = 0✳ ❲❡ ❝❛♥ t❛❦❡ t❤✐s ❡✐❣❡♥✈❡❝t♦r ❢♦r ❢✉rt❤❡r ✉s❡✿   V1 =

2 −1

=⇒ F V1 = 0 .

▲❡t✬s ✐♥st❡❛❞ t✉r♥ t♦ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✿  1−λ 2 = λ2 − 5λ = λ(λ − 5) =⇒ λ1 = 0, λ2 = 5 . det(F − λI) = det 2 4−λ 

▲❡t✬s ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs ❢♦r λ2 = 5✳ ❲❡ ♥❡❡❞ t♦ s♦❧✈❡ t❤❡ ✈❡❝t♦r ❡q✉❛t✐♦♥✿ F V = 5V ,

✷✳✽✳

❇❛s❡s

✶✽✹

✐✳❡✳✱



1 2 FV = 2 4

    x x . =5 y y

❚❤✐s ❣✐✈❡s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ t✇♦ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ✭✐t✬s t❤❡ s❛♠❡ ❡q✉❛t✐♦♥✮✿



x + 2y = 5x ❆◆❉ 2x + 4y = 5y

=⇒



−4x + 2y = 0 ❆◆❉ 2x − y =0

=⇒ y = 2x .

❚❤✐s ❧✐♥❡ ✐s t❤❡ ❡✐❣❡♥s♣❛❝❡✳ ❲❡ ❝❤♦♦s❡ ❛ ✈❡❝t♦r ❛❧♦♥❣ t❤✐s ❧✐♥❡ t♦ ❛s t❤❡ ❡✐❣❡♥✈❡❝t♦r ❢♦r ❢✉rt❤❡r ✉s❡✿

  1 . V2 = 2 ❲❡ s✉♠♠❛r✐③❡ ✇❤❛t

• •

F

❞♦❡s✿

❆ ♣r♦❥❡❝t✐♦♥ ❛❧♦♥❣ t❤❡ ✈❡❝t♦r ❆ str❡t❝❤ ❜② ❛ ❢❛❝t♦r ♦❢

5

< 2, −1 >✿

x = −2y ✐s ❝♦❧❧❛♣s❡❞ t♦ 0✳ < 1, 2 >✿ ❚❤❡ ❧✐♥❡ y = 2x ✐s str❡t❝❤❡❞

❚❤❡ ❧✐♥❡

❛❧♦♥❣ t❤❡ ✈❡❝t♦r

✇✐t❤♦✉t ❛♥②

r♦t❛t✐♦♥✳

❲❡ ❤❛✈❡ ❝♦♥✜r♠❡❞ t❤❡ ✐❧❧✉str❛t✐♦♥ ❛❜♦✈❡✦ ❋✉rt❤❡r♠♦r❡✱ t❤❡ t✇♦ ❡✐❣❡♥✈❡❝t♦rs ❛r❡♥✬t ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✳ ❚❤❛t ✐s ✇❤② ❡✈❡r② ✈❡❝t♦r ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ ❡✐❣❡♥✈❡❝t♦rs ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✐ts ✈❛❧✉❡ ✉♥❞❡r

F

✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡

❡✐❣❡♥✈❡❝t♦rs t♦♦✿

X = xV1 + yV2 =⇒ F X = x · 0 · V1 + y · 5 · V2 . ❲❡ ❞❡r✐✈❡ ✇❤❡r❡

X

❣♦❡s ❢r♦♠ t❤❡ ❛❜♦✈❡ s✉♠♠❛r②✦

❊①❡r❝✐s❡ ✷✳✼✳✼

❋✐♥❞ t❤❡ ❧✐♥❡ ♦❢ t❤❡ ♣r♦❥❡❝t✐♦♥✳

❲❡ ❛r❡ ❛❜❧❡ t♦ s✉♠♠❛r✐③❡ ✇❤❛t t❤❡ ♦♣❡r❛t♦r ❞♦❡s ❢r♦♠ t❤❡ ❛❧❣❡❜r❛ ♦♥❧②✳ ❚❤❡ ✐❞❡❛ ✐s ✉♥❝♦♠♣❧✐❝❛t❡❞✿



❚❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r

✇✐t❤✐♥ t❤❡ ❡✐❣❡♥s♣❛❝❡ ✐s ✏ 1✲❞✐♠❡♥s✐♦♥❛❧✑❀ ✐t ❝❛♥ t❤❡♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛

s✐♥❣❧❡ ♥✉♠❜❡r✳ ❚❤✐s ♥✉♠❜❡r✱ t❤❡ str❡t❝❤✲s❤r✐♥❦ ❢❛❝t♦r✱ ✐s ♦❢ ❝♦✉rs❡ t❤❡ ❡✐❣❡♥✈❛❧✉❡✳ ■❢ ✇❡ ❦♥♦✇ t❤❡s❡ t✇♦ ♥✉♠❜❡rs✱ ❤♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡ r❡st ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r❄ ■♥ t✇♦ st❡♣s✳ ❋✐rst✱ ❡✈❡r② ✉♥❞❡r

F

X

t❤❛t ❧✐❡s ✇✐t❤✐♥ t❤❡ ❡✐❣❡♥s♣❛❝❡✱ ✇❤✐❝❤ ✐s ❛ ❧✐♥❡✱ ✐s ❛ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❡✐❣❡♥✈❡❝t♦r✱ ❛♥❞ ✐ts ✈❛❧✉❡

❝❛♥ ❜❡ ❡❛s✐❧② ❝♦♠♣✉t❡❞✿

X = rV =⇒ U = F (rV ) = rF V = rλV . ❙❡❝♦♥❞✱ t❤❡ r❡st ♦❢ t❤❡ ✈❛❧✉❡s ❛r❡ ❢♦✉♥❞ ❜② ❢♦❧❧♦✇✐♥❣ t❤✐s ✐❞❡❛✿ ❚r② t♦ ❡①♣r❡ss t❤❡ ✈❛❧✉❡ ❛s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t✇♦ ✈❛❧✉❡s ❢♦✉♥❞ s♦ ❢❛r✳ ▲❡t✬s ♣r♦✈✐❞❡ ❛ ❢♦✉♥❞❛t✐♦♥ ❢♦r t❤✐s ✐❞❡❛✳

✷✳✽✳ ❇❛s❡s

❚❤❡ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ✐s ❞❡t❡r♠✐♥❡❞ ❜② ♦✉r ❝❤♦✐❝❡ ♦❢ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠✳

❖♥

t❤❡ ♦t❤❡r ❤❛♥❞✱ ✇❤❛t ✐t ❞♦❡s ♠❛② ❜❡ ❞❡s❝r✐❜❡❞ ✇✐t❤ s✉❝❤ ✇♦r❞s ❛s ✏str❡t❝❤✑✱ ✏r♦t❛t✐♦♥✑✱ ✏✢✐♣✑✱ ❡t❝✳ ❚❤❡s❡

✷✳✽✳

❇❛s❡s

✶✽✺

❞❡s❝r✐♣t✐♦♥s ❤❛✈❡ ♥♦t❤✐♥❣ t♦ ❞♦ ✇✐t❤ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠✳ ❆♥❞ ♥❡✐t❤❡r ❞♦ s✉❝❤ ❛❧❣❡❜r❛✐❝ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ t❤❡ ♦♣❡r❛t♦r ❛s t❤❡ tr❛❝❡✱ t❤❡ ❞❡t❡r♠✐♥❛♥t✱ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✳ ❲❡ ❛r❡ ♦♥ t❤❡ r✐❣❤t tr❛❝❦✦ ❖♥❝❡ ❛❣❛✐♥✱ ❞❡❛❧✐♥❣ ✇✐t❤ ✈❡❝t♦rs ✐♥st❡❛❞ ♦❢ ♣♦✐♥ts r❡q✉✐r❡s ❛ ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤✳ ❚❤❡ st❛♥❞❛r❞

❜❛s✐s ♦❢ R2 ✱ ❛s ❜❡❢♦r❡✱ ❝♦♥s✐sts ♦❢ t❤❡s❡ t✇♦✿ e1 =< 1, 0 >, e2 =< 0, 1 > .

❚❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ ❛ ✈❡❝t♦r X =< a, b > ✇✐t❤ r❡s♣❡❝t t♦ t❤✐s ❜❛s✐s ❛r❡✱ ❛s ❜❡❢♦r❡✱ a, b✿ < a, b >= a < 1, 0 > +b < 0, 1 >= ae1 + be2 .

■♥ ♦t❤❡r ✇♦r❞s✱ ❡✈❡r② ✈❡❝t♦r ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❛ ❧✐♥❡❛r

❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡s❡ t✇♦ ✈❡❝t♦rs✳

❍♦✇❡✈❡r✱ t❤❡② ❛r❡♥✬t t❤❡ ♦♥❧② ♦♥❡s ✇✐t❤ t❤✐s ♣r♦♣❡rt②✦ ❋♦r ❡①❛♠♣❧❡✱ ❧❡t✬s r❡✇r✐t❡ t❤❡ ❛❜♦✈❡ r❡♣r❡s❡♥t❛t✐♦♥✿ < a, b >= a < 1, 0 > +b < 0, 1 >= ae1 + (−b)(−e2 ) .

❙♦✱ t❤✐s ✈❡❝t♦r ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ ✈❡❝t♦rs V1 = e1 ❛♥❞ V2 = −e2 ✳

❊①❡r❝✐s❡ ✷✳✽✳✶

❘❡♣r❡s❡♥t ✈❡❝t♦r < a, b > ✐♥ t❡r♠s ♦❢ e1 ❛♥❞ e1 + e2 ✳ ❆❧❧ ♦❢ t❤❡s❡ ♣❛✐rs ♦❢ ✈❡❝t♦rs ♠❛② s❡r✈❡ ✐♥ s✉❝❤ r❡♣r❡s❡♥t❛t✐♦♥s✿

❊①❡r❝✐s❡ ✷✳✽✳✷ ❙❤♦✇ t❤❛t ✈❡❝t♦rs t❤❛t ❛r❡ ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r ❝❛♥✬t ❜❡ ✉s❡❞ ❢♦r t❤✐s ♣✉r♣♦s❡✳ ❆ ✈❡r② ✐♠♣♦rt❛♥t ❝♦♥❝❡♣t ❜❡❧♦✇ ❝❛♣t✉r❡s t❤✐s ✐❞❡❛✿

❉❡✜♥✐t✐♦♥ ✷✳✽✳✸✿ ❜❛s✐s ❆ ❜❛s✐s ♦❢ R2 ✐s ❛♥② s✉❝❤ ♣❛✐r ♦❢ ✈❡❝t♦rs V1 , V2 t❤❛t ❡✈❡r② ✈❡❝t♦r ❝❛♥ ❜❡ r❡♣r❡✲ s❡♥t❡❞ ❛s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡s❡ ✈❡❝t♦rs✿ X = r1 V 1 + r2 V 2 .

❚❤❡♥ t❤❡ ❝♦❡✣❝✐❡♥ts r1 , r2 ❛r❡ ❝❛❧❧❡❞ t❤❡ ❜❛s✐s✳

❝♦♠♣♦♥❡♥ts ♦❢ X ✇✐t❤ r❡s♣❡❝t t♦ t❤❡

❊①❡r❝✐s❡ ✷✳✽✳✹ Pr♦✈❡ t❤❛t t❤❡r❡ ✐s ♦♥❧② ♦♥❡ s✉❝❤ r❡♣r❡s❡♥t❛t✐♦♥✳ ❚❤❡ r❡❛s♦♥✐♥❣ ✐s t❤❛t t❤❡ ❛❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs ✇❛s ❡st❛❜❧✐s❤❡❞ ❜❡❢♦r❡ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ✇❛s ❛❞❞❡❞ t♦ t❤❡ ✈❡❝t♦r s♣❛❝❡ ❛♥❞ ❜❡❢♦r❡ t❤❡ t✇♦ ♦♣❡r❛t✐♦♥s ✇❡r❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ ✈❡❝t♦rs✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ❤♦✇ ✇❡ ❛❞❞ ✈❡❝t♦rs✿

✷✳✽✳

❇❛s❡s

✶✽✻

❊①❛♠♣❧❡ ✷✳✽✳✺✿ ❝♦♠♣♦♥❡♥ts

❚❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ ❛ ✈❡❝t♦r ✐♥ t❡r♠s ♦❢ t❤❡ st❛♥❞❛r❞ ❜❛s✐s ❛r❡ ❢♦✉♥❞ ✈✐❛ t❤❡ ♦rt❤♦❣♦♥❛❧ ♣r♦❥❡❝t✐♦♥s✿

❆ ✈❡r② ❞✐✛❡r❡♥t ❝❤♦✐❝❡ ♦❢ ❜❛s✐s U, V ✐s s❤♦✇♥ ❜❡❧♦✇✿

❍❡r❡✱ ✈❡❝t♦r X ❤❛s ❝♦♠♣♦♥❡♥ts 2 ❛♥❞ 2 ✇✐t❤ r❡s♣❡❝t t♦ t❤✐s ❜❛s✐s✿ X = 2U + 2V . ❊①❛♠♣❧❡ ✷✳✽✳✻✿ ❝♦♠♣♦♥❡♥t ❛❧❣❡❜r❛

▲✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ❛r❡ ❜❡❤✐♥❞ t❤❡ ❝♦♠♣♦♥❡♥ts ❛❧❣❡❜r❛✳ ❈♦❧❧❡❝t✐♥❣ ❝♦♠♠♦♥ t❡r♠s ❛❢t❡r ❛❞❞✐t✐♦♥ ✐s ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❝♦♠♣♦♥❡♥ts✿ ❝♦♠♣♦♥❡♥ts

❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥

U

=< 2, 3 >

= 2e1 + 3e2

W

=< −1, 2 >

= −1e1 + 2e2

U +V

=< 2, 3 > + < −1, 2 > = (2e1 + 3e2 ) + (−1e1 + 2e2 ) =< 2 − 1, 3 + 2 >

= (2e1 − 1e1 ) + (3e2 + 2e2 )

=< 1, 5 >

= 1e1 + 5e2

❘❡♣❧❛❝✐♥❣ e1 , e2 ✇✐t❤✱ s❛②✱ V1 , V2 ✇♦♥✬t ❝❤❛♥❣❡ ❛♥②t❤✐♥❣ ✐♥ t❤✐s ❝♦♠♣✉t❛t✐♦♥✳ ❙❛♠❡ ❢♦r s❝❛❧❛r ♠✉❧t✐✲

✷✳✽✳

❇❛s❡s

✶✽✼

♣❧✐❝❛t✐♦♥✿ ❝♦♠♣♦♥❡♥ts

❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥

U

=< 2, 3 >

= 2e1 + 3e2

r

= −3

rU = (−3) < 2, 3 >

= (−3)(2e1 + 3e2 )

=< (−3)2, (−3)3 > = (−3)2e1 + (−3)(1e1 ) =< −6, −9 >

= −6e1 − 9e2

❊①❛♠♣❧❡ ✷✳✽✳✼✿ ♥♦♥✲❜❛s✐s

❲❡ ♥❡❡❞ t♦ ❝♦✈❡r ❛❧❧ t❤❡ ✈❡❝t♦rs✿

▲❡t✬s tr② V1 =< 1, 0 >, V2 =< 2, 0 >✳ ❲❤❛t ❛r❡ ❛❧❧ ♣♦ss✐❜❧❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s❄ ❋♦r ❛❧❧ ♣❛✐rs r1 , r2 ✱ ✇❡ ❤❛✈❡ X = r1 V1 + r2 V2 = r1 < 1, 0 > +r2 < 2, 0 >=< r1 + 2r2 , 0 > .

❚❤❡ s❡❝♦♥❞ ❝♦♠♣♦♥❡♥t ✇✐❧❧ r❡♠❛✐♥ 0 ♥♦ ♠❛tt❡r ✇❤❛t t❤❡ ❝♦❡✣❝✐❡♥ts ❛r❡✦ ❚❤✐s ✐s ♥♦t ❛ ❜❛s✐s ❜❡❝❛✉s❡ ✇❡ ❝❛♥✬t r❡♣r❡s❡♥t s♦♠❡ ♦❢ t❤❡ ✈❡❝t♦rs✳ ❚❤❡ ❣❡♥❡r❛❧ r❡s✉❧t ✐s ❛s ❢♦❧❧♦✇s✿ ❚❤❡♦r❡♠ ✷✳✽✳✽✿ ❇❛s✐s ♦♥ t❤❡ P❧❛♥❡ ❆♥② t✇♦ ✈❡❝t♦rs t❤❛t ❛r❡♥✬t ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r ❛♥❞ ♦♥❧② t❤❡② ❢♦r♠ ❛ ❜❛s✐s ♦❢

R2 ❀

✐✳❡✳✱

V1 = rV2 ⇐⇒ {V1 , V2 }

✐s ♥♦t ❛ ❜❛s✐s✳

❊①❡r❝✐s❡ ✷✳✽✳✾

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❊①❛♠♣❧❡ ✷✳✽✳✶✵✿ ❧✐♥❡❛r s②st❡♠

❘❡❝❛❧❧ ❢r♦♠ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ❝❤❛♣t❡r ❤♦✇ t❤❡ s♦❧✉t✐♦♥ t♦ ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ✇❛s s❡❡♥ ❛s ✐❢ t❤❡ t✇♦ ❡q✉❛t✐♦♥s ✇❡r❡ ❡q✉❛t✐♦♥s ❛❜♦✉t t❤❡ ❝♦❡✣❝✐❡♥ts✱ x ❛♥❞ y ✱ ♦❢ ✈❡❝t♦rs ✐♥ t❤❡ ♣❧❛♥❡✿       6 1 1 . = +y x 14 3 2

❚♦ s♦❧✈❡ t❤❡ s②st❡♠ ✐s t♦ ✜♥❞ ❛ ✇❛② t♦ str❡t❝❤ t❤❡s❡ t✇♦ ✈❡❝t♦rs s♦ t❤❛t ❛❢t❡r ❛❞❞✐♥❣ t❤❡♠ t❤❡ r❡s✉❧t ✐s t❤❡ ✈❡❝t♦r ♦♥ t❤❡ r✐❣❤t✿

✷✳✽✳

❇❛s❡s

✶✽✽

◆♦✇ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t ✇❡ ❝❛♥ ❣✉❛r❛♥t❡❡ t❤❛t t❤❡r❡ ✐s ❛ s♦❧✉t✐♦♥ ♦♥❧② ✇❤❡♥ t❤❡ t✇♦ ✈❡❝t♦rs ❢♦r♠ ❛ ❜❛s✐s✦ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ♠✐①t✉r❡ ♣r♦❜❧❡♠ ❞♦❡s♥✬t ❤❛✈❡ ❛ s♦❧✉t✐♦♥✿       6 2 1 . = +y x 14 6 2 ❊①❛♠♣❧❡ ✷✳✽✳✶✶✿ ❝♦♦r❞✐♥❛t❡s

❈♦♥s✐❞❡r ❛♥ ❛❧t❡r♥❛t✐✈❡ ❜❛s✐s ❛❧♦♥❣ ✇✐t❤ t❤❡ st❛♥❞❛r❞ ♦♥❡✿

■t✬s ❡❛s② t♦ ❡①♣r❡ss t❤❡ ♥❡✇ ✈❡❝t♦rs ✐♥ t❡r♠s ♦❢ ♦❧❞✿ V1 =< 1, 1 >, V2 =< −1, 1 > .

❇✉t ✇❤❛t ❛❜♦✉t ✈✐❝❡ ✈❡rs❛❄ ■t✬s ❤❛r❞❡r❀ ✇❡ ♥❡❡❞ t♦ ✜♥❞ a ❛♥❞ b s♦ t❤❛t e1 = aV1 + bV2 ,

❛♥❞ t❤❡ s❛♠❡ ❢♦r e2 ✳ ❲❡ ❞r❛✇ ❛ ❣r✐❞ ❢♦r t❤❡ ♥❡✇ s②st❡♠ t♦ ❤❡❧♣✿

❚❤❡ ❛♥s✇❡r ✐s✿

1 1 1 1 e1 = V1 − V2 , e2 = V1 + V2 . 2 2 2 2

✷✳✽✳

❇❛s❡s

✶✽✾

❲❡ ❝❛♥ t❤❡♥ r❡✲✇r✐t❡ t❤❡s❡ ✈❡❝t♦rs ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ ❝♦♠♣♦♥❡♥ts ✇✐t❤ e1 =



1 1 ,− 2 2



1 = < 1, −1 >, e2 = 2



1 1 , 2 2



=

r❡s♣❡❝t t♦ t❤❡ ♥❡✇ ❜❛s✐s ✿

1 < 1, 1 > . 2

■♥ s✉♠♠❛r②✱ ✇❡ ❤❛✈❡ ❞✐✛❡r❡♥t ❝♦❡✣❝✐❡♥ts ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❞✐✛❡r❡♥t ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ s❛♠❡ ✈❡❝t♦rs ✇✐t❤ r❡s♣❡❝t t♦ ❞✐✛❡r❡♥t ❜❛s❡s✿ ❜❛s❡s✿ {e1 , e2 } V1 = V2 =

{V1 , V2 } < 1, 1 > =< 1, 0 > < −1, 1 > =< 0, 1 >

❛♥❞ ❜❛s❡s✿ {e1 , e2 } e1 = e2 =

{V1 , V2 } + * 1 1 < 1, 0 > = ,− 2 2 * + 1 1 < 0, 1 > = ,− 2 2

■♥ ❣❡♥❡r❛❧✱ t❤❡ ✈❡❝t♦rs ♦❢ t❤❡ ❛❧t❡r♥❛t✐✈❡ ❜❛s✐s ♠✐❣❤t ❤❛✈❡ ❛♥② ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡♠ ✭❛s ❧♦♥❣ ❛s ✐t✬s ♥♦t ③❡r♦✮✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡ ❛ s❦❡✇❡❞ ❣r✐❞✿

❚❤✉s✱ t❤❡ ❝♦♠♣♦♥❡♥t✲✇✐s❡ ❛❧❣❡❜r❛ ✐s ❢✉❧❧② ♦♣❡r❛t✐♦♥❛❧ ✇❤❛t❡✈❡r ❜❛s✐s ✇❡ ❝❤♦♦s❡✿ < a, b > + < c, d >=< a + c, b + d >

❛♥❞ r < a, b >=< ra, rb > . ❲❛r♥✐♥❣✦ ❯♥❧✐❦❡ t❤❡ ❛❧❣❡❜r❛✱ t❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ r❡❧✐❡s ♦♥ t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠✳

❆s ❛

r❡s✉❧t✱ t❤❡ ❢♦r♠✉❧❛s ❢♦r ♠❛❣♥✐t✉❞❡s ❛♥❞ t❤❡ ❞♦t ♣r♦❞✉❝ts ❢❛✐❧ ✐♥ t❤❡ ❝✉rr❡♥t ❢♦r♠ ✐❢ ✉s❡❞ ✇✐t❤ ♥♦♥✲ ♣❡r♣❡♥❞✐❝✉❧❛r ❜❛s❡s✳

◆♦✇✱ t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦rs✳ ❚❤❡② ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❛♣❛rt ❢r♦♠ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ✭t❤❛t ✇❛s ❛❞❞❡❞ t♦ t❤❡ ✈❡❝t♦r s♣❛❝❡✮✱ ✐✳❡✳✱ r♦t❛t✐♦♥✱ str❡t❝❤✐♥❣✱ ❡t❝✳✿

✷✳✽✳

❇❛s❡s

✶✾✵

❍♦✇❡✈❡r✱ ❥✉st ❛s ✈❡❝t♦r ❛❧❣❡❜r❛ ✇♦r❦s ❝♦♠♣♦♥❡♥t✇✐s❡✱ s♦ ❞♦ ❧✐♥❡❛r ♦♣❡r❛t♦rs✳ ❋✉rt❤❡r♠♦r❡✱ t❤✐s ❛♣♣r♦❛❝❤ ✇♦r❦s ✇✐t❤ r❡s♣❡❝t t♦ ❛♥② ❜❛s✐s✿ ❚❤❡♦r❡♠ ✷✳✽✳✶✷✿ ▲✐♥❡❛r ❖♣❡r❛t♦r ✐♥ ❚❡r♠s ♦❢ ❇❛s✐s

❙✉♣♣♦s❡ {V1, V2} ✐s ❛ ❜❛s✐s✳ ❚❤❡♥✱ ❛❧❧ ✈❛❧✉❡s ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r Y = F (X) ❛r❡ ❡①♣r❡ss❡❞ ❛s ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ✐ts ✈❛❧✉❡s ♦♥ t❤❡s❡ ✈❡❝t♦rs❀ ✐✳❡✳✱ ❢♦r ❛♥② ♣❛✐r ♦❢ r❡❛❧ ❝♦❡✣❝✐❡♥ts r1 ❛♥❞ r2✱ ✇❡ ❤❛✈❡✿ X = r1 V1 + r2 V2 =⇒ F (X) = r1 F (V1 ) + r2 F (V2 ) . ❊①❡r❝✐s❡ ✷✳✽✳✶✸

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♦♣❡r❛t♦r ✐s ❢✉❧❧② ❞❡t❡r♠✐♥❡❞ ❜② ✐ts ✈❛❧✉❡s ♦♥ t❤❡ ❜❛s✐s ✈❡❝t♦rs ✕ ❥✉st ❛s ✇✐t❤ t❤❡ st❛♥❞❛r❞ ❜❛s✐s✳

❇✉t✱ ❥✉st ❛s ✇✐t❤ ✈❡❝t♦rs✱ ✇❡ ❤❛✈❡ ❞✐✛❡r❡♥t ♠❛tr✐❝❡s ❢♦r t❤❡ s❛♠❡ ❧✐♥❡❛r ♦♣❡r❛t♦r ✇✐t❤ r❡s♣❡❝t t♦

❞✐✛❡r❡♥t ❜❛s❡s✳ ❚❤❡ ❝♦❧✉♠♥s ♦❢ t❤❡ ♠❛tr✐① ♦❢

A

❛r❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❜❛s✐s ✈❡❝t♦rs ✉♥❞❡r t❤❡ ♦♣❡r❛t♦r✿

A(V1 )

❛♥❞

A(V2 ) .

❊①❛♠♣❧❡ ✷✳✽✳✶✹✿ ♠❛tr✐❝❡s

❲❡✱ ❛❣❛✐♥✱ ❝♦♥s✐❞❡r t❤✐s ❛❧t❡r♥❛t✐✈❡ ❜❛s✐s ❛❧♦♥❣ ✇✐t❤ t❤❡ st❛♥❞❛r❞ ♦♥❡✿

❚❤❡ ♠✉t✉❛❧ r❡♣r❡s❡♥t❛t✐♦♥s ❛r❡✿

1 1 1 1 e1 = V1 − V2 , e2 = V1 + V2 , 2 2 2 2 ❛♥❞

V1 = e1 + e2 , V2 = −e1 + e2 .

A str❡t❝❤❡s t❤❡ ♣❧❛♥❡ ❛❧♦♥❣ t❤❡ x✲❛①✐s ♦❢ A ✇✐t❤ r❡s♣❡❝t t♦ {e1 , e2 } ✐s✿   2 0 . A= 0 1

❙✉♣♣♦s❡ ❛♥ ♦♣❡r❛t♦r

2✳

❚❤❡♥ t❤❡ ♠❛tr✐①

❲❤❛t ❛❜♦✉t t❤❡ ♦t❤❡r ❜❛s✐s✱

{V1 , V2 }❄

✭✐♥ ♦t❤❡r ✇♦r❞s✱ ❛❧♦♥❣

e1 ✮

❜② ❛ ❢❛❝t♦r ♦❢

■t ✐s ❤❛r❞ t♦ ❣✉❡ss t❤✐s t✐♠❡ ❜❡❝❛✉s❡ ✕ ✉♥❧✐❦❡ ❢♦r t❤❡ st❛♥❞❛r❞

❜❛s✐s ✕ t❤❡ ❝❤❛♥❣❡ ✐s ♥♦t ❛❧✐❣♥❡❞ ✇✐t❤ t❤❡ ❜❛s✐s ✈❡❝t♦rs✳

▲❡t✬s ✉s❡ t❤❡ ✏❝♦♥✈❡♥✐❡♥t✑ ❜❛s✐s ❛♥❞ t❤❡♥

s✇✐t❝❤ t♦ t❤❡ ♦♥❡ ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥✳ ❲❡ ✇r✐t❡ t❤❡ ❢♦r♠✉❧❛s ✐♥ t❡r♠s ♦❢

A(V1 ) = A(e1 + e2 )

= A(e1 ) + A(e2 )

e 1 , e2

✜rst✿

= 2e1 + e2

A(V2 ) = A(−e1 + e2 ) = −A(e1 ) + A(e2 ) = −2e1 + e2 ❚❤❡♥ ✇❡ s✉❜st✐t✉t❡ t❤❡ ✈❛❧✉❡s ♦❢

A(V1 ) = 2e1 + e2 A(V2 ) = −2e1 + e2

e 1 , e2

V1 , V2 ❛s ✇r✐tt❡♥ ❛❜♦✈❡✿     3 1 1 1 1 1 = V1 − V2 V1 − V2 + V1 + V2 =2 2 2 2 2 2 2     1 1 1 1 3 1 = − V1 + V2 V1 − V2 + V1 + V2 = −2 2 2 2 2 2 2 ✐♥ t❡r♠s ♦❢

✷✳✾✳

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❡✐❣❡♥✈❛❧✉❡s

❚❤❡r❡❢♦r❡✱ t❤❡ ♠❛tr✐① ♦❢ t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r ✇✐t❤ r❡s♣❡❝t t♦

{V1 , V2 }

✶✾✶

✐s✿



 3 1 −   2 2 A=  1 3 . − 2 2 ❲❤❛t ✐❢ t❤❡ str❡t❝❤ ✇❛s ❛❧♦♥❣

V1 ❄

❚❤✐s ♦♣❡r❛t♦r✬s ♠❛tr✐① ✇✐t❤ r❡s♣❡❝t t♦

{V1 , V2 }

✐s s✐♠♣❧❡✿

 2 0 . B= 0 1 

❊①❡r❝✐s❡ ✷✳✽✳✶✺ ❘❡✇r✐t❡ t❤❡ ❛❜♦✈❡ ❝♦♠♣✉t❛t✐♦♥ ✐♥ t❡r♠s ♦❢ ❝♦♠♣♦♥❡♥ts✳

✷✳✾✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❡✐❣❡♥✈❛❧✉❡s ❲❡ ❛♣♣❧② t❤❡ ❧❛st t❤❡♦r❡♠ t♦ ❡✐❣❡♥✈❡❝t♦rs✳

❈♦r♦❧❧❛r② ✷✳✾✳✶✿ ❘❡♣r❡s❡♥t❛t✐♦♥ ✐♥ ❚❡r♠s ♦❢ ❊✐❣❡♥✈❡❝t♦rs

❙✉♣♣♦s❡ V1 ❛♥❞ V2 ❛r❡ t✇♦ ❡✐❣❡♥✈❡❝t♦rs ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r F t❤❛t ❝♦rr❡s♣♦♥❞ t♦ t✇♦ ✭♣♦ss✐❜❧② ❡q✉❛❧✮ ❡✐❣❡♥✈❛❧✉❡s λ1 ❛♥❞ λ2✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t V1 ❛♥❞ V2 ❛r❡♥✬t ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✳ ❚❤❡♥✱ ❛❧❧ ✈❛❧✉❡s ♦❢ t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r Y = F (X) ❛r❡ r❡♣r❡s❡♥t❡❞ ❛s ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ✐ts ✈❛❧✉❡s ♦♥ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✿ X = r1 V1 + r2 V2 =⇒ F (X) = r1 λ1 V1 + r2 λ2 V2 ,

✇✐t❤ s♦♠❡ r❡❛❧ ❝♦❡✣❝✐❡♥ts r1 ❛♥❞ r2✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♠❛tr✐① ♦❢ F ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❜❛s✐s {V1, V2} ♦❢ ❡✐❣❡♥✈❡❝t♦rs ✐s ❞✐❛❣♦♥❛❧ ✇✐t❤ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦♥ t❤❡ ❞✐❛❣♦♥❛❧✿ 

 λ1 0 F = . 0 λ2 ❊①❛♠♣❧❡ ✷✳✾✳✷✿ str❡t❝❤✲s❤r✐♥❦ ▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ❢✉♥❝t✐♦♥✿



u = −x − 2y v =x − 4y

 −1 −2 . =⇒ F = 1 −4 

✷✳✾✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❡✐❣❡♥✈❛❧✉❡s

✶✾✷

▲❡t✬s ❝♦♥✜r♠ ✇❤❛t ✐s s❤♦✇♥ ❛❜♦✈❡✳ ❚❤❡ ❛♥❛❧②s✐s st❛rts ✇✐t❤ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✿  −1 − λ −2 = λ2 − 5λ + 6 . det(F − λI) = det 1 −4 − λ 

❚❤❡r❡❢♦r❡✱ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡✿ λ1 = −3, λ2 = −2 .

❚♦ ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✱ ✇❡ s♦❧✈❡ t❤❡ t✇♦ ✈❡❝t♦r ❡q✉❛t✐♦♥s✿

F Vi = λi Vi , i = 1, 2 .

❚❤❡ ✜rst✱ λ1 = −3✿



−1 −2 F V1 = 1 −4

    x x . = −3 y y

❚❤✐s ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ 

−x − 2y = −3x x − 4y = −3y

=⇒



2x − 2y = 0 x − y =0

=⇒ x = y .

❲❡ ❤❛✈❡ ❞✐s❝♦✈❡r❡❞✱ ❛❣❛✐♥✱ t❤❛t t❤✐s ✐s t❤❡ s❛♠❡ ❡q✉❛t✐♦♥❀ t❤✐s ❧✐♥❡ ❣✐✈❡s ✉s t❤❡ ❡✐❣❡♥s♣❛❝❡✳ ❲❡ ❝❤♦♦s❡ ♦♥❡ ❡✐❣❡♥✈❡❝t♦r✿   1 . 1

V1 =

❚❤❡ s❡❝♦♥❞ ❡✐❣❡♥✈❛❧✉❡✿



−1 −2 F V2 = 1 −4

❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ✭s❛♠❡ ❡q✉❛t✐♦♥✮✿ 

−x − 2y = −2x x − 4y = −2y

=⇒

    x x . = −2 y y



x − 2y = 0 x − 2y = 0

=⇒ x = 2y .

❚❤✐s ❧✐♥❡ ✐s t❤❡ ❡✐❣❡♥s♣❛❝❡ ♦❢ λ2 = −2✳ ❲❡ ❝❤♦♦s❡ ♦♥❡ ❡✐❣❡♥✈❡❝t♦r✿   2 . V2 = 1

❚❤❡ ♣❛✐r {V1 , V2 } ✐s ❛ ❜❛s✐s✦ ❲❡ s✉♠♠❛r✐③❡ ✇❤❛t F ❞♦❡s✿ ✶✳ ❆ ✢✐♣ ❛♥❞ str❡t❝❤ ❛❧♦♥❣ t❤❡ ✈❡❝t♦r < 1, 1 >✿ ❚❤❡ ❧✐♥❡ y = x r❡♠❛✐♥s ✐♥t❛❝t✳ ✷✳ ❆ ✢✐♣ ❛♥❞ str❡t❝❤ ❛❧♦♥❣ t❤❡ ✈❡❝t♦r < 2, 1 >✿ ❚❤❡ ❧✐♥❡ x = 2y r❡♠❛✐♥s ✐♥t❛❝t✳

✷✳✾✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❡✐❣❡♥✈❛❧✉❡s

✶✾✸

❲❡ ❛❧s♦ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡r❡ ✐s ♥♦ ❝❤❛♥❣❡ ♦❢ ♦r✐❡♥t❛t✐♦♥✳ ❙tr❡t❝❤✐♥❣ ❛s✐❞❡✱ t❤✐s ❧♦♦❦s ❧✐❦❡ ❝❡♥tr❛❧ s②♠♠❡tr②✿

❲❡ ♦❜s❡r✈❡ ❢❛♥♥✐♥❣ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ❧✐♥❡s✳ ❋♦r t❤❡ r❡st ♦❢ t❤❡ ✈❡❝t♦rs✱ ✇❡ ❤❛✈❡✿ X = xV1 + yV2

   1 2 . − 2y =⇒ F X = −3x 2 −1 

❚❤❡r❡❢♦r❡✱ ♠❛tr✐① ♦❢ F ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❜❛s✐s {V1 , V2 } ✐s ❞✐❛❣♦♥❛❧✿ 

 −3 0 F = . 0 −2 ❊①❛♠♣❧❡ ✷✳✾✳✸✿ str❡t❝❤✲s❤r✐♥❦

▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ❧✐♥❡❛r ♦♣❡r❛t♦r✿ 

u = x + 2y v = 3x + 2y

 1 2 . =⇒ F = 3 2 

▲❡t✬s ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✿  1−λ 2 = λ2 − 3λ − 4 . det(F − λI) = det 3 2−λ 

❚❤❡r❡❢♦r❡✱ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡✿

λ1 = −1, λ2 = 4 .

◆♦✇ ✇❡ ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✳ ❲❡ s♦❧✈❡ t❤❡ t✇♦ ❡q✉❛t✐♦♥s✿

F Vi = λi Vi , i = 1, 2 .

❚❤❡ ✜rst✿



1 2 F V1 = 3 2

    x x . = −1 y y

❚❤✐s ❣✐✈❡s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ 

x + 2y = −x 3x + 2y = −y

=⇒



2x + 2y = 0 3x + 3y = 0

=⇒ x = −y .

✷✳✾✳

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❡✐❣❡♥✈❛❧✉❡s

❲❡ ❝❤♦♦s❡✿

✶✾✹

 1 . V1 = −1 

y = −x✮

❊✈❡r② ✈❛❧✉❡ ✇✐t❤✐♥ t❤✐s ❡✐❣❡♥s♣❛❝❡ ✭t❤❡ ❧✐♥❡

✐s ❛ ♠✉❧t✐♣❧❡ ♦❢ t❤✐s ❡✐❣❡♥✈❡❝t♦r✿

 1 . X = λ1 V 1 = − −1 

❚❤❡ s❡❝♦♥❞ ❡✐❣❡♥✈❛❧✉❡✿



1 2 F V2 = 3 2

    x x . =4 y y

❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠✿



x + 2y = 4x 3x + 2y = 4y



=⇒

❲❡ ❝❤♦♦s❡✿

−3x + 2y = 0 3x − 2y = 0

 1 . V2 = 3/2 

y = 3x/2✮ ✐s ❛ ♠✉❧t✐♣❧❡   1 . X = λ2 V 2 = 4 3/2

❊✈❡r② ✈❛❧✉❡ ✇✐t❤✐♥ t❤✐s ❡✐❣❡♥s♣❛❝❡ ✭t❤❡ ❧✐♥❡

❚❤❡ ♣❛✐r

=⇒ x = 2y/3 .

{V1 , V2 }

♦❢ t❤✐s ❡✐❣❡♥✈❡❝t♦r✿

✐s ❛ ❜❛s✐s✦ ❚❤❡♥✱

X = xV1 + yV2 =⇒ U = F (X) = −xV1 + 4yV2 . ❚❤❡ ♠❛tr✐① ♦❢

F

✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❜❛s✐s

{V1 , V2 } ✐s✿   −1 0 . F = 0 4

❲❡ ♦❜s❡r✈❡ ❢❛♥♥✐♥❣ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ❧✐♥❡s✳

▲❡t✬s s✉♠♠❛r✐③❡ t❤❡ r❡s✉❧ts✳

❚❤❡♦r❡♠ ✷✳✾✳✹✿ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r ❖♣❡r❛t♦rs ✕ ❘❡❛❧ ❊✐❣❡♥✈❛❧✉❡s F ❤❛s t✇♦ r❡❛❧ ♥♦♥✲③❡r♦ ❡✐❣❡♥✈❛❧✉❡s λ1 ❛♥❞ λ2 ✳ ❚❤❡♥✱ t❤❡ ❢✉♥❝t✐♦♥ U = F (X) str❡t❝❤❡s✴s❤r✐♥❦s t❤❡ t✇♦ ❡✐❣❡♥s♣❛❝❡ ❜② ❢❛❝t♦rs |λ1 | ❛♥❞ |λ2 | ❙✉♣♣♦s❡ ♠❛tr✐①

r❡s♣❡❝t✐✈❡❧② ❛♥❞✱ ❢✉rt❤❡r♠♦r❡✿



■❢



■❢

λ1

λ2

❛♥❞

❤❛✈❡ t❤❡ s❛♠❡ s✐❣♥✱ ✐t ♣r❡s❡r✈❡s t❤❡ ♦r✐❡♥t❛t✐♦♥ ♦❢ ❛ ❝❧♦s❡❞

❝✉r✈❡ ❛r♦✉♥❞ t❤❡ ♦r✐❣✐♥✳

λ1

❛♥❞

λ2

❤❛✈❡ t❤❡ ♦♣♣♦s✐t❡ s✐❣♥s✱ ✐t r❡✈❡rs❡s t❤❡ ♦r✐❡♥t❛t✐♦♥ ♦❢ ❛ ❝❧♦s❡❞

❝✉r✈❡ ❛r♦✉♥❞ t❤❡ ♦r✐❣✐♥✳

❊①❡r❝✐s❡ ✷✳✾✳✺ ❆♣♣❧② t❤❡ t❤❡♦r❡♠ t♦ t❤❡ ❧❛st ❡①❛♠♣❧❡✳

❊①❛♠♣❧❡ ✷✳✾✳✻✿ s❦❡✇✐♥❣✲s❤❡❛r✐♥❣ ❈♦♥s✐❞❡r t❤✐s ♠❛tr✐①✿

 1 1 . F = 0 1 

✷✳✾✳

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❡✐❣❡♥✈❛❧✉❡s

✶✾✺

❇❡❧♦✇✱ ✇❡ r❡♣❧❛❝❡ ❛ ❝✐r❝❧❡ ✇✐t❤ ❛♥ ❡❧❧✐♣s❡ t♦ s❡❡ ✇❤❛t ❤❛♣♣❡♥s t♦ ✐t ✉♥❞❡r s✉❝❤ ❛ ❢✉♥❝t✐♦♥✿

❚❤❡r❡ ✐s st✐❧❧ ❛♥❣✉❧❛r str❡t❝❤✲s❤r✐♥❦ ❜✉t t❤✐s t✐♠❡ ✐t ✐s ❜❡t✇❡❡♥ t❤❡ t✇♦ ❡♥❞s ♦❢ t❤❡ s❛♠❡ ❧✐♥❡✳ ❲❡ s❡❡ ✏❢❛♥♥✐♥❣ ♦✉t✑ ❛❣❛✐♥✿

❚❤✐s t✐♠❡✱ ❤♦✇❡✈❡r✱ t❤❡ ❢❛♥ ✐s ❢✉❧❧② ♦♣❡♥✦ ■t ♠❛❦❡s ❛ ❞✐✛❡r❡♥❝❡ t❤❛t t❤❡ ❢❛♥♥✐♥❣ ❤❛♣♣❡♥s t♦ ❛ ✇❤♦❧❡ ❤❛❧❢✲♣❧❛♥❡✳ ❚♦ s❡❡ ♠♦r❡ ❝❧❡❛r❧②✱ ❝♦♥s✐❞❡r ✇❤❛t ❤❛♣♣❡♥s t♦ ❛ sq✉❛r❡✿

❚❤✐s ✐s t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✿

 1−λ 1 = (1 − λ)2 . det(F − λI) = det 0 1−λ 

❚❤❡r❡❢♦r❡✱ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡

λ1 = λ2 = 1 . ❲❤❛t ❛r❡ t❤❡ ❡✐❣❡♥✈❡❝t♦rs❄



1 1 FV = 0 1

    x x . =1 y y

❚❤✐s ❣✐✈❡s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿



x + y = x ❆◆❉ y =y

=⇒ x

❛♥②,

y = 0.

✷✳✾✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❡✐❣❡♥✈❛❧✉❡s

✶✾✻

❚❤❡ ♦♥❧② ❡✐❣❡♥✈❡❝t♦rs ❛r❡ ❤♦r✐③♦♥t❛❧✦ ❚❤❡r❡❢♦r❡✱ ♦✉r ❝❧❛ss✐✜❝❛t✐♦♥ t❤❡♦r❡♠ ❞♦❡s♥✬t ❛♣♣❧②✳ ❚❤❡r❡ ✐s ♥♦ ❞✐❛❣♦♥❛❧ ♠❛tr✐① ❢♦r t❤✐s ♦♣❡r❛t♦r✳ ❊①❛♠♣❧❡ ✷✳✾✳✼✿ r♦t❛t✐♦♥s

❚❤❡r❡ ❛r❡ ♦t❤❡r ♦✉t❝♦♠❡s t❤❛t t❤❡ t❤❡♦r❡♠ ❞♦❡s♥✬t ❝♦✈❡r✳ ❘❡❝❛❧❧ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ♦❢ t❤❡ ♠❛tr✐① A ♦❢ t❤❡ 90✲❞❡❣r❡❡ r♦t❛t✐♦♥✿  −λ −1 = λ2 + 1 . χA (λ) = det 1 −λ 

❇✉t t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❡q✉❛t✐♦♥✱

x2 + 1 = 0 ,

❤❛s ♥♦ s♦❧✉t✐♦♥s✦ ❆r❡ ✇❡ ❞♦♥❡ t❤❡♥❄ ◆♦t ✐❢ ✇❡ ❛r❡ ✇✐❧❧✐♥❣ t♦ ✉s❡ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✭♥❡①t ❝❤❛♣t❡r✮✿ λ1,2 = i ❛♥❞ − i .

❚❤✐s ✐s t❤❡ ❡✛❡❝t ♦❢ r♦t❛t✐♦♥✿

▲❡t✬s ❝♦♥s✐❞❡r ❛ r♦t❛t✐♦♥ t❤r♦✉❣❤ ❛♥ ❛r❜✐tr❛r② ❛♥❣❧❡ θ✿      cos θ − sin θ x u . = y sin θ cos θ v

❚❤❡ ❛♥❣❧❡ ❝❛♥ ❜❡ s❡❡♥ ✐♥ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✿ χA (λ) = (cos θ − λ)2 + sin2 θ = cos2 θ − 2 cos θ λ + λ2 + sin2 θ = λ2 − 2 cos θ λ + 1 .

❚❤❡ ❞✐s❝r✐♠✐♥❛♥t ♦❢ t❤✐s ♣♦❧②♥♦♠✐❛❧ ✐s ♥❡❣❛t✐✈❡✳ ❚❤❡r❡❢♦r❡✱ ✐t ❤❛s ♥♦ r❡❛❧ r♦♦ts✳ ❚❤❡ r❡s✉❧t ♠❛❦❡s s❡♥s❡✿ ❆ r♦t❛t✐♦♥ ❝❛♥♥♦t ♣♦ss✐❜❧② ❤❛✈❡ ❡✐❣❡♥✈❡❝t♦rs ❜❡❝❛✉s❡ ❛❧❧ ✈❡❝t♦rs ❛r❡ r♦t❛t❡❞✦

✷✳✾✳

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❡✐❣❡♥✈❛❧✉❡s

✶✾✼

❊①❛♠♣❧❡ ✷✳✾✳✽✿ r♦t❛t✐♦♥ ✇✐t❤ str❡t❝❤✲s❤r✐♥❦ ▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ❧✐♥❡❛r ♦♣❡r❛t♦r✿



u = 3x −13y , v = 5x +y ,

❛♥❞

 3 −13 . F = 5 1 

❇❡❧♦✇ ✇❡ ❝❛♥ r❡❝♦❣♥✐③❡ ❜♦t❤ r♦t❛t✐♦♥ ❛♥❞ r❡✲s❝❛❧✐♥❣✿

❚❤✐s ✐s ♦✉r ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✿

 3 − λ −13 = λ2 − 4λ + 68 . χ(λ) = det(F − λI) = det 5 1−λ 

❊①❡r❝✐s❡ ✷✳✾✳✾ ❲❤❛t ❞♦❡s ✐t t❡❧❧ ✉s❄

❖✉r ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ✐♥ t❡r♠s ♦❢ t❤❡ tr❛❝❡ ♦❢ t❤❡ ♠❛tr✐①✿

χ(λ) = λ2 − tr F λ + det F . ❛❧❧♦✇s ✉s t♦ ♣r♦✈❡ ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ❢♦r t❤❡ ❝❛s❡ ♦❢ ♥♦ r❡❛❧ ❡✐❣❡♥✈❛❧✉❡s✿

❈♦r♦❧❧❛r② ✷✳✾✳✶✵✿ ❚r❛❝❡ ❛♥❞ ❉✐s❝r✐♠✐♥❛♥t

❙✉♣♣♦s❡ t❤❡ ❞✐s❝r✐♠✐♥❛♥t ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❛ ♠❛tr✐① F s❛t✐s✜❡s✿ D = (tr F )2 − 4 det F ≤ 0 .

❚❤❡♥✱ t❤❡ ♦♣❡r❛t♦r U = F X ❞♦❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ✶✳ ■t r♦t❛t❡s t❤❡ r❡❛❧ ♣❧❛♥❡ t❤r♦✉❣❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛♥❣❧❡✿ θ = sin−1

1 2

r

4 − (tr F )2 det F

!

.

✷✳ ■t r❡✲s❝❛❧❡s t❤❡ ♣❧❛♥❡ ✉♥✐❢♦r♠❧② ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝t♦r✿ s= ❊①❡r❝✐s❡ ✷✳✾✳✶✶ ❆♣♣❧② t❤❡ ❝♦r♦❧❧❛r② t♦ t❤❡ ❧❛st ❡①❛♠♣❧❡✳



det F .

❈❤❛♣t❡r ✸✿ ❱❡❝t♦r ❛♥❞ ❝♦♠♣❧❡① ✈❛r✐❛❜❧❡s

❈♦♥t❡♥ts

✸✳✶ ❆❧❣❡❜r❛ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛♥❞ ♠❛tr✐❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✸ ❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✺ ❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ R2 ✳ ✳ ✸✳✻ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿ C ✐s♥✬t ❥✉st R2 ✸✳✼ ❈♦♠♣❧❡① ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✽ ❈♦♠♣❧❡① ❧✐♥❡❛r ♦♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✾ ▲✐♥❡❛r ♦♣❡r❛t♦rs ✇✐t❤ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✵ ❈♦♠♣❧❡① ❝❛❧❝✉❧✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✶ ❙❡r✐❡s ❛♥❞ ♣♦✇❡r s❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✷ ❙♦❧✈✐♥❣ ❖❉❊s ✇✐t❤ ♣♦✇❡r s❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✾✽ ✷✵✹ ✷✵✾ ✷✶✻ ✷✶✾ ✷✷✸ ✷✷✽ ✷✸✸ ✷✸✻ ✷✹✷ ✷✹✹ ✷✺✵

✸✳✶✳ ❆❧❣❡❜r❛ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛♥❞ ♠❛tr✐❝❡s

❲❡ ✇✐❧❧ t❛❦❡ ❛ ❜r♦❛❞❡r ✈✐❡✇ ❛t ❧✐♥❡❛r ♦♣❡r❛t♦r ❛♥❞ ✐♥❝❧✉❞❡ t❤❡ ❧♦✇❡r ❞✐♠❡♥s✐♦♥s✳ ♣♦ss✐❜✐❧✐t✐❡s✿

❚❤❡s❡ ❛r❡ t❤❡ ❢♦✉r

✸✳✶✳

❆❧❣❡❜r❛ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛♥❞ ♠❛tr✐❝❡s

✶✾✾

❲❡ ✇❛♥t t♦ ✉♥❞❡rst❛♥❞ ❤♦✇ t❤❡s❡ ♦♣❡r❛t♦rs ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② ♠❛tr✐❝❡s ❛♥❞ ❤♦✇ t❤❡s❡ ♠❛tr✐❝❡s ❛r❡ ❝♦♠❜✐♥❡❞ t♦ ♣r♦❞✉❝❡ ❝♦♠♣♦s✐t✐♦♥s✳ ❚❤❡ r✉❧❡ r❡♠❛✐♥s✿ ◮ ❚❤❡ ✈❛❧✉❡ ✉♥❞❡r F ♦❢ ❡❛❝❤ ❜❛s✐s ✈❡❝t♦r ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ F ❜❡❝♦♠❡s ❛ ❝♦❧✉♠♥ ✐♥ t❤❡ ♠❛tr✐① ♦❢ F ✳

▲❡t✬s ❛♣♣❧② t❤❡ r✉❧❡ t♦ t❤❡s❡ ❢♦✉r s✐t✉❛t✐♦♥s ✉s✐♥❣ t❤❡ st❛♥❞❛r❞ ❜❛s❡s✳ ❊①❛♠♣❧❡ ✸✳✶✳✶✿ ❞✐♠❡♥s✐♦♥s ✶ ❛♥❞ ✶

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✱ ✇❤✐❝❤ ✐s ❥✉st ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✿ f : R → R ❞❡✜♥❡❞ ❜② f (x) = 3x .

■t ✐s ❛ str❡t❝❤ ❜② ❛ ❢❛❝t♦r ♦❢ 3✳ ❲❤❛t ✐s ✐ts ♠❛tr✐①❄ ❚❤❡ ❜❛s✐s ♦❢ t❤❡ x✲❛①✐s ✐s < 1 > ❛♥❞ t❤❡ ❜❛s✐s ♦❢ t❤❡ u✲❛①✐s ✐s < 1 >✳ ❚❤❡ ♦♣❡r❛t♦r ✇♦r❦s ❛s ❢♦❧❧♦✇s✿ f (< 1 >) = 3 < 1 > .

❚❤❡r❡❢♦r❡✱ ✐ts ♠❛tr✐① ✐s f = [3] . ❊①❛♠♣❧❡ ✸✳✶✳✷✿ ❞✐♠❡♥s✐♦♥s ✶ ❛♥❞ ✷

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✱ ✇❤✐❝❤ ✐s ❥✉st ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿ F : R → R2 ❞❡✜♥❡❞ ❜② F (x) =< 3x, 2x > .

■t str❡t❝❤❡s t❤❡ x✲❛①✐s ♦♥ t❤❡ uv ✲♣❧❛♥❡ ❛❧♦♥❣ t❤❡ ✈❡❝t♦r < 3, 2 >✳ ❲❤❛t ✐s ✐ts ♠❛tr✐①❄ ❚❤❡ ❜❛s✐s ♦❢ t❤❡ x✲❛①✐s ✐s < 1 > ❛♥❞ t❤❡ ❜❛s✐s ♦❢ t❤❡ uv ✲♣❧❛♥❡ ✐s < 1, 0 > ❛♥❞ < 0, 1 >✳ ❚❤❡ ♦♣❡r❛t♦r ✇♦r❦s ❛s ❢♦❧❧♦✇s✿ F (< 1 >) =< 3, 2 > .

❚❤❡r❡❢♦r❡✱ ✐ts ♠❛tr✐① ✐s

  3 . F = 2

❊①❛♠♣❧❡ ✸✳✶✳✸✿ ❞✐♠❡♥s✐♦♥s ✷ ❛♥❞ ✶

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✱ ✇❤✐❝❤ ✐s ❥✉st ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✿ f : R2 → R ❞❡✜♥❡❞ ❜② f (x, y) = 3x + 2y .

■t r♦❧❧s t❤❡ xy ✲♣❧❛♥❡ ♦♥ t❤❡ u✲❛①✐s✳ ❲❤❛t ✐s ✐ts ♠❛tr✐①❄ ❚❤❡ ❜❛s✐s ♦❢ t❤❡ xy ✲♣❧❛♥❡ ✐s < 1, 0 > ❛♥❞ < 0, 1 > ❛♥❞ t❤❡ ❜❛s✐s ♦❢ t❤❡ u✲❛①✐s ✐s < 1 >✳ ❚❤❡ ♦♣❡r❛t♦r ✇♦r❦s ❛s ❢♦❧❧♦✇s✿ f (< 1, 0 >) =< 3 >

❛♥❞ f (< 0, 1 >) =< 2 > .

❚❤❡r❡❢♦r❡✱ ✐ts ♠❛tr✐① ✐s f = [3 2] . ❊①❛♠♣❧❡ ✸✳✶✳✹✿ ❞✐♠❡♥s✐♦♥s ✷ ❛♥❞ ✷

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✱ ✇❤✐❝❤ ✐s ❥✉st ❛ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡✿ F : R2 → R2 ❞❡✜♥❡❞ ❜② f (x, y) =< 3x + 2y, 5x − y > .

❲❡✬❞ ♥❡❡❞ t❤❡ ❡✐❣❡♥✈❡❝t♦r ❛♥❛❧②s✐s ✐♥ ♦r❞❡r t♦ ❞❡t❡r♠✐♥❡ ✇❤❛t ✐t ❞♦❡s✳✳✳ ❲❤❛t ✐s ✐ts ♠❛tr✐①❄ ❚❤❡

✸✳✶✳

❆❧❣❡❜r❛ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛♥❞ ♠❛tr✐❝❡s

✷✵✵

❜❛s✐s ♦❢ t❤❡ xy ✲♣❧❛♥❡ ✐s < 1, 0 > ❛♥❞ < 0, 1 > ❛♥❞ t❤❡ ❜❛s✐s ♦❢ t❤❡ u✲❛①✐s ✐s < 1, 0 > ❛♥❞ < 0, 1 >✳ ❚❤❡ ♦♣❡r❛t♦r ✇♦r❦s ❛s ❢♦❧❧♦✇s✿ F (< 1, 0 >) =< 3, 5 >

❚❤❡r❡❢♦r❡✱ ✐ts ♠❛tr✐① ✐s

❛♥❞ F (< 0, 1 >) =< 2, −1 > . 

 3 2 F = . 5 −1

❆s ②♦✉ ❝❛♥ s❡❡✱ ✇❡ ❝❛♥ ❥✉♠♣ ❛❤❡❛❞ ♦❢ t❤❡ r✉❧❡ ❞❡s❝r✐❜❡❞ ❛❜♦✈❡ ❛♥❞ ✇r✐t❡ t❤❡ ❝♦❡✣❝✐❡♥ts ♣r❡s❡♥t ✐♥ t❤❡ ❢♦r♠✉❧❛ ♦❢ t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r str❛✐❣❤t ✐♥t♦ t❤❡ ♠❛tr✐①✳

❲❛r♥✐♥❣✦ ■ts ♠❛tr✐① ✐s ❥✉st ❛♥ ❛❜❜r❡✈✐❛t❡❞ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✳

❊①❡r❝✐s❡ ✸✳✶✳✺ ■♥❝❧✉❞❡ t❤❡ ❞✐♠❡♥s✐♦♥ 0 ❢♦r ❞♦♠❛✐♥s ❛♥❞ ❝♦❞♦♠❛✐♥s ✐♥ t❤❡ ❛❜♦✈❡ ❛♥❛❧②s✐s✳ ❲❤❡♥❡✈❡r t❤❡r❡ ✐s ❛❧❣❡❜r❛ ✐♥ t❤❡ ♦✉t♣✉t s♣❛❝❡✱ ✇❡ ❝❛♥ ✉s❡ ✐t t♦ ❞♦ ❛❧❣❡❜r❛ ♦❢ t❤❡ ❢✉♥❝t✐♦♥s✳ ■❢ t❤❡ ❝♦❞♦♠❛✐♥ ♦❢ ❢✉♥❝t✐♦♥s ✐s ❛ ✈❡❝t♦r s♣❛❝❡✱ ✇❡ ❝❛♥ ❛❞❞ t❤❡s❡ ❢✉♥❝t✐♦♥s ❛♥❞ ♠✉❧t✐♣❧② t❤❡♠ ❜② ❛ ❝♦♥st❛♥t✳ ❲❡ ❥✉st ♥❛rr♦✇ ❞♦✇♥ t❤✐s ✐❞❡❛ t♦ ❧✐♥❡❛r ♦♣❡r❛t♦rs✿

❉❡✜♥✐t✐♦♥ ✸✳✶✳✻✿ ❛❞❞✐t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ●✐✈❡♥ t✇♦ ❧✐♥❡❛r ♦♣❡r❛t♦rs✿

t❤❡✐r

s✉♠ ✐s ❧✐♥❡❛r ♦♣❡r❛t♦r✿

F, G : Rn → Rm ,

F + G : Rn → Rm ,

❞❡✜♥❡❞ ❜②✿ (F + G)(x) = F (x) + G(x) .

❲❡ ✐❧❧✉str❛t❡ t❤✐s ♦♣❡r❛t✐♦♥ ❥✉st ❛s ❜❡❢♦r❡✿

❇✉t ✐❢ t❤❡s❡ ❛r❡ ❧✐♥❡❛r ♦♣❡r❛t♦rs✱ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡✐r ♠❛tr✐❝❡s❄ ❲❡ ❣♦ t❤♦✉❣❤ t❤❡ s❛♠❡ ❢♦✉r ❝❛s❡s ❜❡❧♦✇✳

✸✳✶✳

❆❧❣❡❜r❛ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛♥❞ ♠❛tr✐❝❡s

❊①❛♠♣❧❡ ✸✳✶✳✼✿ ❞✐♠❡♥s✐♦♥s

1

❛♥❞

✷✵✶

1

●✐✈❡♥ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✭♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✮✿

f, g : R → R ❞❡✜♥❡❞ ❜② f (x) = 3x ❛♥❞ g(x) = 2x . ❚❤❡✐r ♠❛tr✐❝❡s ❛r❡✿

f = [3] ❛♥❞ g = [2] .

❲❤❛t ❛❜♦✉t t❤❡✐r s✉♠❄ ■t ✐s ❛♥ ♦♣❡r❛t♦r ✇✐t❤ t❤❡ s❛♠❡ ❞♦♠❛✐♥ ❛♥❞ ❝♦❞♦♠❛✐♥ ❛♥❞ ✐t ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿ f + g : R → R ❞❡✜♥❡❞ ❜② (f + g)(x) = f (x) + g(x) = 3x + 2x = 5x .

■ts ♠❛tr✐① ✐s

f + g = [5] . ❖❢ ❝♦✉rs❡✱ t❤✐s ♥❡✇ ♥✉♠❜❡r ✐s ❥✉st t❤❡ s✉♠ ♦❢ t❤❡ t✇♦ ♦r✐❣✐♥❛❧ ♥✉♠❜❡rs✳ ❊①❛♠♣❧❡ ✸✳✶✳✽✿ ❞✐♠❡♥s✐♦♥s

1

❛♥❞

2

●✐✈❡♥ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✭♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✮✿

F, G : R → R2 ❞❡✜♥❡❞ ❜② F (x) =< 3x, 2x > ❚❤❡✐r ♠❛tr✐❝❡s ❛r❡✿

  3 F = 2

❛♥❞ G(x) =< 5x, −x > .

 5 . ❛♥❞ G = −1 

❲❤❛t ❛❜♦✉t t❤❡✐r s✉♠❄ ■t ✐s ❛♥ ♦♣❡r❛t♦r ✇✐t❤ t❤❡ s❛♠❡ ❞♦♠❛✐♥ ❛♥❞ ❝♦❞♦♠❛✐♥ ❛♥❞ ✐t ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿

F + G : R → R2 ❞❡✜♥❡❞ ❜② (F + G)(x) = F (x) + G(x) =< 3x, 2x > + < 5x, −x >=< 8x, x > . ■ts ♠❛tr✐① ✐s

  8 . F +G= 1

❖❢ ❝♦✉rs❡✱ t❤✐s ✐s ❥✉st t❤❡ s✉♠ ♦❢ t❤❡ t✇♦ ❛s ✐❢ t❤❡② ✇❡r❡ ✈❡❝t♦rs✳ ❊①❛♠♣❧❡ ✸✳✶✳✾✿ ❞✐♠❡♥s✐♦♥s

2

❛♥❞

1

●✐✈❡♥ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✭❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✮✿

f, g : R2 → R ❞❡✜♥❡❞ ❜② f (x, y) = 3x + 2y ❚❤❡✐r ♠❛tr✐❝❡s ❛r❡✿

❛♥❞ g(x, y) = 5x − 2y .

f = [3, 2] ❛♥❞ g = [5, −2] .

❚❤❡✐r s✉♠ ✐s ❛♥ ♦♣❡r❛t♦r ✇✐t❤ t❤❡ s❛♠❡ ❞♦♠❛✐♥ ❛♥❞ ❝♦❞♦♠❛✐♥ ❛♥❞ ✐t ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s ✭t❤✐s ✐s ✈❡❝t♦r ❛❞❞✐t✐♦♥✮✿

f + g : R2 → R ❞❡✜♥❡❞ ❜② (f + g)(x, y) = f (x, y) + g(x, y) = 3x + 2y + 5x − 2y = 8x . ■ts ♠❛tr✐① ✐s

f = [8, 0] , t❤❡ s✉♠ ✕ ❝♦♠♣♦♥❡♥t✇✐s❡ ✕ ♦❢ t❤❡ t✇♦✳

✸✳✶✳

❆❧❣❡❜r❛ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛♥❞ ♠❛tr✐❝❡s

✷✵✷

❊①❛♠♣❧❡ ✸✳✶✳✶✵✿ ❞✐♠❡♥s✐♦♥s 2 ❛♥❞ 2 ●✐✈❡♥ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✭tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡✮✿

F, G : R2 → R2 ❞❡✜♥❡❞ ❜② F (x, y) =< 3x + 2y, 5x − y > ❚❤❡✐r ♠❛tr✐❝❡s ❛r❡✿



3 2 F = 5 −1



❛♥❞ G(x, y) =< 5x + y, x + y > .

 5 1 . ❛♥❞ G = 1 1 

❚❤❡✐r s✉♠ ✐s ❛♥ ♦♣❡r❛t♦r ✇✐t❤ t❤❡ s❛♠❡ ❞♦♠❛✐♥ ❛♥❞ ❝♦❞♦♠❛✐♥ ❛♥❞ ✐t ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s ✭t❤✐s ✐s ✈❡❝t♦r ❛❞❞✐t✐♦♥✮✿

F +G : R2 → R2 ❞❡✜♥❡❞ ❜② (F +G)(x, y) =< 3x+2y, 5x−y > + < 5x+y, x+y >=< 8x+3y, 6x > . ■ts ♠❛tr✐① ✐s

 8 3 , F +G= 6 0 

t❤❡ s✉♠ ✕ ❝♦♠♣♦♥❡♥t✇✐s❡ ✕ ♦❢ t❤❡ t✇♦✿         8 3 3+5 2+1 5 1 3 2 . = = + F +G= 6 0 5 + 1 (−1) + 1 1 1 5 −1 ❚❤✐s ♦♣❡r❛t✐♦♥ ♦❢ ♠❛tr✐① ❛❞❞✐t✐♦♥ ✐s ❝♦♠♣♦♥❡♥t✇✐s❡✿ ❚❤❡ t✇♦ ♦♣❡r❛t♦rs ❤❛✈❡ t❤❡ s❛♠❡ ❞♦♠❛✐♥ ❛♥❞ ❝♦❞♦♠❛✐♥✱ s♦ t❤❛t t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡ ♠❛tr✐❝❡s ❛r❡ ❡q✉❛❧ t♦♦✳ ❚❤❡② ❝❛♥ t❤❡♥ ❜❡ ♦✈❡r❧❛♣♣❡❞ s♦ t❤❛t t❤❡ ❡♥tr✐❡s ❛r❡ ❛❧✐❣♥❡❞ ❛♥❞ ❛❞❞❡❞ ❛❝❝♦r❞✐♥❣❧②✿

❉❡✜♥✐t✐♦♥ ✸✳✶✳✶✶✿ ❛❞❞✐t✐♦♥ ♦❢ ♠❛tr✐❝❡s ❙✉♣♣♦s❡ A ❛♥❞ B ❛r❡ t✇♦ m × n ♠❛tr✐❝❡s✳ ❚❤❡♥ t❤❡✐r ❞❡♥♦t❡❞ ❜②✿

s✉♠ ✐s t❤❡ m × n ♠❛tr✐①✱

A+B t❤❡ ij ✲❡♥tr② ♦❢ ✇❤✐❝❤ ✐s t❤❡ s✉♠ ♦❢ t❤❡ ij ✲❡♥tr✐❡s ♦❢ A ❛♥❞ B ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ A = aij ✱ B = bij ✱ ❛♥❞ C = A + B = cij ✱ t❤❡♥

cij = aij + bij , ❢♦r ❡❛❝❤ i = 1, 2, ..., m ❛♥❞ ❡❛❝❤ j = 1, 2, ..., n✳

✸✳✶✳

❆❧❣❡❜r❛ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛♥❞ ♠❛tr✐❝❡s

✷✵✸

■t ✐s s✐♠♣❧❡r ❢♦r s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✿

❉❡✜♥✐t✐♦♥ ✸✳✶✳✶✷✿ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦r ●✐✈❡♥ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✿ ✐ts s❝❛❧❛r

F : Rn → R m ,

♣r♦❞✉❝t ✇✐t❤ ❛ r❡❛❧ ♥✉♠❜❡r r ✐s ❧✐♥❡❛r ♦♣❡r❛t♦r✿ rF : Rn → Rm ,

❞❡✜♥❡❞ ❜②✿ (rF )(x) = rF (x) .

❚❤✐s ♦♣❡r❛t✐♦♥ ✐s ❝♦♠♣♦♥❡♥t✲✇✐s❡ ❛❣❛✐♥✿ ❊✈❡r② ❡♥tr② ✐s ♠✉❧t✐♣❧✐❡❞ ❜② t❤❡ s❛♠❡ ♥✉♠❜❡r✳

❉❡✜♥✐t✐♦♥ ✸✳✶✳✶✸✿ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ♠❛tr✐❝❡s ❙✉♣♣♦s❡ A ✐s ❛♥ m × n ♠❛tr✐①✳ ❚❤❡♥ ✐ts ❞❡♥♦t❡❞ ❜②✿

s❝❛❧❛r ♠✉❧t✐♣❧❡

❜② ❛ r❡❛❧ ♥✉♠❜❡r r✱

rA

✐s ❛♥ m × n ♠❛tr✐①✱ t❤❡ ij ✲❡♥tr② ♦❢ ✇❤✐❝❤ ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ij ✲❡♥tr② ♦❢ A ❜② r✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ A = aij ❛♥❞ C = rA = cij ✱ t❤❡♥ cij = raij ,

❢♦r ❡❛❝❤ i = 1, 2, ..., m ❛♥❞ ❡❛❝❤ j = 1, 2, ..., n✳

❲❛r♥✐♥❣✦ ❚❤❡ ♠❛tr✐① ✐s ❥✉st ❛♥ ❛❜❜r❡✈✐❛t❡❞ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✳ ❆❝❝♦r❞✐♥❣❧②✱ t❤❡ ♠❛tr✐① ♦♣❡r✲ ❛t✐♦♥s ❛r❡ ❥✉st ❛❜❜r❡✈✐❛t❡❞ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ t❤❡ ♦♣❡r❛t✐♦♥s ♦♥ ❧✐♥❡❛r ♦♣❡r❛t♦rs✳

✸✳✷✳ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs

✷✵✹

✸✳✷✳ ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs

▲❡t✬s t❛❦❡ t❤❡ ♣r♦❜❧❡♠ ❛❜♦✉t ♠✐①t✉r❡s t♦ t❤❡ ♥❡①t ❧❡✈❡❧✳ ❲❡ ❤❛✈❡✿ ✶✳ n ✐♥❣r❡❞✐❡♥ts ❛♥❞✱ t❤❡r❡❢♦r❡✱ n ✉♥❦♥♦✇♥s ♦r ✈❛r✐❛❜❧❡s x1, ..., xn r❡♣r❡s❡♥t✐♥❣ t❤❡ ❛♠♦✉♥ts ♦❢ ❡❛❝❤❀ ❛♥❞ ✷✳ m r❡q✉✐r❡♠❡♥ts ♦r r❡str✐❝t✐♦♥s✱ ✐✳❡✳✱ m ❧✐♥❡❛r ❡q✉❛t✐♦♥s ✐♥✈♦❧✈✐♥❣ t❤❡s❡ ✈❛r✐❛❜❧❡s ✭k = 1, 2, ..., m✮✿ ak1 x1 + ... + akn xn = bk .

❚❤✐s s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ✐s ✈❡r② ❝✉♠❜❡rs♦♠❡✳ ❆s ❜❡❢♦r❡✱ ✇❡ tr❛♥s❧❛t❡ t❤✐s ❛ s②st❡♠ ✐♥t♦ ❛ ✈❡❝t♦r✲♠❛tr✐① ❡q✉❛t✐♦♥✿ FX = B

✇❤❡r❡ ✶✳ X =< x1, ..., xn > ✐s t❤❡ ✈❡❝t♦r ♦❢ t❤❡ ✉♥❦♥♦✇♥s✱ ✷✳ B =< b1, ..., bm > ✐s t❤❡ ✈❡❝t♦r ♦❢ t❤❡ t♦t❛❧s✱ ❛♥❞ ✸✳ F = aij ✐s t❤❡ m × n ♠❛tr✐① ♠❛❞❡ ✉♣ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✳ ■♥ ❧✐❣❤t ♦❢ t❤❡ r❡❝❡♥t ❞❡✈❡❧♦♣♠❡♥t✱ ✇❡ ♣r❡❢❡r t♦ ❧♦♦❦ ❛t t❤❡ ❡q✉❛t✐♦♥ ❛s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ✐✳❡✳✱ ✇❡ ❤❛✈❡ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✿

F (X) = B F : Rn → Rm .

❆♥❞ t❤❡ ❡q✉❛t✐♦♥ ♥❡❡❞s t♦ ❜❡ s♦❧✈❡❞✦ ■♥ ❞✐♠❡♥s✐♦♥ 1✱ t❤❡ ❡q✉❛t✐♦♥ kx = b ✐s s♦❧✈❡❞ ❜② ✉♥❞♦✐♥❣ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② k ❜② ❞✐✈✐s✐♦♥ ❜② k✿ kx = b =⇒ x =

❙✐♠✐❧❛r❧②✱ ✇❡ ♥❡❡❞ t❤❡ ✐♥✈❡rs❡ ♦❢ F t♦ s♦❧✈❡ ♦✉r ❡q✉❛t✐♦♥✿

b . k

F −1 : Rm → Rn F (X) = B =⇒ X = F −1 (B)

❆s ❛♥ ✐❧❧✉str❛t✐♦♥✱ t❤❡ ♦♣❡r❛t♦r F tr❛♥s❢♦r♠s t❤❡ xy✲♣❧❛♥❡ ✐♥t♦ t❤❡✱ s❛②✱ uv✲♣❧❛♥❡✿

✸✳✷✳

❈♦♠♣♦s✐t✐♦♥s ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs

✷✵✺

❖♥❡ ♣❛rt✐❝✉❧❛r ✈❡❝t♦r ✐♥ t❤❡ uv ✲♣❧❛♥❡✱ B ✱ ♥❡❡❞s t♦ ❜❡ tr❛❝❡❞ ❜❛❝❦ t♦ t❤❡ xy ✲♣❧❛♥❡✳ ❖❢ ❝♦✉rs❡✱ ✐❢ ✇❡ ❤❛✈❡ F −1 ✱ ✇❡✬❧❧ ✜♥❞ t❤❡ ❝♦✉♥t❡r♣❛rts ❢♦r ❛❧❧ B ✬s✳ ❊①❛♠♣❧❡ ✸✳✷✳✶✿ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡

❲❡ ❦♥♦✇ s♦♠❡ ♦❢ t❤❡ ❛♥s✇❡rs✿ ✶✳ ■❢ F ✐s t❤❡ ✉♥✐❢♦r♠ str❡t❝❤ ❜② 2✱ t❤❡♥ F −1 ✐s t❤❡ ✉♥✐❢♦r♠ s❤r✐♥❦ ❜② 2✳ ✷✳ ■❢ F ✐s t❤❡ str❡t❝❤ ❜② 2 ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ❛ ✈❡❝t♦r e1 =< 1, 0 >✱ t❤❡♥ F −1 ✐s t❤❡ ✉♥✐❢♦r♠ s❤r✐♥❦ ❜② 2 ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ e1 ✳ ✸✳ ■❢ F ✐s t❤❡ r♦t❛t✐♦♥ ❜② 90 ❞❡❣r❡❡s ❝❧♦❝❦✇✐s❡✱ t❤❡♥ F −1 ✐s t❤❡ r♦t❛t✐♦♥ ❜② 90 ❞❡❣r❡❡s ❝♦✉♥t❡r❝❧♦❝❦✲ ✇✐s❡✳ ✹✳ ■❢ F ✐s t❤❡ ✢✐♣ ❛❜♦✉t t❤❡ x✲❛①✐s✱ t❤❡♥ F −1 ✐s t❤❡ ✢✐♣ ❛❜♦✉t t❤❡ x✲❛①✐s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✶✳ F (X) = 2X =⇒ F −1 (Y ) = 21 Y 

✷✳ F = 



✸✳ F = 



✹✳ F = 

2 0 0 1 0

 

1

−1 0 1

0

0 −1

=⇒ F −1 = 



1 2

0 1

 



0 −1 1

0



1

0

 =⇒ F −1 =  

0

 =⇒ F −1 = 

0 −1

 

 

▲❡t✬s r❡❝❛❧❧ t❤❛t t❤❡ ✐♥✈❡rs❡ ✐s ❞❡✜♥❡❞ ✈✐❛ ❝♦♠♣♦s✐t✐♦♥s✳ ■t ♠✉st s❛t✐s❢②✿ F (F −1 (Y )) = Y ,

❢♦r ❛❧❧ Y ✱ ❛♥❞

F −1 (F (X)) = X ,

❢♦r ❛❧❧ X ✳ ■♥ ♦t❤❡r ✇♦r❞s✱

F ◦ F −1 = I ,

❛♥❞

F −1 ◦ F = I ,

✇❤❡r❡ I ✐s t❤❡ ✐❞❡♥t✐t② ♠❛tr✐①✳ ❲❡ ♥❡❡❞ t♦ ✉♥❞❡rst❛♥❞ ❝♦♠♣♦s✐t✐♦♥s ❜❡tt❡r✳

❲❡ ❦♥♦✇ ❤♦✇ t♦ ❝♦♠♣✉t❡ ❝♦♠♣♦s✐t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s✳ ❚❤✐s ✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥✿ F

G

Rn −−−−→ Rm −−−−→ Rk

■t ✐s ❝♦♠♣✉t❡❞ ✈✐❛ s✉❜st✐t✉t✐♦♥✿ (G ◦ F )(X) = G(F (X)) .

❇✉t ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ♠❛tr✐❝❡s❄ ❍♦✇ ❛r❡ t❤❡ ♠❛tr✐❝❡s ♦❢ F ❛♥❞ G ❝♦♠❜✐♥❡❞ t♦ ♣r♦❞✉❝❡ t❤❡ ♠❛tr✐① ♦❢ G ◦ F ❄ ■t ✐s ❝❛❧❧❡❞ ♠❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ❍❡r❡ ✐s ✇❤②✳ ❊①❛♠♣❧❡ ✸✳✷✳✷✿ ❝♦♠♣♦s✐t✐♦♥s

R1 → R1 → R1

●✐✈❡♥ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✭♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✮✿ ❛♥❞

f : R → R ❞❡✜♥❡❞ ❜② f (x) = 3x , g : R → R ❞❡✜♥❡❞ ❜② g(y) = 2y .

✸✳✷✳

❈♦♠♣♦s✐t✐♦♥s ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs

❚❤❡✐r ♠❛tr✐❝❡s ❛r❡✿

✷✵✻

f = [3] ❛♥❞ g = [2] .

❲❤❛t ❛❜♦✉t t❤❡✐r ❝♦♠♣♦s✐t✐♦♥❄ ❚❤❡ ❝♦❞♦♠❛✐♥ ♦❢ t❤❡ ❢♦r♠❡r ❛♥❞ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❧❛tt❡r ♠❛t❝❤✦ ❚❤❡ ❝♦♠♣♦s✐t✐♦♥ ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿ g ◦ f : R → R ❞❡✜♥❡❞ ❜② (g ◦ f )(x) = g(f (x)) = 2(3x) = 6x .

■ts ♠❛tr✐① ✐s g ◦ f = [6] .

❖❢ ❝♦✉rs❡✱ t❤✐s ♥❡✇ ♥✉♠❜❡r ✐s ❥✉st t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ t✇♦ ♦r✐❣✐♥❛❧ ♥✉♠❜❡rs✳ ❊①❛♠♣❧❡ ✸✳✷✳✸✿ ❝♦♠♣♦s✐t✐♦♥s

R1 → R1 → R2

●✐✈❡♥ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✭❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ❛♥❞ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✮✿ f : R → R ❞❡✜♥❡❞ ❜② f (x) = 3x ,

❛♥❞ ❚❤❡✐r ♠❛tr✐❝❡s ❛r❡✿

G : R → R2 ❞❡✜♥❡❞ ❜② G(y) =< 3y, 2y > .   3 . f = [3] ❛♥❞ G = 2

❲❤❛t ❛❜♦✉t t❤❡✐r ❝♦♠♣♦s✐t✐♦♥❄ ■t ✐s ❛♥ ♦♣❡r❛t♦r ✭❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✮ ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿ G ◦ f : R → R2 ,

❞❡✜♥❡❞ ❜② ■ts ♠❛tr✐① ✐s

(G ◦ f )(x) = G(f (x)) =< 3(3x), 2(3x) >=< 9x, 6x > .   9 . G◦f = 6

❖❢ ❝♦✉rs❡✱ t❤✐s ✐s ❥✉st t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ t✇♦ ❛s ✐❢ t❤❡ ✜rst ✐s ❛ ♥✉♠❜❡r ❛♥❞ t❤❡ s❡❝♦♥❞ ❛ ✈❡❝t♦r✿       9 3·3 3 . = [3] = Gf = 6 2·3 2 ❊①❛♠♣❧❡ ✸✳✷✳✹✿ ❝♦♠♣♦s✐t✐♦♥s

R1 → R2 → R1

●✐✈❡♥ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✭❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❛♥❞ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✮✿ ❛♥❞ ❚❤❡✐r ♠❛tr✐❝❡s ❛r❡✿

F : R → R2 ❞❡✜♥❡❞ ❜② F (x) =< 3x, 2x > , g : R2 → R ❞❡✜♥❡❞ ❜② g(u, v) = 5u − 2v .   3 F = 2

❛♥❞ g = [5, −2] .

❚❤❡✐r s✉♠ ✐s ❛ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥ ❛♥❞ ✐t ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿ ❞❡✜♥❡❞ ❜②

g ◦ F : R → R, (g ◦ F )(x) = g(F (x)) = 5(3x) − 2(2x) = 11x .

✸✳✷✳

❈♦♠♣♦s✐t✐♦♥s ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs

✷✵✼

■ts ♠❛tr✐① ✐s g ◦ F = [11] .

■t✬s t❤❡ ❞♦t ♣r♦❞✉❝t ♦❢ t❤❡ t✇♦ ✈❡❝t♦r✲❧✐❦❡ ♠❛tr✐❝❡s✿

  3 = [5 · 3 + (−2) · 2] = [11] . gF = [5, −2] 2 ❊①❛♠♣❧❡ ✸✳✷✳✺✿ ❝♦♠♣♦s✐t✐♦♥s

R2 → R2 → R2

●✐✈❡♥ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✭tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡s✮✿ F, G : R2 → R2

❞❡✜♥❡❞ ❜② F (x, y) =< u, v >=< 3x + 2y, 5x − y >

❚❤❡✐r ♠❛tr✐❝❡s ❛r❡✿



3 2 F = 5 −1



❛♥❞ G(u, v) =< 5u + v, u + v > .

 5 1 . ❛♥❞ G = 1 1 

❚❤❡✐r ❝♦♠♣♦s✐t✐♦♥ ✐s ❛♥ ♦♣❡r❛t♦r ❝♦♠♣✉t❡❞ ❜② s✉❜st✐t✉t✐♦♥✿ G ◦ F : R2 → R 2 ,

❞❡✜♥❡❞ ❜② (G ◦ F )(x, y) =< 5(3x + 2y) + (5x − y), (3x + 2y) + (5x − y) >=< 20x + 9y, 8x + y > .

■ts ♠❛tr✐① ✐s

 20 9 . G◦F = 8 1 

■t ✐s s❡❡♥ ❛s ❝♦♠♣✉t❡❞ ✈✐❛ ❢♦✉r ❞♦t ♣r♦❞✉❝ts ♦❢ t❤❡ ❢♦✉r ♣❛✐rs ♦❢ r♦✇s ✭❢r♦♠ t❤❡ ✜rst ♠❛tr✐①✮ ❛♥❞ ❝♦❧✉♠♥s ✭❢r♦♠ t❤❡ s❡❝♦♥❞✮✿ 

5  1  5  1  5  1  5  1

 

1 3 · 1 5   1 3 · 1 5   1 3 · 1 5   1 3 · 1 5



2 → −1  2 → −1  2 → −1  2 → −1

20 → 8  10 5 · 2 + 1 · (−1) = 9 →  8  10 1·3+1·5=8 → 8  10 1 · 2 + 1 · (−1) = 1 →  8 5 · 3 + 1 · 5 = 20

■♥ ❣❡♥❡r❛❧✱ ✇❡ ❤❛✈❡ ❛ s✐♥❣❧❡ ❢♦r♠✉❧❛✿ 



     ae + bg af + bh e f a b = · ce + dg cf + dh g h c d



9  1  9  1  9  1  9  1

✸✳✷✳

❈♦♠♣♦s✐t✐♦♥s ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs

✷✵✽

❲❛r♥✐♥❣✦ ❆s t❤❡ ♠❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ❥✉st ❛♥ ❛❜❜r❡✈✐❛t❡❞ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❧✐♥❡❛r ♦♣❡r❛✲ t♦rs✱ ✐t ✐s s❡❝♦♥❞❛r② t♦ t❤❡ s✉❜st✐t✉t✐♦♥ ♦❢ t❤❡ ❢♦r✲ ♠✉❧❛s ✐t ❝♦♠❡s ❢r♦♠✳

◆♦✇ ✇❡ ❝♦♥s✐❞❡r ❧✐♥❡❛r ♦♣❡r❛t♦rs ✐♥ ❛r❜✐tr❛r② ❞✐♠❡♥s✐♦♥s✿

R n → Rm → R k . ❋♦r t❤❡✐r ♠❛tr✐❝❡s✱ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝♦❧✉♠♥ ✐♥ t❤❡ ✜rst ♠✉st ❜❡ ❡q✉❛❧ t♦ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ r♦✇ ✐♥ t❤❡ s❡❝♦♥❞✳ ❚❤❛t✬s

m✦

❲❡ ❞♦ t❤✐s

❚❤❡♥✱ ✇❡ ❝❛♥ ❝❛rr② ♦✉t ❛ ❞♦t ♣r♦❞✉❝t✿

nk

t✐♠❡s ❛♥❞ ♣r♦❞✉❝❡ ❛♥

n×k

♠❛tr✐①✳

❊①❡r❝✐s❡ ✸✳✷✳✻ ▼✉❧t✐♣❧②✿

   1 −1 1 2 3 4 5 6 · −1 1  . 1 −1 7 8 9  ❊①❛♠♣❧❡ ✸✳✷✳✼✿ s♣r❡❛❞s❤❡❡t ❖♥❡ ❝❛♥ ✉t✐❧✐③❡ ❛ s♣r❡❛❞s❤❡❡t ❛♥❞ ♦t❤❡r s♦❢t✇❛r❡ t♦ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❢♦r ♠❛tr✐❝❡s ♦❢ ❛♥② ❞✐♠❡♥s✐♦♥s✳ ■♥ ♦r❞❡r t♦ ♠❛❦❡ t❤✐s ✇♦r❦✱ t❤❡ s❡❝♦♥❞ ♠❛tr✐①

❚❤✐s ✐s t❤❡ ❝♦❞❡ ❢♦r t❤❡ tr❛♥s♣♦s❡ ♦❢

B

❤❛s t♦ ❜❡ ✏tr❛♥s♣♦s❡❞✑ ✭❜♦tt♦♠✮✿

X✿

❂❚❘❆◆❙P❖❙❊✭❘❬✲✺❪❈✿❘❬✲✸❪❈❬✶❪✮ ❚❤✐s ✐s t❤❡ ❝♦❞❡ ❢♦r

Y✿

❂❙❯▼P❘❖❉❯❈❚✭❘❈✷✿❘❈✹✱❘✽❈❬✲✸❪✿❘✽❈❬✲✶❪✮

❋✐♥❞✐♥❣ t❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ♠❛tr✐①✱ ❡s♣❡❝✐❛❧❧② ♦❢ ❤✐❣❤ ❞✐♠❡♥s✐♦♥✱ ✐s ❛ ❝❤❛❧❧❡♥❣✐♥❣ ♣r♦❜❧❡♠✳ ❚❤❡r❡ ✐s ❛ s✐♠♣❧❡ ❢♦r♠✉❧❛ ❢♦r t❤❡

2×2

♠❛tr✐❝❡s✿

✸✳✸✳

❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡

✷✵✾

❚❤❡♦r❡♠ ✸✳✷✳✽✿ ■♥✈❡rs❡ ♦❢

2×2

❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛♥ ✐♥✈❡rt✐❜❧❡ ♠❛tr✐①



a b c d

−1

▼❛tr✐①

A

✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿

  1 d −b = det A −c a

❊①❡r❝✐s❡ ✸✳✷✳✾ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

❊①❡r❝✐s❡ ✸✳✷✳✶✵

$3 ♣❡r ❝❤✐❧❞ ❛♥❞ $3.20 ♣❡r ❛❞✉❧t ❢♦r ❛ t♦t❛❧ ♦❢ $118.40✳ ❚❤❡② t♦♦❦ t❤❡ tr❛✐♥ ❜❛❝❦ ❛t $3.50 ♣❡r ❝❤✐❧❞ ❛♥❞ $3.60 ♣❡r ❛❞✉❧t ❢♦r ❛ t♦t❛❧ ♦❢ $135.20✳ ❍♦✇ ♠❛♥② ❝❤✐❧❞r❡♥✱ ❛♥❞ ❤♦✇ ♠❛♥② ❛❞✉❧ts❄ ❯s❡ t❤❡ t❤❡♦r❡♠ t♦ s♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠✿ ❆ t♦✉r✐st ❣r♦✉♣ t♦♦❦ ❛ tr❛✐♥ tr✐♣ ❛t

❊①❡r❝✐s❡ ✸✳✷✳✶✶ ❯s❡ t❤❡ t❤❡♦r❡♠ t♦ s♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠✿ ❆ t♦✉r✐st ❣r♦✉♣ ✇✐t❤ ❛ tr❛✐♥ tr✐♣ ❢♦r ❛ t♦t❛❧ ♦❢ ❢♦r ❛ t♦t❛❧ ♦❢

$145✳

$110✳

❆♥♦t❤❡r t♦✉r✐st ❣r♦✉♣ ✇✐t❤

15

10

❝❤✐❧❞r❡♥ ❛♥❞

❝❤✐❧❞r❡♥ ❛♥❞

25

20

❛❞✉❧ts t♦♦❦

❛❞✉❧ts t♦♦❦ ❛ tr❛✐♥ tr✐♣

❲❤❛t ✇❡r❡ t❤❡ t✐❝❦❡t ♣r✐❝❡s❄

❚♦ s✉♠♠❛r✐③❡✱ ✇❡ ❣♦ ❜❛❝❦ t♦ t❤❡ ❣❡♥❡r❛❧ s❡t✉♣ ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛♣♣❧✐❡❞ ❝♦♥s❡❝✉t✐✈❡❧②✿

G

F

Rn −−−−→ Rm −−−−→ Rk ❲❡ ❤❛✈❡ ✐♠♣❧✐❝✐t❧② ✉s❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝t✿

❚❤❡♦r❡♠ ✸✳✷✳✶✷✿ ❈♦♠♣♦s✐t✐♦♥ ♦❢ ▲✐♥❡❛r ❖♣❡r❛t♦rs ❚❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✐s ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✳

❊①❡r❝✐s❡ ✸✳✷✳✶✸ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

❲❡ ❤❛✈❡ ❛❧s♦ ✐♠♣❧✐❝✐t❧② ✉s❡❞ t❤❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t r❡s✉❧t✿

❚❤❡♦r❡♠ ✸✳✷✳✶✹✿ ▼❛tr✐① ♦❢ ❈♦♠♣♦s✐t✐♦♥ ❚❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♠❛tr✐❝❡s t❤❛t r❡♣r❡s❡♥t t✇♦ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✐s t❤❡ ♠❛tr✐① ♦❢ t❤❡✐r ❝♦♠♣♦s✐t✐♦♥✳

✸✳✸✳ ❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡

❚❤❡ ❡q✉❛t✐♦♥

x2 + 1 = 0 ❤❛s ♥♦ s♦❧✉t✐♦♥s✳ ■♥❞❡❡❞✱ ✇❡ ♦❜s❡r✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

x2 ≥ 0 =⇒ x2 + 1 > 0 =⇒ x2 + 1 6= 0 .

✸✳✸✳

❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡

✷✶✵

■❢ ✇❡ tr② t♦ s♦❧✈❡ ✐t t❤❡ ✉s✉❛❧ ✇❛②✱ ✇❡ ❣❡t t❤❡s❡✿

x= ❚❤❡r❡ ❛r❡ ♥♦ s✉❝❤



r❡❛❧ ♥✉♠❜❡rs✳

−1

❛♥❞

√ x = − −1 .

❍♦✇❡✈❡r✱ ❧❡t✬s ✐❣♥♦r❡ t❤✐s ❢❛❝t ❢♦r ❛ ♠♦♠❡♥t✳ ▲❡t✬s s✉❜st✐t✉t❡ ✇❤❛t ✇❡ ❤❛✈❡ ❜❛❝❦ ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥ ❛♥❞ ✕ ❜❧✐♥❞❧② ✕ ❢♦❧❧♦✇ t❤❡ r✉❧❡s ♦❢ ❛❧❣❡❜r❛✳ ❲❡ ✏❝♦♥✜r♠✑ t❤❛t t❤✐s ✏♥✉♠❜❡r✑ ✐s ❛ ✏s♦❧✉t✐♦♥✑✿

√ x2 + 1 = ( −1)2 + 1 = (−1) + 1 = 0 . ❲❡ ❝❛❧❧ t❤✐s ❡♥t✐t② t❤❡

✐♠❛❣✐♥❛r② ✉♥✐t✱ ❞❡♥♦t❡❞ ❜② i✳

❲❡ ❥✉st ❛❞❞ t❤✐s ✏♥✉♠❜❡r✑ t♦ t❤❡ s❡t ♦❢ ♥✉♠❜❡rs ✇❡ ❞♦ ❛❧❣❡❜r❛ ✇✐t❤✿

❆♥❞ s❡❡ ✇❤❛t ❤❛♣♣❡♥s✳✳✳ ▼❛❦✐♥❣

i

❛ ♣❛rt ♦❢ ❛❧❣❡❜r❛ ✇✐❧❧ ♦♥❧② r❡q✉✐r❡ t❤✐s t❤r❡❡✲♣❛rt ❝♦♥✈❡♥t✐♦♥✿

✶✳

i

✷✳

i ❝❛♥ ♣❛rt✐❝✐♣❛t❡ ✐♥ t❤❡ ✭❢♦✉r✮ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ✇✐t❤ r❡❛❧ ♥✉♠❜❡rs ❜② ❢♦❧❧♦✇✐♥❣ t❤❡ s❛♠❡ r✉❧❡s❀

✸✳

i2 = −1✳

✐s ♥♦t ❛ r❡❛❧ ♥✉♠❜❡r ✭❛♥❞✱ ✐♥ ♣❛rt✐❝✉❧❛r✱

i 6= 0✮✱

❜✉t ❛❧s♦

❲❤❛t ❛❧❣❡❜r❛✐❝ r✉❧❡s ❛r❡ t❤♦s❡❄ ❆ ❢❡✇ ✈❡r② ❜❛s✐❝ ♦♥❡s✿

x + y = y + x, x · y = y · x, x(y + z) = xy + xz, ❲❡ ❛❧❧♦✇ ♦♥❡ ♦r s❡✈❡r❛❧ ♦❢ t❤❡s❡ ♣❛r❛♠❡t❡rs t♦ ❜❡

i✳

❡t❝✳

❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❤❛✈❡✿

i + y = y + i, i · y = y · i, i(y + z) = iy + iz,

❡t❝✳

❲❤❛t ♠❛❦❡s t❤✐s ❡①tr❛ ❡✛♦rt ✇♦rt❤✇❤✐❧❡ ✐s ❛ ♥❡✇ ❧♦♦❦ ❛t q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ❤♦✇ ✇❡ ♠❛② ❢❛❝t♦r ♦♥❡✿

x2 − 1 = (x − 1)(x + 1) . ❚❤❡♥

x=1

❛♥❞

x = −1

❛r❡ t❤❡

❇✉t s♦♠❡ ♣♦❧②♥♦♠✐❛❧s✱ ❝❛❧❧❡❞

x✲✐♥t❡r❝❡♣ts

♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧✿

✐rr❡❞✉❝✐❜❧❡✱ ❝❛♥♥♦t ❜❡ ❢❛❝t♦r❡❞❀ t❤❡r❡ ❛r❡ ♥♦ a, b s✉❝❤ t❤❛t✿ x2 + 1 = (x − a)(x − b) .

❚❤❡r❡ ❛r❡ ♥♦

r❡❛❧ a, b✱ t❤❛t ✐s✦ ❯s✐♥❣ ♦✉r r✉❧❡s✱ ✇❡ ❞✐s❝♦✈❡r✿

(x − i)(x + i) = x2 − ix + ix − i2 = x2 + 1 .

✸✳✸✳ ❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡

❖❢ ❝♦✉rs❡✱ t❤❡ ♥✉♠❜❡r

i

✐s ♥♦t ❛♥

✷✶✶

x✲✐♥t❡r❝❡♣t

♦❢

f (x) = x2 + 1

❛s t❤❡

x✲❛①✐s

✭✏t❤❡ r❡❛❧ ❧✐♥❡✑✮ ❝♦♥s✐sts ♦❢ ♦♥❧②

✭❛♥❞ ❛❧❧✮ r❡❛❧ ♥✉♠❜❡rs✳ ❙♦✱ ♠✉❧t✐♣❧❡s ♦❢

i

❛♣♣❡❛r ✐♠♠❡❞✐❛t❡❧② ❛s ✇❡ st❛rt ❞♦✐♥❣ ❛❧❣❡❜r❛ ✇✐t❤ ✐t✳

❉❡✜♥✐t✐♦♥ ✸✳✸✳✶✿ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs ❚❤❡ r❡❛❧ ♠✉❧t✐♣❧❡s ♦❢ t❤❡ ✐♠❛❣✐♥❛r② ✉♥✐t✱ ✐✳❡✳✱

z = ri, r

r❡❛❧,

❛r❡ ❝❛❧❧❡❞ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs✳

❲❡ ❤❛✈❡ ❝r❡❛t❡❞ ❛ ✇❤♦❧❡ ❝❧❛ss ♦❢ ♥♦♥✲r❡❛❧ ♥✉♠❜❡rs✦ ❖❢ ❝♦✉rs❡✱

ri✱

✇❤❡r❡

r

✐s r❡❛❧✱ ❝❛♥✬t ❜❡ r❡❛❧✿

(ri)2 = r2 i2 = −r2 < 0 . ❚❤❡ ♦♥❧② ❡①❝❡♣t✐♦♥ ✐s

0i = 0❀

✐t✬s r❡❛❧✦

❚❤❡r❡ ❛r❡ ❛s ♠❛♥② ♦❢ t❤❡♠ ❛s t❤❡ r❡❛❧ ♥✉♠❜❡rs✿

❊①❛♠♣❧❡ ✸✳✸✳✷✿ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s ❚❤❡ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs ♠❛② ❛❧s♦ ❝♦♠❡ ❢r♦♠ s♦❧✈✐♥❣ t❤❡ s✐♠♣❧❡st q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❡q✉❛t✐♦♥

x2 + 4 = 0 ❣✐✈❡s ✉s ✈✐❛ ♦✉r s✉❜st✐t✉t✐♦♥✿

■♥❞❡❡❞✱ ✐❢ ✇❡ s✉❜st✐t✉t❡

p √ √ √ x = ± −4 = ± 4(−1) = ± 4 −1 = ±2i .

x = 2i

✐♥t♦ t❤❡ ❡q✉❛t✐♦♥✱ ✇❡ ❤❛✈❡✿

(2i)2 + 4 = (2)2 (i)2 + 4 = 4(−1) + 4 = 0 . ▼♦r❡ ❣❡♥❡r❛❧ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s ❛r❡ ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳

■♠❛❣✐♥❛r② ♥✉♠❜❡rs ♦❜❡② t❤❡ ❧❛✇s ♦❢ ❛❧❣❡❜r❛ ❛s ✇❡ ❦♥♦✇ t❤❡♠✦ ■❢ ✇❡ ♥❡❡❞ t♦ s✐♠♣❧✐❢② t❤❡ ❡①♣r❡ss✐♦♥✱ ✇❡ tr② t♦ ♠❛♥✐♣✉❧❛t❡ ✐t ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t r❡❛❧ ♥✉♠❜❡rs ❛r❡ ❝♦♠❜✐♥❡❞ ✇✐t❤ r❡❛❧ ✇❤✐❧❡ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ ❥✉st ❢❛❝t♦r

i

i

✐s ♣✉s❤❡❞ ❛s✐❞❡✳

♦✉t ♦❢ ❛❧❧ ❛❞❞✐t✐♦♥ ❛♥❞ s✉❜tr❛❝t✐♦♥✿

5i + 3i = (5 + 3)i = 8i . ■t ❧♦♦❦s ❡①❛❝t❧② ❧✐❦❡ ♠✐❞❞❧❡ s❝❤♦♦❧ ❛❧❣❡❜r❛✿

5x + 3x = (5 + 3)x = 8x . ❆❢t❡r ❛❧❧✱

x

❝♦✉❧❞ ❜❡ i✳ ❆♥♦t❤❡r s✐♠✐❧❛r✐t② ✐s ✇✐t❤ t❤❡ ❛❧❣❡❜r❛ ♦❢ q✉❛♥t✐t✐❡s t❤❛t ❤❛✈❡ ✉♥✐ts✿

5

✐♥✳

+3

✐♥✳

= (5 + 3)

✐♥✳

=8

✐♥✳ .

❙♦✱ t❤❡ ♥❛t✉r❡ ♦❢ t❤❡ ✉♥✐t ❞♦❡s♥✬t ♠❛tt❡r ✭✐❢ ✇❡ ❝❛♥ ♣✉s❤ ✐t ❛s✐❞❡✮✳ ❊✈❡♥ s✐♠♣❧❡r✿

5

❛♣♣❧❡s

+3

❛♣♣❧❡s

= (5 + 3)

❛♣♣❧❡s

=8

❛♣♣❧❡s

.

✸✳✸✳

❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡

✷✶✷

■t✬s ✏ 8 ❛♣♣❧❡s✑ ♥♦t ✏ 8✑✦ ❆♥❞ s♦ ♦♥✳ ❚❤✐s ✐s ❤♦✇ ✇❡ ♠✉❧t✐♣❧② ❛♥ ✐♠❛❣✐♥❛r② ♥✉♠❜❡r ❜② ❛ r❡❛❧ ♥✉♠❜❡r✿

2 · (3i) = (2 · 3)i = 6i . ❲❡ ❤❛✈❡ ❛ ♥❡✇ ✐♠❛❣✐♥❛r② ♥✉♠❜❡r✳ ❍♦✇ ❞♦ ✇❡ ♠✉❧t✐♣❧② t✇♦ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs❄ ■t✬s ❞✐✛❡r❡♥t❀ ❛❢t❡r ❛❧❧✱ ✇❡ ❞♦♥✬t ✉s✉❛❧❧② ♠✉❧t✐♣❧② ❛♣♣❧❡s ❜② ❛♣♣❧❡s✦ ■♥ ❝♦♥tr❛st t♦ t❤❡ ❛❜♦✈❡✱ ❡✈❡♥ t❤♦✉❣❤ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥ ❢♦❧❧♦✇ t❤❡ s❛♠❡ r✉❧❡ ❛s ❛❧✇❛②s✱ ✇❡ ❝❛♥✱ ✇❤❡♥ ♥❡❝❡ss❛r②✱ ❛♥❞ ♦❢t❡♥ ❤❛✈❡ t♦✱ s✐♠♣❧✐❢② t❤❡ ♦✉t❝♦♠❡ ♦❢ ♦✉r ❛❧❣❡❜r❛ ✉s✐♥❣ ♦✉r

❢✉♥❞❛♠❡♥t❛❧ ✐❞❡♥t✐t② ✿

i2 = −1 . ❋♦r ❡①❛♠♣❧❡✿

(5i) · (3i) = (5 · 3)(i · i) = 15i2 = 15(−1) = −15 . ■t✬s r❡❛❧✦ ❲❡ ❛❧s♦ s✐♠♣❧✐❢② t❤❡ ♦✉t❝♦♠❡ ❜② ✉s✐♥❣ t❤❡ ♦t❤❡r

❢✉♥❞❛♠❡♥t❛❧ ❢❛❝t

❛❜♦✉t t❤❡ ✐♠❛❣✐♥❛r② ✉♥✐t✿

i 6= 0 . ❲❡ ❝❛♥ ❞✐✈✐❞❡ ❜②

i✦

❋♦r ❡①❛♠♣❧❡✱

5i 5 5 5i = = ·1= . 3i 3i 3 3

❆s ②♦✉ ❝❛♥ s❡❡✱ ❞♦✐♥❣ ❛❧❣❡❜r❛ ✇✐t❤ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs ✇✐❧❧ ♦❢t❡♥ ❜r✐♥❣ ✉s ❜❛❝❦ t♦ r❡❛❧ ♥✉♠❜❡rs✳ ❚❤❡s❡ t✇♦ ❝❧❛ss❡s ♦❢ ♥✉♠❜❡rs ❝❛♥♥♦t ❜❡ s❡♣❛r❛t❡❞ ❢r♦♠ ❡❛❝❤ ♦t❤❡r✦ ❚❤❡② ❛r❡♥✬t✳ ▲❡t✬s t❛❦❡ ❛♥♦t❤❡r ❧♦♦❦ ❛t q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s✳ ❚❤❡ ❡q✉❛t✐♦♥

ax2 + bx + c = 0, a 6= 0 , ✐s s♦❧✈❡❞ ✇✐t❤ t❤❡ ❢❛♠✐❧✐❛r

◗✉❛❞r❛t✐❝ ❋♦r♠✉❧❛ ✿ x=

−b ±



b2 − 4ac . 2a

▲❡t✬s ❝♦♥s✐❞❡r

x2 + 2x + 10 = 0 . ❚❤❡♥ t❤❡ r♦♦ts ❛r❡ s✉♣♣♦s❡❞ t♦ ❜❡✿



22 − 4 · 10 √2 −2 ± −36 = 2 √ = −1 ± −9 √ √ = −1 ± 9 −1

x =

−2 ±

❚❤❡r❡ ✐s ♥♦ r❡❛❧ s♦❧✉t✐♦♥✦ ❇✉t ✇❡ ❣♦ ♦♥✳

= −1 ± 3i . ❲❡ ❡♥❞ ✉♣ ❛❞❞✐♥❣ r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs✦ ❆s t❤❡r❡ ✐s ♥♦ ✇❛② t♦ s✐♠♣❧✐❢② t❤✐s✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿



❆ ♥✉♠❜❡r

a + bi✱

✇❤❡r❡

a, b 6= 0

❛r❡ r❡❛❧✱ ✐s ♥❡✐t❤❡r r❡❛❧ ♥♦r ✐♠❛❣✐♥❛r②✳

✸✳✸✳

❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡

✷✶✸

❊①❡r❝✐s❡ ✸✳✸✳✸ ❊①♣❧❛✐♥ ✇❤②✳

❚❤✐s ❛❞❞✐t✐♦♥ ✐s ♥♦t ❧✐t❡r❛❧✳ ■t✬s ❧✐❦❡ ✏❛❞❞✐♥❣✑ ❛♣♣❧❡s t♦ ♦r❛♥❣❡s✿

5 ■t✬s ♥♦t

❛♣♣❧❡s

+3

♦r❛♥❣❡s

= ...

8 ❛♥❞ ✐t✬s ♥♦t 8 ❢r✉✐t ❜❡❝❛✉s❡ ✇❡ ✇♦✉❧❞♥✬t ❜❡ ❛❜❧❡ t♦ r❡❛❞ t❤✐s ❡q✉❛❧✐t② ❜❛❝❦✇❛r❞s✳

❚❤❡ ❛❧❣❡❜r❛ ✇✐❧❧✱

❤♦✇❡✈❡r✱ ❜❡ ♠❡❛♥✐♥❣❢✉❧✿

(5a + 3o) + (2a + 4o) = (5 + 3)a + (3 + 4)o = 8a + 7o . ■t ✐s ❛s ✐❢ ✇❡ ❝♦❧❧❡❝t

s✐♠✐❧❛r t❡r♠s✱ ❧✐❦❡ t❤✐s✿ (5 + 3x) + (2 + 4x) = (5 + 2) + (3 + 4)x = 8 + 7x .

❚❤✐s ✐❞❡❛ ❡♥❛❜❧❡s ✉s t♦ ❞♦ t❤✐s✿

(5 + 3i) + (2 + 4i) = (5 + 3) + (3 + 4)i = 8 + 7i . ❊❛❝❤ ♦❢ t❤❡ ♥✉♠❜❡rs ✇❡ ❛r❡ ❢❛❝✐♥❣ ❝♦♥t❛✐♥ ❜♦t❤ r❡❛❧ ♥✉♠❜❡rs ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts✳ ❚❤✐s ❢❛❝t ♠❛❦❡s t❤❡♠ ✏❝♦♠♣❧❡①✑✳✳✳

❉❡✜♥✐t✐♦♥ ✸✳✸✳✹✿ ❝♦♠♣❧❡① ♥✉♠❜❡r ❆♥② s✉♠ ♦❢ r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs ✐s ❝❛❧❧❡❞ ❛

❝♦♠♣❧❡① ♥✉♠❜❡r✳ ❚❤❡ s❡t ♦❢

❛❧❧ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

C = {z = a + bi : a, b

r❡❛❧}

❲❛r♥✐♥❣✦ ❆❧❧ r❡❛❧ ♥✉♠❜❡rs ❛r❡ ❝♦♠♣❧❡① ✭b

= 0✮✳

❆❞❞✐t✐♦♥ ❛♥❞ s✉❜tr❛❝t✐♦♥ ❛r❡ ❡❛s②❀ ✇❡ ❥✉st ❝♦♠❜✐♥❡ s✐♠✐❧❛r t❡r♠s ❥✉st ❧✐❦❡ ✐♥ ♠✐❞❞❧❡ s❝❤♦♦❧✳ ❋♦r ❡①❛♠♣❧❡✱ (1 + 5i) + (3 − i) = 1 + 5i + 3 − i = (1 + 3) + (5i − i) = 4 + 4i . ❚♦ s✐♠♣❧✐❢②

♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✱ ✇❡ ❡①♣❛♥❞ ❛♥❞ t❤❡♥ ✉s❡ i2 = −1✱ ❛s ❢♦❧❧♦✇s✿ (1 + 5i) · (3 − i) = 1 · 3 + 5i · 3 + 1 · (−i) + 5i · (−i) = 3 + 15i − i − 5i2 = (3 + 5) + (15i − i) = 8 + 14i .

■t✬s ❛ ❜✐t tr✐❝❦✐❡r ✇✐t❤

❞✐✈✐s✐♦♥ ✿

1 + 5i 1 + 5i 3 + i = 3−i 3−i 3+i (1 + 5i)(3 + i) = (3 − i)(3 + i) −2 + 8i = 2 3 − i2 −2 + 8i = 2 3 +1 1 = (−2 + 8i) 10 = −0.2 + 0.8i .

✸✳✸✳

❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡

✷✶✹

❚❤❡ s✐♠♣❧✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❞❡♥♦♠✐♥❛t♦r ✐s ♠❛❞❡ ♣♦ss✐❜❧❡ ❜② t❤❡ tr✐❝❦ ♦❢ ♠✉❧t✐♣❧②✐♥❣ ❜② 3 + i✳ ■t ✐s t❤❡ s❛♠❡ tr✐❝❦ ✇❡ ✉s❡❞ ✐♥ ❱♦❧✉♠❡ ✶ t♦ s✐♠♣❧✐❢② ❢r❛❝t✐♦♥s ✇✐t❤ r♦♦ts t♦ ❝♦♠♣✉t❡ t❤❡✐r ❧✐♠✐ts✿ √ √ 1 1 1+ x 1+ x √ = √ √ = . 1−x 1− x 1− x1+ x

❉❡✜♥✐t✐♦♥ ✸✳✸✳✺✿ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ❚❤❡

❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ♦❢ z = a + bi ✐s ❞❡✜♥❡❞ ❛♥❞ ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ z¯ = a + bi = a − bi .

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❝r✉❝✐❛❧✳

❚❤❡♦r❡♠ ✸✳✸✳✻✿ ❆❧❣❡❜r❛ ♦❢ ❈♦♠♣❧❡① ◆✉♠❜❡rs ❚❤❡ r✉❧❡s ♦❢ t❤❡ ❛❧❣❡❜r❛ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛r❡ ✐❞❡♥t✐❝❛❧ t♦ t❤♦s❡ ♦❢ r❡❛❧ ♥✉♠✲ ❜❡rs✿

• • • • •

z+u=u+z ❆ss♦❝✐❛t✐✈✐t② ♦❢ ❛❞❞✐t✐♦♥✿ (z + u) + v = z + (u + v) ❈♦♠♠✉t❛t✐✈✐t② ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ z · u = u · z ❆ss♦❝✐❛t✐✈✐t② ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ (z · u) · v = z · (u · v) ❉✐str✐❜✉t✐✈✐t②✿ z · (u + v) = z · u + z · v ❈♦♠♠✉t❛t✐✈✐t② ♦❢ ❛❞❞✐t✐♦♥✿

❚❤✐s ✐s t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠ ❀ ✐t ❢♦❧❧♦✇s t❤❡ r✉❧❡s ♦❢ t❤❡ r❡❛❧ ♥✉♠❜❡r s②st❡♠ ❜✉t ❛❧s♦ ❝♦♥t❛✐♥s ✐t✳ ❚❤✐s t❤❡♦r❡♠ ✇✐❧❧ ❛❧❧♦✇ ✉s t♦ ❜✉✐❧❞ ❝❛❧❝✉❧✉s ❢♦r ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥s t❤❛t ✐s ❛❧♠♦st ✐❞❡♥t✐❝❛❧ t♦ ❝❛❧❝✉❧✉s ❢♦r r❡❛❧ ❢✉♥❝t✐♦♥s ❜✉t ❛❧s♦ ❝♦♥t❛✐♥s ✐t✳

❉❡✜♥✐t✐♦♥ ✸✳✸✳✼✿ st❛♥❞❛r❞ ❢♦r♠ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡r ❊✈❡r② ❝♦♠♣❧❡① ♥✉♠❜❡r x ❤❛s t❤❡ st❛♥❞❛r❞

r❡♣r❡s❡♥t❛t✐♦♥ ✿

z = a + bi ,

✇❤❡r❡ a ❛♥❞ b ❛r❡ t✇♦ r❡❛❧ ♥✉♠❜❡rs✳ ❚❤❡ t✇♦ ❝♦♠♣♦♥❡♥ts ❛r❡ ♥❛♠❡❞ ❛s ❢♦❧❧♦✇s✿ • a ✐s t❤❡ r❡❛❧ ♣❛rt ♦❢ z ✱ ✇✐t❤ ♥♦t❛t✐♦♥✿ a = Re(z) ; • bi ✐s t❤❡

✐♠❛❣✐♥❛r② ♣❛rt ♦❢ z ✱ ✇✐t❤ ♥♦t❛t✐♦♥✿ b = Im(z) .

❚❤❡♥✱ t❤❡ ♣✉r♣♦s❡ ♦❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥s ❛❜♦✈❡ ✇❛s t♦ ✜♥❞ t❤❡ st❛♥❞❛r❞ ❢♦r♠ ♦❢ ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r t❤❛t ❝♦♠❡s ❢r♦♠ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ✇✐t❤ ♦t❤❡r ❝♦♠♣❧❡① ♥✉♠❜❡rs✳ ❚❤❡② ✇❡r❡ ❧✐t❡r❛❧❧② s✐♠♣❧✐✜❝❛t✐♦♥s✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♠❛❦❡s s❡♥s❡ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t✿

❚❤❡♦r❡♠ ✸✳✸✳✽✿ ❙t❛♥❞❛r❞ ❋♦r♠ ♦❢ ❈♦♠♣❧❡① ◆✉♠❜❡r ❚✇♦ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛r❡ ❡q✉❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❜♦t❤ t❤❡✐r r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts ❛r❡ ❡q✉❛❧✳

❙♦✱ ✇❡ ❤❛✈❡✿ z = Re(z) + Im(z)i .

■♥ ♦r❞❡r t♦ s❡❡ t❤❡ ❣❡♦♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✱ ✇❡ ♥❡❡❞ t♦ ❝♦♠❜✐♥❡ t❤❡ r❡❛❧ ♥✉♠❜❡r ❧✐♥❡ ❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ♥✉♠❜❡r ❧✐♥❡✳ ❍♦✇❄ ❲❡ r❡❛❧✐③❡ t❤❛t t❤❡② ❤❛✈❡ ♥♦t❤✐♥❣ ✐♥ ❝♦♠♠♦♥✳✳✳ ❡①❝❡♣t 0 = 0i

✸✳✸✳

❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡

✷✶✺

❜❡❧♦♥❣s t♦ ❜♦t❤✿

❲❡ ❝❛♥ tr② t♦ ❝♦♠❜✐♥❡ t❤❡♠ ❧✐❦❡ t❤❛t✱ ♦r ❧✐❦❡ t❤✐s✿

❖r ✇❡ ❝❛♥ tr② t♦ ❝♦♠❜✐♥❡ t❤❡♠ ✐♥ t❤❡ s❛♠❡ ♠❛♥♥❡r ✇❡ ❜✉✐❧t t❤❡

xy ✲♣❧❛♥❡✿

❚❤✐s r❡♣r❡s❡♥t❛t✐♦♥ ❤❡❧♣s ✉s ✉♥❞❡rst❛♥❞ t❤❡ ♠❛✐♥ ✐❞❡❛✿

◮ ■❢

❈♦♠♣❧❡① ♥✉♠❜❡rs ❛r❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t❤❡ r❡❛❧ ✉♥✐t✱

z = a + bi✱

t❤❡♥

a

❛♥❞

b

1✱

❛r❡ t❤♦✉❣❤t ♦❢ ❛s t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ ✈❡❝t♦r

❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ✉♥✐t✱

z

i✳

✐♥ t❤❡ ♣❧❛♥❡✳ ❲❡ ❤❛✈❡ ❛ ♦♥❡✲t♦✲♦♥❡

❝♦rr❡s♣♦♥❞❡♥❝❡✿

C ←→ R2 , ❣✐✈❡♥ ❜②

a + bi ←→ < a, b > . ❚❤❡♥ t❤❡

x✲❛①✐s

■t ✐s ❝❛❧❧❡❞ t❤❡

♦❢ t❤✐s ♣❧❛♥❡ ❝♦♥s✐sts ♦❢ t❤❡ r❡❛❧ ♥✉♠❜❡rs ❛♥❞ t❤❡

y ✲❛①✐s

♦❢ t❤❡ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs✳

❝♦♠♣❧❡① ♣❧❛♥❡



❲❛r♥✐♥❣✦ ❚❤✐s ✐s ❥✉st ❛ ✈✐s✉❛❧✐③❛t✐♦♥✳

✸✳✹✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s

✷✶✻

❚❤❡♥ t❤❡ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ♦❢ z ✐s t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r ✇✐t❤ t❤❡ s❛♠❡ r❡❛❧ ♣❛rt ❛s z ❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ♣❛rt ✇✐t❤ t❤❡ ♦♣♣♦s✐t❡ s✐❣♥✿ Re(¯ z ) = Re(z) ❛♥❞ Im(¯ z ) = − Im(z) .

❲❛r♥✐♥❣✦ ❆❧❧ ♥✉♠❜❡rs ✇❡ ❤❛✈❡ ❡♥❝♦✉♥t❡r❡❞ s♦ ❢❛r ❛r❡ r❡❛❧ ♥♦♥✲❝♦♠♣❧❡①✱ ❛♥❞ s♦ ❛r❡ ❛❧❧ q✉❛♥t✐t✐❡s ♦♥❡ ❝❛♥ ❡♥✲ ❝♦✉♥t❡r ✐♥ ❞❛②✲t♦✲❞❛② ❧✐❢❡ ♦r s❝✐❡♥❝❡✿ t✐♠❡✱ ❧♦❝❛t✐♦♥✱ ❧❡♥❣t❤✱ ❛r❡❛✱ ✈♦❧✉♠❡✱ ♠❛ss✱ t❡♠♣❡r❛t✉r❡✱ ♠♦♥❡②✱ ❡t❝✳

✸✳✹✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s ❚❤❡ ❣❡♥❡r❛❧ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥ ✇✐t❤ r❡❛❧ ❝♦❡✣❝✐❡♥ts✱ ax2 + bx + c = 0, a 6= 0 ,

❝❛♥ ❜❡ s✐♠♣❧✐✜❡❞✳ ▲❡t✬s ❞✐✈✐❞❡ ❜② a ❛♥❞ st✉❞② t❤❡ r❡s✉❧t✐♥❣ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✿ f (x) = x2 + px + q ,

✇❤❡r❡ p = b/a ❛♥❞ q = c/a✳ ❚❤❡ ◗✉❛❞r❛t✐❝ ❋♦r♠✉❧❛ t❤❡♥ ♣r♦✈✐❞❡s t❤❡ x✲✐♥t❡r❝❡♣ts ♦❢ t❤✐s ❢✉♥❝t✐♦♥✿ p x=− ± 2

p p2 − 4q . 2

❖❢ ❝♦✉rs❡✱ t❤❡ x✲✐♥t❡r❝❡♣ts ❛r❡ t❤❡ r❡❛❧ s♦❧✉t✐♦♥s ♦❢ t❤✐s ❡q✉❛t✐♦♥ ❛♥❞ t❤❛t ✐s ✇❤② t❤❡ r❡s✉❧t ♦♥❧② ♠❛❦❡s s❡♥s❡ ✇❤❡♥ t❤❡ ❞✐s❝r✐♠✐♥❛♥t ♦❢ t❤❡ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✱ D = p2 − 4q ,

✐s ♥♦♥✲♥❡❣❛t✐✈❡✳ ◆♦✇✱ ✐♥❝r❡❛s✐♥❣ t❤❡ ✈❛❧✉❡ ♦❢ q ♠❛❦❡s t❤❡ ❣r❛♣❤ ♦❢ y = f (x) s❤✐❢t ✉♣✇❛r❞ ❛♥❞✱ ❡✈❡♥t✉❛❧❧②✱ ♣❛ss t❤❡ x✲❛①✐s ❡♥t✐r❡❧②✳ ❲❡ ❝❛♥ ♦❜s❡r✈❡ ❤♦✇ ✐ts t✇♦ x✲✐♥t❡r❝❡♣ts st❛rt t♦ ❣❡t ❝❧♦s❡r t♦ ❡❛❝❤ ♦t❤❡r✱ t❤❡♥ ♠❡r❣❡✱ ❛♥❞ ✜♥❛❧❧② ❞✐s❛♣♣❡❛r✿

✸✳✹✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s

✷✶✼

❚❤✐s ♣r♦❝❡ss ✐s ❡①♣❧❛✐♥❡❞ ❜② ✇❤❛t ✐s ❤❛♣♣❡♥✐♥❣✱ ✇✐t❤ t❤❡ ❣r♦✇t❤ ♦❢ q ✱ t♦ t❤❡ r♦♦ts ❣✐✈❡♥ ❜② t❤❡ ◗✉❛❞r❛t✐❝ ❋♦r♠✉❧❛ ✿ √ p D x1,2 = − ± . 2 2 √ D ❞❡❝r❡❛s❡s❀ t❤❡♥ • ❙t❛rt✐♥❣ ✇✐t❤ ❛ ♣♦s✐t✐✈❡ ✈❛❧✉❡✱ D ❞❡❝r❡❛s❡s ❛♥❞ 2 √ D = 0❀ t❤❡♥ • D = 0 ❛♥❞ 2 √ √ D −D • D ❜❡❝♦♠❡s ♥❡❣❛t✐✈❡ ❛♥❞ ❜❡❝♦♠❡s ✐♠❛❣✐♥❛r② ✭❜✉t ✐s r❡❛❧✮✳ ❚❤❡ r♦♦ts ❛r❡✱ r❡s♣❡❝t✐✈❡❧②✿ 2 2

❞✐s❝r✐♠✐♥❛♥t D>0 D=0 D 0✮ ❛r❡ ✉♥r❡❧❛t❡❞ ✇❤✐❧❡ t❤❡ ❝♦♠♣❧❡① ♦♥❡s ✭D < 0✮ ❛r❡ ❧✐♥❦❡❞ s♦ ♠✉❝❤ t❤❛t ❦♥♦✇✐♥❣ ♦♥❡ t❡❧❧s ✉s ✇❤❛t t❤❡ ♦♥❡ ✐s✿ ❥✉st ✢✐♣ t❤❡ s✐❣♥❀ t❤❡② ❛r❡ ❝♦♥❥✉❣❛t❡ ♦❢ ❡❛❝❤ ♦t❤❡r✿

❚❤❡② ❛❧✇❛②s ❝♦♠❡ ✐♥ ♣❛✐rs✦ ❆s ❛ s✉♠♠❛r②✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❛ss✐✜❝❛t✐♦♥ t❤❡ r♦♦ts ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s ✐♥ t❡r♠s ♦❢ t❤❡ s✐❣♥ ♦❢ t❤❡ ❞✐s❝r✐♠✐♥❛♥t✳

❚❤❡♦r❡♠ ✸✳✹✳✶✿ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❘♦♦ts ■

❚❤❡ t✇♦ r♦♦ts ♦❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ✇✐t❤ r❡❛❧ ❝♦❡✣❝✐❡♥ts ❛r❡✿ • ❞✐st✐♥❝t r❡❛❧ ✇❤❡♥ ✐ts ❞✐s❝r✐♠✐♥❛♥t D ✐s ♣♦s✐t✐✈❡❀ • ❡q✉❛❧ r❡❛❧ ✇❤❡♥ ✐ts ❞✐s❝r✐♠✐♥❛♥t D ✐s ③❡r♦❀ • ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ♦❢ ❡❛❝❤ ♦t❤❡r ✇❤❡♥ ✐ts ❞✐s❝r✐♠✐♥❛♥t D ✐s ♥❡❣❛t✐✈❡✳

✸✳✹✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s

✷✶✽

❚♦ ✉♥❞❡rst❛♥❞ ❖❉❊s✱ ✇❡ ✇✐❧❧ ♥❡❡❞ ❛ ♠♦r❡ ♣r❡❝✐s❡ ✇❛② t♦ ❝❧❛ss✐❢② t❤❡ ♣♦❧②♥♦♠✐❛❧s✿ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ s✐❣♥s ♦❢ t❤❡ r❡❛❧ ♣❛rts ♦❢ t❤❡✐r r♦♦ts✳ ❚❤❡ s✐❣♥s ✇✐❧❧ ❞❡t❡r♠✐♥❡ ✐♥❝r❡❛s✐♥❣ ❛♥❞ ❞❡❝r❡❛s✐♥❣ ❜❡❤❛✈✐♦r ♦❢ ❝❡rt❛✐♥ s♦❧✉t✐♦♥s✳ ❖♥❝❡ ❛❣❛✐♥✱ t❤❡s❡ ❛r❡ t❤❡ ♣♦ss✐❜✐❧✐t✐❡s✿

❚❤❡♦r❡♠ ✸✳✹✳✷✿ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❘♦♦ts ■■

❙✉♣♣♦s❡ x1 ❛♥❞ x2 ❛r❡ t❤❡ t✇♦ r♦♦ts ♦❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ f (x) = x2 +px+q ✇✐t❤ r❡❛❧ ❝♦❡✣❝✐❡♥ts✳ ❚❤❡♥ t❤❡ s✐❣♥s ♦❢ t❤❡ r❡❛❧ ♣❛rts Re(x1) ❛♥❞ Re(x2) ♦❢ x1 ❛♥❞ x2 ❛r❡✿ • s❛♠❡ ✇❤❡♥ p2 > 4q ❛♥❞ q ≥ 0❀ • ♦♣♣♦s✐t❡ ✇❤❡♥ p2 > 4q ❛♥❞ q < 0❀ • s❛♠❡ ❛♥❞ ♦♣♣♦s✐t❡ ♦❢ t❤❛t ♦❢ p ✇❤❡♥ p2 ≤ 4q ✳ Pr♦♦❢✳ ❚❤❡ ❝♦♥❞✐t✐♦♥ p2 ≤ 4q ✐s ❡q✉✐✈❛❧❡♥t t♦ D ≤ 0✳ ❲❡ ❝❛♥ s❡❡ ✐♥ t❤❡ t❛❜❧❡ ❛❜♦✈❡ t❤❛t✱ ✐♥ t❤❛t ❝❛s❡✱ ✇❡ p ❤❛✈❡ Re(x1 ) = Re(x2 ) = − ✳ ❲❡ ❛r❡ ❧❡❢t ✇✐t❤ t❤❡ ❝❛s❡ D > 0 ❛♥❞ r❡❛❧ r♦♦ts✳ ❚❤❡ ❝❛s❡ ♦❢ ❡q✉❛❧ s✐❣♥s 2 ✐s s❡♣❛r❛t❡❞ ❢r♦♠ t❤❡ ❝❛s❡ ♦❢ ♦♣♣♦s✐t❡ s✐❣♥s ♦❢ x1 ❛♥❞ x2 ❜② t❤❡ ❝❛s❡ ✇❤❡♥ ❜♦t❤ ❛r❡ ❡q✉❛❧ t♦ ③❡r♦✿ x1 = x2 = 0✳ ❲❡ s♦❧✈❡✿ √ p p D − − = 0 =⇒ p = − p2 − 4q =⇒ p2 = p2 − 4q =⇒ q = 0 . 2 2

❊①❡r❝✐s❡ ✸✳✹✳✸

❋✐♥✐s❤ t❤❡ ♣r♦♦❢✳ ▲❡t✬s ✈✐s✉❛❧✐③❡ ♦✉r ❝♦♥❝❧✉s✐♦♥✳ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ s❤♦✇ t❤❡ ♠❛✐♥ s❝❡♥❛r✐♦s ♦❢ ✇❤❛t ❦✐♥❞s ♦❢ r♦♦ts t❤❡ ♣♦❧②♥♦♠✐❛❧ ♠✐❣❤t ❤❛✈❡ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ✈❛❧✉❡s ♦❢ ✐ts t✇♦ ❝♦❡✣❝✐❡♥ts✱ p ❛♥❞ q ✳ ❋✐rst✱ ❤♦✇ ❞♦ ✇❡ ✈✐s✉❛❧✐③❡ ♣❛✐rs ♦❢ ♥✉♠❜❡rs ❄ ❆s ♣♦✐♥ts ♦♥ ❛ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡ ♦❢ ❝♦✉rs❡✳✳✳ ❜✉t ♦♥❧② ✇❤❡♥ t❤❡② ❛r❡ r❡❛❧✳ ❙✉♣♣♦s❡ ❢♦r ♥♦✇ t❤❛t t❤❡② ❛r❡✳ ❲❡ st❛rt ✇✐t❤ ❛ ♣❧❛♥❡✱ t❤❡ x1 x2 ✲♣❧❛♥❡ t♦ ❜❡ ❡①❛❝t✱ ❛s ❛ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ♣❛✐rs ♦❢ r❡❛❧ r♦♦ts ✭❧❡❢t✮✳ ❚❤❡♥ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ t❤✐s ♣❧❛♥❡ r❡♣r❡s❡♥ts t❤❡ ❝❛s❡ ♦❢ ❡q✉❛❧ ✭❛♥❞ st✐❧❧ r❡❛❧✮ r♦♦ts✱ x1 = x2 ✱ ✐✳❡✳✱ D = 0✳ ❙✐♥❝❡ t❤❡ ♦r❞❡r ♦❢ t❤❡ r♦♦ts ❞♦❡s♥✬t ♠❛tt❡r ✕ (x1 , x2 ) ✐s ❛s ❣♦♦❞ ❛s (x2 , x1 ) ✕ ✇❡ ♥❡❡❞ ♦♥❧② ❤❛❧❢ ♦❢ t❤❡ ♣❧❛♥❡✳ ❲❡ ❢♦❧❞ t❤❡ ♣❧❛♥❡ ❛❧♦♥❣ t❤❡ ❞✐❛❣♦♥❛❧ ✭♠✐❞❞❧❡✮✳

✸✳✺✳

❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ R2

✷✶✾

❚❤❡ ❞✐❛❣♦♥❛❧ ✕ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥ D = 0 ✕ ❡①♣♦s❡❞ t❤✐s ✇❛② ❝❛♥ ♥♦✇ s❡r✈❡ ✐ts ♣✉r♣♦s❡ ♦❢ s❡♣❛✲ r❛t✐♥❣ t❤❡ ❝❛s❡ ♦❢ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① r♦♦ts✳ ◆♦✇✱ ❧❡t✬s ❣♦ t♦ t❤❡ pq ✲♣❧❛♥❡✳ ❍❡r❡✱ t❤❡ ♣❛r❛❜♦❧❛ p2 = 4q ❛❧s♦ r❡♣r❡s❡♥ts t❤❡ ❡q✉❛t✐♦♥ D = 0✳ ▲❡t✬s ❜r✐♥❣ t❤❡♠ t♦❣❡t❤❡r✦ ❲❡ t❛❦❡ ♦✉r ❤❛❧❢✲♣❧❛♥❡ ❛♥❞ ❜❡♥❞ ✐ts ❞✐❛❣♦♥❛❧ ❡❞❣❡ ✐♥t♦ t❤❡ ♣❛r❛❜♦❧❛ p2 = 4q ✭r✐❣❤t✮✳ ❈❧❛ss✐❢②✐♥❣ ♣♦❧②♥♦♠✐❛❧s t❤✐s ✇❛② ❛❧❧♦✇s ♦♥❡ t♦ ❝❧❛ss✐❢② ♠❛tr✐❝❡s ❛♥❞ ✉♥❞❡rst❛♥❞ ✇❤❛t ❡❛❝❤ ♦❢ t❤❡♠ ❞♦❡s ❛s ❛ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡✱ ✇❤✐❝❤ ✐♥ t✉r♥ ✇✐❧❧ ❤❡❧♣ ✉s ✉♥❞❡rst❛♥❞ s②st❡♠s ♦❢ ❖❉❊s✳

✸✳✺✳ ❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡

C

✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡

R2

♥✉♠❜❡rs✱ t❤❡② ♠✉st ❜❡ s✉❜❥❡❝t t♦ s♦♠❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✳ ❲❡ ✇✐❧❧ ✐♥✐t✐❛❧❧② ❧♦♦❦ ❛t t❤❡♠ t❤r♦✉❣❤ t❤❡ ❧❡♥s ♦❢ ✈❡❝t♦r ❛❧❣❡❜r❛ ♦❢ t❤❡ ♣❧❛♥❡ R2 ✳ ❆ ❝♦♠♣❧❡① ♥✉♠❜❡r z ❤❛s t❤❡ st❛♥❞❛r❞ r❡♣r❡s❡♥t❛t✐♦♥ ✿ ■❢ ✇❡ ❝❛❧❧ ❝♦♠♣❧❡① ♥✉♠❜❡r

z = a + bi , ✇❤❡r❡ a ❛♥❞ b ❛r❡ t✇♦ r❡❛❧ ♥✉♠❜❡rs✳ ❚❤❡s❡ t✇♦ ❝❛♥ ❜❡ s❡❡♥ ✐♥ t❤❡ ♥✉♠❜❡rs✿

❣❡♦♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥

♦❢ ❝♦♠♣❧❡①

❚❤❡r❡❢♦r❡✱ a ❛♥❞ b ❛r❡ t❤♦✉❣❤t ♦❢ ❛s t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ z ❛s ❛ ♣♦✐♥t ♦♥ t❤❡ ♣❧❛♥❡✳ ❇✉t ❛♥② ❝♦♠♣❧❡① ♥✉♠❜❡r ✐s ♥♦t ♦♥❧② ❛ ♣♦✐♥t ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ ❜✉t ❛❧s♦ ❛ ✈❡❝t♦r✳ ❲❡ ❤❛✈❡ ❛ ❝♦rr❡s♣♦♥❞❡♥❝❡✿

C ←→ R2 , ❣✐✈❡♥ ❜②

a + bi ←→ < a, b >

✸✳✺✳

❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ R2

✷✷✵

❚❤❡r❡ ✐s ♠♦r❡ t♦ t❤✐s t❤❛♥ ❥✉st ❛ ♠❛t❝❤❀ t❤❡ ❛❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs ✐♥

R2

❛♣♣❧✐❡s✦

❲❛r♥✐♥❣✦

■♥ s♣✐t❡ ♦❢ t❤✐s ❢✉♥❞❛♠❡♥t❛❧ ❝♦rr❡s♣♦♥❞❡♥❝❡✱ ✇❡ ✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ t❤✐♥❦ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛s ♥✉♠❜❡rs ✭❛♥❞ ✉s❡ t❤❡ ❧♦✇❡r ❝❛s❡ ❧❡tt❡rs✮✳ ▲❡t✬s s❡❡ ❤♦✇ t❤✐s ❛❧❣❡❜r❛ ♦❢ ♥✉♠❜❡rs ✇♦r❦s ✐♥ ♣❛r❛❧❧❡❧ ✇✐t❤ t❤❡ ❛❧❣❡❜r❛ ♦❢ ❋✐rst✱ t❤❡ ❛❞❞✐t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✐s ❞♦♥❡

2✲✈❡❝t♦rs✳

❝♦♠♣♦♥❡♥t✇✐s❡ ✿

(a + bi) + (c + di) = (a + c) + (b + d)i < a, b > + < c, d > = < a + c , b + d > ■t ❝♦rr❡s♣♦♥❞s t♦ ❛❞❞✐t✐♦♥ ♦❢ ✈❡❝t♦rs✿

❙❡❝♦♥❞✱ ✇❡ ❝❛♥ ❡❛s✐❧② ♠✉❧t✐♣❧② ❝♦♠♣❧❡① ♥✉♠❜❡rs ❜② r❡❛❧ ♦♥❡s✿

(a + bi) c = (ac) + (bc)i < a, b > c = < ac , bc > ■t ❝♦rr❡s♣♦♥❞s t♦ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ✈❡❝t♦rs✳

❲❛r♥✐♥❣✦

❱❡❝t♦r ❛❧❣❡❜r❛ ♦❢ R2 ✐s ❝♦♠♣❧❡① ❛❧❣❡❜r❛✱ ❜✉t ♥♦t ✈✐❝❡ ✈❡rs❛✳ ❈♦♠♣❧❡① ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ✇❤❛t ♠❛❦❡s ✐t ❞✐✛❡r❡♥t✳ ❊①❛♠♣❧❡ ✸✳✺✳✶✿ ❝✐r❝❧❡ ❲❡ ❝❛♥ ❡❛s✐❧② r❡♣r❡s❡♥t ❝✐r❝❧❡s ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡✿

z = r cos θ + r sin θ · i .

❖✉r st✉❞② ♦❢ ❝❛❧❝✉❧✉s ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs st❛rts ✇✐t❤ t❤❡ st✉❞② ♦❢ t❤❡ t♦♣♦❧♦❣② ✐s t❤❡ s❛♠❡ ❛s t❤❛t ♦❢

t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡

R

2



t♦♣♦❧♦❣②

♦❢ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡✳ ❚❤✐s

✸✳✺✳

❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ R2

❏✉st ❛s ❜❡❢♦r❡✱ ❡✈❡r② ❢✉♥❝t✐♦♥

z = f (t)

✷✷✶

✇✐t❤ ❛♥ ❛♣♣r♦♣r✐❛t❡ ❞♦♠❛✐♥ ❝r❡❛t❡s ❛ s❡q✉❡♥❝❡✿

zk = f (k) . ❆ ❢✉♥❝t✐♦♥ ✇✐t❤ ❝♦♠♣❧❡① ✈❛❧✉❡s ❞❡✜♥❡❞ ♦♥ ❛ r❛② ✐♥ t❤❡ s❡t ♦❢ ✐♥t❡❣❡rs✱

s❡q✉❡♥❝❡✱ ♦r s✐♠♣❧② s❡q✉❡♥❝❡✳

{p, p + 1, ...}✱

✐s ❝❛❧❧❡❞ ❛♥

✐♥✜♥✐t❡

❊①❛♠♣❧❡ ✸✳✺✳✷✿ s♣✐r❛❧

❆ ❣♦♦❞ ❡①❛♠♣❧❡ ✐s t❤❛t ♦❢ t❤❡ s❡q✉❡♥❝❡ ♠❛❞❡ ♦❢ t❤❡ r❡❝✐♣r♦❝❛❧s✿

zk = ■t

cos k sin k + i. k k

t❡♥❞s t♦ 0 ✇❤✐❧❡ s♣✐r❛❧✐♥❣ ❛r♦✉♥❞ ✐t✳

❚❤❡ st❛rt✐♥❣ ♣♦✐♥t ♦❢ ❝❛❧❝✉❧✉s ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛

♥✉♠❜❡rs

s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣❧❡①

✐s t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ✐ts r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts ♦r✱ ✇❤✐❝❤ ✐s ❡q✉✐✈❛❧❡♥t✱ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♣♦✐♥ts

✭♦r ✈❡❝t♦rs✮ ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ s❡❡♥ ❛s ❛♥② ♣❧❛♥❡✿ ❚❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡

k t❤ ♣♦✐♥t t♦ t❤❡ ❧✐♠✐t ✐s ❣❡tt✐♥❣

s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r✳

❲❡ ✉s❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r ✈❡❝t♦rs ♦♥ t❤❡ ♣❧❛♥❡ ❜② s✐♠♣❧② r❡♣❧❛❝✐♥❣ ✈❡❝t♦rs ✇✐t❤ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛♥❞ ✏♠❛❣♥✐t✉❞❡✑ ✇✐t❤ ✏♠♦❞✉❧✉s✑✳

✸✳✺✳

❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ R2

✷✷✷

❉❡✜♥✐t✐♦♥ ✸✳✺✳✸✿ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ❙✉♣♣♦s❡

C✳

{zk : k = 1, 2, 3, ...}

✐s ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✱ ✐✳❡✳✱ ♣♦✐♥ts ✐♥

❝♦♥✈❡r❣❡s

❲❡ s❛② t❤❛t t❤❡ s❡q✉❡♥❝❡

♣♦✐♥t ✐♥

C✱

❝❛❧❧❡❞ t❤❡

❧✐♠✐t

t♦ ❛♥♦t❤❡r ❝♦♠♣❧❡① ♥✉♠❜❡r

z✱

✐✳❡✳✱ ❛

♦❢ t❤❡ s❡q✉❡♥❝❡✱ ✐❢✿

||zk − z|| → 0

❛s

k → ∞,

❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

zk → z

❛s

k → ∞,

♦r

z = lim zk . k→∞

❝♦♥✈❡r❣❡♥t ❞✐✈❡r❣❡s✳

■❢ ❛ s❡q✉❡♥❝❡ ❤❛s ❛ ❧✐♠✐t✱ ✇❡ ❝❛❧❧ t❤❡ s❡q✉❡♥❝❡

✈❡r❣❡s ❀ ♦t❤❡r✇✐s❡ ✐t ✐s ❞✐✈❡r❣❡♥t

❛♥❞ ✇❡ s❛② ✐t

❛♥❞ s❛② t❤❛t ✐t

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♣♦✐♥ts st❛rt t♦ ❛❝❝✉♠✉❧❛t❡ ✐♥ s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r ❝✐r❝❧❡s ❛r♦✉♥❞ ❛ tr❡♥❞ ✐♥ ❛ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ✐s t♦ ❡♥❝❧♦s❡ t❤❡ t❛✐❧ ♦❢ t❤❡ s❡q✉❡♥❝❡ ✐♥ ❛

❞✐s❦ ✿

z✳

❝♦♥✲

❆ ✇❛② t♦ ✈✐s✉❛❧✐③❡

❚❤❡♦r❡♠ ✸✳✺✳✹✿ ❯♥✐q✉❡♥❡ss ♦❢ ▲✐♠✐t ❆ s❡q✉❡♥❝❡ ❝❛♥ ❤❛✈❡ ♦♥❧② ♦♥❡ ❧✐♠✐t ✭✜♥✐t❡ ♦r ✐♥✜♥✐t❡✮❀ ✐✳❡✳✱ ✐❢ a ❛♥❞ b ❛r❡ ❧✐♠✐ts ♦❢ t❤❡ s❛♠❡ s❡q✉❡♥❝❡✱ t❤❡♥ a = b✳

❉❡✜♥✐t✐♦♥ ✸✳✺✳✺✿ s❡q✉❡♥❝❡ t❡♥❞s t♦ ✐♥✜♥✐t② ❲❡ s❛② t❤❛t ❛ s❡q✉❡♥❝❡ ❡❛❝❤ r❡❛❧ ♥✉♠❜❡r ♥✉♠❜❡r

k > N✱

zk t❡♥❞s t♦ ✐♥✜♥✐t②

✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥ ❤♦❧❞s✿ ❋♦r

R✱ t❤❡r❡ ❡①✐sts s✉❝❤ ❛ ♥❛t✉r❛❧ ♥✉♠❜❡r N

t❤❛t✱ ❢♦r ❡✈❡r② ♥❛t✉r❛❧

✇❡ ❤❛✈❡

||zk || > R . ❲❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥✿

zk → ∞

❛s

k → ∞.

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛♥♦t❤❡r ❛♥❛❧♦❣ ♦❢ ❛ ❢❛♠✐❧✐❛r t❤❡♦r❡♠ ❛❜♦✉t t❤❡ t♦♣♦❧♦❣② ♦❢ t❤❡ ♣❧❛♥❡✳

❚❤❡♦r❡♠ ✸✳✺✳✻✿ ❈♦♠♣♦♥❡♥t✇✐s❡ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❙❡q✉❡♥❝❡s ❆ s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs zk ✐♥ C ❝♦♥✈❡r❣❡s t♦ ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r z ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❜♦t❤ t❤❡ r❡❛❧ ❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ zk ❝♦♥✈❡r❣❡ t♦ t❤❡ r❡❛❧ ❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ z r❡s♣❡❝t✐✈❡❧②❀ ✐✳❡✳✱ zk → z ⇐⇒ Re(zk ) → Re(z) ❛♥❞ Im(zk ) → Im(z) .

C

✸✳✻✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿

✐s♥✬t ❥✉st

R2

✷✷✸

❚❤❡ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✇✐❧❧ ❛❧s♦ ❧♦♦❦ ❢❛♠✐❧✐❛r✿

❚❤❡♦r❡♠ ✸✳✺✳✼✿ ❙✉♠ ❘✉❧❡ ❢♦r ❈♦♠♣❧❡① ❙❡q✉❡♥❝❡s ■❢ s❡q✉❡♥❝❡s

zk , uk

❝♦♥✈❡r❣❡✱ t❤❡♥ s♦ ❞♦❡s

z k + uk ✱

❛♥❞ ✇❡ ❤❛✈❡✿

lim (zk + uk ) = lim zk + lim uk .

k→∞

k→∞

k→∞

❚❤❡♦r❡♠ ✸✳✺✳✽✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❈♦♠♣❧❡① ❙❡q✉❡♥❝❡s ■❢ s❡q✉❡♥❝❡

zk

❝♦♥✈❡r❣❡s✱ t❤❡♥ s♦ ❞♦❡s

czk

❢♦r ❛♥② ❝♦♠♣❧❡① ♥✉♠❜❡r

c✱

❛♥❞ ✇❡

❤❛✈❡✿

lim c zk = c · lim zk .

k→∞

k→∞

❲♦✉❧❞♥✬t ❝❛❧❝✉❧✉s ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❜❡ ❥✉st ❛ ❝♦♣② ♦❢ ❝❛❧❝✉❧✉s ♦♥ t❤❡ ♣❧❛♥❡❄ ◆♦✱ ♥♦t ✇✐t❤ t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t✳

✸✳✻✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿

❙♦✱ t❤❡ ✈❡❝t♦r ❛❧❣❡❜r❛ ♦❢ ❏✉st ❧✐❦❡ ✐♥ ♣❧❛♥❡

C✳

R2 ✱

R2

✐s ✐♥❝❧✉❞❡❞ ✐♥ t❤❡ ❝♦♠♣❧❡① ❛❧❣❡❜r❛ ♦❢

♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛ r❡❛❧ ♥✉♠❜❡r

r

▲❡t✬s st❛rt ✇✐t❤ ❞❡❣r❡❡s✿

1

1

❜❡❝♦♠❡s

i✱

✇❤✐❧❡

c

R2

❚❤❡r❡ ✐s ♠♦r❡ t♦ t❤❡ ❧❛tt❡r✳

✇✐❧❧ ❛❧s♦ r♦t❛t❡ ❡❛❝❤ ✈❡❝t♦r✳

i

i s❡✈❡r❛❧ t✐♠❡s✳ i ❜❡❝♦♠❡s −1✱ ❡t❝✳✿

❛♥❞ ♠✉❧t✐♣❧② ✐t ❜②

✐s♥✬t ❥✉st

✇✐❧❧ str❡t❝❤✴s❤r✐♥❦ ❛❧❧ ✈❡❝t♦rs ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ❝♦♠♣❧❡①

❍♦✇❡✈❡r✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r

❊①❛♠♣❧❡ ✸✳✻✳✶✿ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜②

C✳

C

▼✉❧t✐♣❧✐❝❛t✐♦♥ ❜②

i

r♦t❛t❡s t❤❡ ♥✉♠❜❡r ❜②

90

✸✳✻✳

▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿

C

✐s♥✬t ❥✉st

1·i=i i · i = i2 = −1 −1 · i = −i −i · i = −i2 = 1

✷✷✹

R2

r♦t❛t✐♦♥ ❢r♦♠ ✵ ❞❡❣r❡❡s t♦ ✾✵ r♦t❛t✐♦♥ ❢r♦♠ ✾✵ ❞❡❣r❡❡s t♦ ✶✽✵ r♦t❛t✐♦♥ ❢r♦♠ ✶✽✵ ❞❡❣r❡❡s t♦ ✷✼✵ r♦t❛t✐♦♥ ❢r♦♠ ✷✼✵ ❞❡❣r❡❡s t♦ ✸✻✵ ❛♥❞ s♦ ♦♥✳

❊①❛♠♣❧❡ ✸✳✻✳✷✿ ❝♦♠♣❧❡① ♠✉❧t✐♣❧✐❝❛t✐♦♥

❆ ♠♦r❡ ❝♦♠♣❧❡① ❡①❛♠♣❧❡✿

u = 1 + 2i v =2+i uv = 2 + 4i + i + 2i2 = (2 − 2) + (4 + 1)i = 0 + 5i

❚❤❡ r♦t❛t✐♦♥ ♦❢ v ✐s ✈✐s✐❜❧❡✿

■♥ ❝♦♥tr❛st✱ ✇❡ ❝❛♥ s❡❡ t❤❡ r❡s✉❧t ♦❢ ♠✉❧t✐♣❧②✐♥❣ v ❜② w = 2✿ ♥♦ r♦t❛t✐♦♥✳ ❙♦✱ t❤❡ ✐♠❛❣✐♥❛r② ♣❛rt ♦❢ c ✐s r❡s♣♦♥s✐❜❧❡ ❢♦r r♦t❛t✐♦♥✳ ❍♦✇ ❞♦❡s ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛✛❡❝t t♦♣♦❧♦❣②❄

❚❤❡♦r❡♠ ✸✳✻✳✸✿ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ❈♦♠♣❧❡① ❙❡q✉❡♥❝❡s ■❢ s❡q✉❡♥❝❡s

zk , uk

❝♦♥✈❡r❣❡✱ t❤❡♥ s♦ ❞♦❡s

z k · uk ✱

❛♥❞

lim (zk · uk ) = lim zk · lim uk .

k→∞

k→∞

k→∞

Pr♦♦❢✳

❙✉♣♣♦s❡

zk = ak + bk i → a + bi ❛♥❞ uk = pk + qk i = p + qi .

✸✳✻✳

▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿

C

❚❤❡♥✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❈♦♠♣♦♥❡♥t✇✐s❡

✐s♥✬t ❥✉st

✷✷✺

R2

❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠

❛❜♦✈❡✱ ✇❡ ❤❛✈❡✿

ak → a, bk → b ❛♥❞ pk → p, qk → q .

❚❤❡♥✱ ❜② t❤❡ Pr♦❞✉❝t

✱ ✇❡ ❤❛✈❡✿

❘✉❧❡ ❢♦r ♥✉♠❡r✐❝❛❧ s❡q✉❡♥❝❡s

ak pk → ap, ak qk → aq, bk pk → bp, bk qk → bq .

❚❤❡♥✱ ❛s ✇❡ ❦♥♦✇✱ zk · uk = (ak pk − bk qk ) + (ak qk + bk pk )i → (ap − bq) + (aq + bq)i = (a + bi)(p + qi) ,

❜② t❤❡ ❙✉♠



❘✉❧❡ ❢♦r ♥✉♠❡r✐❝❛❧ s❡q✉❡♥❝❡s

❚❤❡♦r❡♠ ✸✳✻✳✹✿ ◗✉♦t✐❡♥t ❘✉❧❡ ❢♦r ❈♦♠♣❧❡① ❙❡q✉❡♥❝❡s ■❢ s❡q✉❡♥❝❡s

z k , uk

❝♦♥✈❡r❣❡ ✭✇✐t❤

lim

k→∞

uk 6= 0✮✱

t❤❡♥ s♦ ❞♦❡s

zk /uk ✱

❛♥❞

limk→∞ zk zk = , uk limk→∞ uk

♣r♦✈✐❞❡❞

lim uk 6= 0 .

k→∞

❏✉st ❧✐❦❡ r❡❛❧ ♥✉♠❜❡rs✦

❊①❡r❝✐s❡ ✸✳✻✳✺ Pr♦✈❡ t❤❡ ❧❛st t❤❡♦r❡♠✳ ■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ st❛♥❞❛r❞✱ ❈❛rt❡s✐❛♥✱ r❡♣r❡s❡♥t❛t✐♦♥✱ ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r x = a + bi ❝❛♥ ❜❡ ❞❡✜♥❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s✳

❲❡ ❥✉st ❛♣♣❡♥❞ ♦✉r ❝♦rr❡s♣♦♥❞❡♥❝❡ ✇✐t❤ ❛ ♥❡✇ ♦♥❡✿ a + bi ←→ (a, b) ←→ (θ, r)

❚❤❡ t✇♦ q✉❛♥t✐t✐❡s θ ❛♥❞ r ❜❡❝♦♠❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❉❡✜♥✐t✐♦♥ ✸✳✻✳✻✿ ♠♦❞✉❧✉s ❛♥❞ ❛r❣✉♠❡♥t ❙✉♣♣♦s❡ z ✐s ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r✳ ✶✳ ❚❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❧♦❝❛t✐♦♥ ♦❢ z ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ t♦ t❤❡ ♦r✐❣✐♥ O ✐s ❝❛❧❧❡❞ t❤❡ ♠♦❞✉❧✉s ♦❢ z ❞❡♥♦t❡❞ ❜②✿ ||z||

✷✳ ❚❤❡ ❛♥❣❧❡ ♦❢ t❤❡ ❧✐♥❡ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t z ❢r♦♠ t❤❡ ♦r✐❣✐♥ O ✇✐t❤ t❤❡ x✲❛①✐s

✸✳✻✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿

C

✐s♥✬t ❥✉st

R2

✐s ❝❛❧❧❡❞ t❤❡ ❛r❣✉♠❡♥t ♦❢

✷✷✻

z

❞❡♥♦t❡❞ ❜②✿

Arg(z)

❆ s✐♠♣❧❡ ❡①❛♠✐♥❛t✐♦♥ t❡❧❧s ✉s ❤♦✇ t♦ tr❛♥s✐t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❝♦♦r❞✐♥❛t❡ s②st❡♠s✿

❚❤❡♦r❡♠ ✸✳✻✳✼✿ ❈♦♥✈❡rs✐♦♥ ♦❢ ❈♦♠♣❧❡① ◆✉♠❜❡rs ❙✉♣♣♦s❡

x = a + bi

✐s ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡✿

✶✳ ❚❤❡ ♠♦❞✉❧✉s ♦❢

z

✐s ❢♦✉♥❞ ❜②✿

||z|| = ✷✳ ❚❤❡ ❛r❣✉♠❡♥t ♦❢

z



a2 + b2

✐s ❢♦✉♥❞ ❜②✿

Arg(z) = arctan

❆♥② t✇♦ r❡❛❧ ♥✉♠❜❡rs t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r✿

r≥0

❛♥❞

0 ≤ θ < 2π

b a

❝❛♥ s❡r✈❡ ❛s t❤♦s❡✳ ■t ✐s ❝❛❧❧❡❞ t❤❡ ❣❡♦♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢

  z = r cos θ + i sin θ ❚❤❡ ❛❧❣❡❜r❛ t❛❦❡s ❛ ♥❡✇ ❢♦r♠ t♦♦✳ ❲❡ ❞♦♥✬t ♥❡❡❞ t❤❡ ♥❡✇ r❡♣r❡s❡♥t❛t✐♦♥ t♦ ❝♦♠♣✉t❡ ❛❞❞✐t✐♦♥ ❛♥❞ ♠✉❧t✐✲ ♣❧✐❝❛t✐♦♥ ❜② r❡❛❧ ♥✉♠❜❡rs✱ ❜✉t ✇❡ ♥❡❡❞ ✐t ❢♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ❲❤❛t ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿

❈♦♥s✐❞❡r✿

  z1 = r1 cos ϕ1 + i sin ϕ1

❛♥❞

  z2 = r2 cos ϕ2 + i sin ϕ2 ?

    z1 z2 = r1 cos  ϕ1 + i sin ϕ1 ) · r2 cos ϕ2 + i sin ϕ2 = r1 r2 cos ϕ1 + i sin ϕ1 · cos ϕ2 + i sin ϕ2  = r1 r2 cos ϕ1 cos ϕ2 + i sin ϕ1 cos ϕ2 + cos ϕ1 sin ϕ2 + i2 sin ϕ1 sin ϕ2 .

❲❡ ✉t✐❧✐③❡ t❤❡ ❢♦❧❧♦✇✐♥❣ tr✐❣♦♥♦♠❡tr✐❝ ✐❞❡♥t✐t✐❡s ✭❱♦❧✉♠❡ ✶✮✿

cos a cos b − sin a sin b = cos(a + b)

❛♥❞

cos a sin b + sin a cos b = sin(a + b) .

❚❤❡♥✱

  z1 z2 = r1 r2 cos(ϕ1 + ϕ2 ) + i sin(ϕ1 + ϕ2 ) .

✸✳✻✳

▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿

C

✐s♥✬t ❥✉st

R2

✷✷✼

❊①❛♠♣❧❡ ✸✳✻✳✽✿ ❣❡♦♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥

❲❡ ❝❛♥ s❡❡ t❤❡ ❛❜♦✈❡ ❝♦♠♣✉t❛t✐♦♥ ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡✿

❲❡ ❤❛✈❡ ♣r♦✈❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✸✳✻✳✾✿ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❈♦♠♣❧❡① ◆✉♠❜❡rs ❲❤❡♥ t✇♦ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛r❡ ♠✉❧t✐♣❧✐❡❞✱ t❤❡✐r ♠♦❞✉❧✐ ❛r❡ ♠✉❧t✐♣❧✐❡❞ ❛♥❞ t❤❡ ❛r❣✉♠❡♥ts ❛r❡ ❛❞❞❡❞✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

    cos ϕ + i sin ϕ ϕ + i sin ϕ ) · r r1 cos 2 2 1 1 2   = r1 r2 cos(ϕ1 + ϕ2 ) + i sin(ϕ1 + ϕ2 ) ❊①❡r❝✐s❡ ✸✳✻✳✶✵

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✳ ❈♦♥s✐❞❡r t❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ ♠♦❞✉❧✐ ❛♥❞ t❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ ❛r❣✉♠❡♥ts ♦❢ t❤❡ t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡ ❛♥❞ ♣r♦✈❡ t❤❛t t❤❡② ❝♦♥✈❡r❣❡✳ ❊①❡r❝✐s❡ ✸✳✻✳✶✶

✭❛✮ ❘❡♣r❡s❡♥t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣❧❡① ♥✉♠❜❡r ✐♥ t❤❡ st❛♥❞❛r❞ ❢♦r♠✿

(2 + 3i)(−1 + 2i)✳

■♥❞✐❝❛t❡ t❤❡ r❡❛❧

❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts✳ ✭❜✮ ❋✐♥❞ ✐ts ♠♦❞✉❧✉s ❛♥❞ ❛r❣✉♠❡♥t✳ ❊①❡r❝✐s❡ ✸✳✻✳✶✷

❙✐♠♣❧✐❢②

(1 + i)2 ✳

❊①❡r❝✐s❡ ✸✳✻✳✶✸

✭❛✮ ❋✐♥❞ t❤❡ r♦♦ts ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧

x2 + 2x + 2✳

✭❜✮ ❋✐♥❞ ✐ts

x✲✐♥t❡r❝❡♣ts✳

✭❝✮ ❋✐♥❞ ✐ts ❢❛❝t♦rs✳

❊①❡r❝✐s❡ ✸✳✻✳✶✹

❲❤❛t ❝❛♥ ②♦✉ s❛② ❛❜♦✉t t❤❡ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ t❤❡ r♦♦ts ♦❢ t❤❡s❡ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s❄

✸✳✼✳ ❈♦♠♣❧❡① ❢✉♥❝t✐♦♥s

✷✷✽

✸✳✼✳ ❈♦♠♣❧❡① ❢✉♥❝t✐♦♥s

❆ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥ ✐s s✐♠♣❧② ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ ❜♦t❤ ✐♥♣✉t ❛♥❞ ♦✉t♣✉t ❝♦♠♣❧❡① ♥✉♠❜❡rs✿ F : C → C.

❍♦✇ ❞♦ ✇❡ ✈✐s✉❛❧✐③❡ t❤❡s❡ ❢✉♥❝t✐♦♥s❄ ❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ F ❧✐❡s ✐♥ t❤❡ 4✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ ❛♥❞ ✐s♥✬t ♦❢ ♠✉❝❤ ❤❡❧♣✦ ❚♦ ❜❡❣✐♥ ✇✐t❤✱ ✇❡ ❝❛♥ r❡❝❛st s♦♠❡ ♦❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r ❛s ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥s✳ ❊①❛♠♣❧❡ ✸✳✼✳✶✿ ❝♦♠♣❧❡① ❛❞❞✐t✐♦♥

❚❤❡ s❤✐❢t ❜② ✈❡❝t♦r V =< a, b > ❜❡❝♦♠❡s ❛❞❞✐t✐♦♥ ♦❢ ❛ ✜①❡❞ ❝♦♠♣❧❡① ♥✉♠❜❡r✿ F (x, y) = (x + a, y + b) ✱ r❡✲✇r✐tt❡♥ F (z) = z + z0 ,

✇❤❡r❡ z0 = a + bi✳

❊①❛♠♣❧❡ ✸✳✼✳✷✿ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t✐♦♥

❚❤❡ ✈❡rt✐❝❛❧ ✢✐♣ ❜❡❝♦♠❡s ❝♦♥❥✉❣❛t✐♦♥✿ F (x, y) = (x, −y) ✱ r❡✲✇r✐tt❡♥ F (z) = z¯ .

✸✳✼✳ ❈♦♠♣❧❡① ❢✉♥❝t✐♦♥s

✷✷✾

❊①❡r❝✐s❡ ✸✳✼✳✸

❋✐♥❞ ❛ ❝♦♠♣❧❡① ❢♦r♠✉❧❛ ❢♦r t❤❡ ❤♦r✐③♦♥t❛❧ ✢✐♣ ✿ F (x, y) = (−x, y) .

❊①❛♠♣❧❡ ✸✳✼✳✹✿ ❝♦♠♣❧❡① ♠✉❧t✐♣❧✐❝❛t✐♦♥

❚❤❡ ✉♥✐❢♦r♠ str❡t❝❤ ❜❡❝♦♠❡s ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛ r❡❛❧ ♥✉♠❜❡r✿ F (x, y) = (kx, ky) , r❡✲✇r✐tt❡♥ F (z) = kz .

❖❢ ❝♦✉rs❡✱ t❤❡ ✢✐♣ ❛❜♦✉t t❤❡ ♦r✐❣✐♥ ✐s ❥✉st ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② −1✳ ❊①❡r❝✐s❡ ✸✳✼✳✺

❋✐♥❞ ❝♦♠♣❧❡① ❢♦r♠✉❧❛s ❢♦r t❤❡ ✈❡rt✐❝❛❧ str❡t❝❤ ✿ F (x, y) = (x, ky)

❛♥❞ t❤❡ ❤♦r✐③♦♥t❛❧ str❡t❝❤✿ F (x, y) = (kx, y) . ❊①❛♠♣❧❡ ✸✳✼✳✻✿ r♦t❛t✐♦♥

❆ r♦t❛t✐♦♥ ✐s ❝❛rr✐❡❞ ♦✉t ✈✐❛ ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛ ✜①❡❞ ❝♦♠♣❧❡① ♥✉♠❜❡r✿ F (z) = z0 z .

❙♣❡❝✐✜❝❛❧❧②✱ ✐t ❤❛s t♦ ❜❡ ❛ ♥✉♠❜❡r ✇✐t❤ ♠♦❞✉❧✉s ❡q✉❛❧ t♦ 1✿ z0 = cos α + i sin α .

▼❡❛♥✇❤✐❧❡✱ ✇✐t❤ z0 = 2i✱ ✇❡ ❤❛✈❡ t❤❡ 90 ❞❡❣r❡❡s r♦t❛t✐♦♥ ✇✐t❤ ❛ str❡t❝❤ ✇✐t❤ ❛ ❢❛❝t♦r ♦❢ 2✿

✸✳✼✳

❈♦♠♣❧❡① ❢✉♥❝t✐♦♥s

✷✸✵

❋♦r ❛♥② ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥✱ ✇❡ r❡♣r❡s❡♥t ❜♦t❤ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❛♥❞ t❤❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ✐♥ t❡r♠s ♦❢ t❤❡✐r r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts✱ ❥✉st ❛s ✈❡❝t♦r ❢✉♥❝t✐♦♥s✳ ❋✐rst✿ x = u + iv ,

✇❤❡r❡ u, v ❛r❡ r❡❛❧ ♥✉♠❜❡rs✳ ❙❡❝♦♥❞✿ z = F (x) = f (u, v) + ig(u, v) , f (u, v), g(u, v) ❛r❡ r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳

❚❤❡ t✇♦ ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s✱ f ❛♥❞ g ✱ ❝❛♥ ❜❡ ♣❧♦tt❡❞✳ ❆♥❞ s♦ ❝❛♥ t❤❡ ❛r❣✉♠❡♥t arg F ❛♥❞ t❤❡ ♠♦❞✉❧❡

||F || ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳

❊①❛♠♣❧❡ ✸✳✼✳✼✿ ♣r♦❥❡❝t✐♦♥s

❚❤❡ ♣r♦❥❡❝t✐♦♥s ♦♥ t❤❡ x✲ ❛♥❞ y ✲❛①❡s ❛r❡ t❤❡s❡✿ F (z) = Re z ❛♥❞ F (z) = Im z .

❊①❛♠♣❧❡ ✸✳✼✳✽✿ q✉❛❞r❛t✐❝

▲❡t✬s ❝♦♥s✐❞❡r ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ❛❣❛✐♥✿ f (x) = x2 + px + q .

❘❡❝❛❧❧ t❤❛t ✐♥❝r❡❛s✐♥❣ t❤❡ ✈❛❧✉❡ ♦❢ q ♠❛❦❡s t❤❡ ❣r❛♣❤ ♦❢ y = f (x) s❤✐❢t ✉♣✇❛r❞✿ ✐ts t✇♦ x✲✐♥t❡r❝❡♣ts st❛rt t♦ ❣❡t ❝❧♦s❡r t♦ ❡❛❝❤ ♦t❤❡r✱ t❤❡♥ ♠❡r❣❡✱ ❛♥❞ ✜♥❛❧❧② ❞✐s❛♣♣❡❛r✿

✸✳✼✳

❈♦♠♣❧❡① ❢✉♥❝t✐♦♥s

✷✸✶

◆♦t❡ t❤❛t ❛♥ ✐❞❡♥t✐❝❛❧ r❡s✉❧t ✐s s❡❡♥ ✐♥ ❛ s❡❡♠✐♥❣❧② ❞✐✛❡r❡♥t s✐t✉❛t✐♦♥✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♣❛r❛❜♦❧♦✐❞✱ t❤❡♥ ✐t ♣r♦❞✉❝❡s ❛ ♣❛r❛❜♦❧❛ ❛s ✐t ✐s ❝✉t ❜② ❛ ✈❡rt✐❝❛❧ ♣❧❛♥❡✳ ❙✉♣♣♦s❡ t❤❡ ♣❛r❛❜♦❧♦✐❞ ✐s ♠♦✈✐♥❣ ❤♦r✐✲ ③♦♥t❛❧❧②✳ ■❢ ✐t ✐s ❢❛❞✐♥❣ ❛✇❛②✱ t❤❡ ♣❛r❛❜♦❧❛ ✐s ♠♦✈✐♥❣ ✉♣✇❛r❞✳

❚❤✐s ✐❧❧✉str❛t❡s ✇❤❛t ❤❛♣♣❡♥s ✇❤❡♥ ♦✉r q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s s❡❡♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❛ ❝♦♠♣❧❡① ✈❛r✐❛❜❧❡✳ ❲❡ ❛r❡ ♣❧♦tt✐♥❣ t❤❡ r❡❛❧ ♣❛rt ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❲❛r♥✐♥❣✦

❱✐s✉❛❧✐③✐♥❣ F (x) ✈✐❛ t❤♦s❡ ❢✉♥❝t✐♦♥s✱ ♦r ❛s ❛ ✈❡❝t♦r ✜❡❧❞✱ ♠❛② ❜❡ ♠✐s❧❡❛❞✐♥❣✳

❈♦♠♣❧❡① ❢✉♥❝t✐♦♥s ❛r❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡✦ ❊①❛♠♣❧❡ ✸✳✼✳✾✿ ❝♦♠♣❧❡① ♣♦✇❡rs

❚❤✐s ✐s ❛ ✈✐s✉❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ♣♦✇❡r ❢✉♥❝t✐♦♥ ♦✈❡r ❝♦♠♣❧❡① ♥✉♠❜❡rs✳ ❋♦r s❡✈❡r❛❧ ✈❛❧✉❡s ♦❢ z ✱ t❤❡ ✈❛❧✉❡s z, z 2 , z 3 , ...

❛r❡ ♣❧♦tt❡❞ ❛s s❡q✉❡♥❝❡s✳

❖♥❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡ r❡❛❧ ♣❛rt ♦❢ z ♠❛❦❡s t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② z str❡t❝❤ ♦r s❤r✐♥❦ t❤❡ ♥✉♠❜❡r ✇❤✐❧❡

✸✳✼✳

❈♦♠♣❧❡① ❢✉♥❝t✐♦♥s

✷✸✷

t❤❡ ✐♠❛❣✐♥❛r② ♣❛rt ♦❢ x ✐s r❡s♣♦♥s✐❜❧❡ ❢♦r r♦t❛t✐♥❣ t❤❡ ♥✉♠❜❡r ❛r♦✉♥❞ 0✳ ❆ s♣❡❝✐❛❧✱ sq✉❛r❡ ♣❛t❤ ✐s ♣r♦❞✉❝❡❞ ❜② z = i✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ✐s ❛ ❝♦♣② ♦❢ t❤❡ ♦♥❡ ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✷✳

❉❡✜♥✐t✐♦♥ ✸✳✼✳✶✵✿ ❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥ ❚❤❡

❧✐♠✐t ♦❢ ❛ ❢✉♥❝t✐♦♥ z = F (x) ❛t ❛ ♣♦✐♥t x = a ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❧✐♠✐t lim F (xn )

n→∞

❝♦♥s✐❞❡r❡❞ ❢♦r ❛❧❧ s❡q✉❡♥❝❡s {xn } ✇✐t❤✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ F ❡①❝❧✉❞✐♥❣ a t❤❛t ❝♦♥✲ ✈❡r❣❡ t♦ a✱ a 6= xn → a ❛s n → ∞ ,

✇❤❡♥ ❛❧❧ t❤❡s❡ ❧✐♠✐ts ❡①✐st ❛♥❞ ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✳ ■♥ t❤❛t ❝❛s❡✱ ✇❡ ✉s❡ t❤❡ ♥♦t❛t✐♦♥✿ lim F (x) .

x→a

❖t❤❡r✇✐s❡✱

t❤❡ ❧✐♠✐t ❞♦❡s ♥♦t ❡①✐st✳

❚❤❡♦r❡♠ ✸✳✼✳✶✶✿ ▲♦❝❛❧✐t② ❙✉♣♣♦s❡ t✇♦ ❢✉♥❝t✐♦♥s

f

❛♥❞

g

❝♦✐♥❝✐❞❡ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ ♣♦✐♥t

f (x) = g(x) ❢♦r s♦♠❡

ε > 0✳

❢♦r ❛❧❧

❚❤❡♥✱ t❤❡✐r ❧✐♠✐ts ❛t

a

x

✇✐t❤

a✿

||x − a|| < ε ,

❝♦✐♥❝✐❞❡ t♦♦✿

lim f (x) = lim g(x) .

x→a

x→a

▲✐♠✐ts ✉♥❞❡r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✳✳✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s t♦ ♣r♦✈❡ ✈✐rt✉❛❧❧② ✐❞❡♥t✐❝❛❧ ❢❛❝ts ❛❜♦✉t ❧✐♠✐ts ♦❢ ❢✉♥❝t✐♦♥s✳ ▲❡t✬s r❡✲✇r✐t❡ t❤❡ ♠❛✐♥ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ✉s✐♥❣ t❤❡ ❛❧t❡r♥❛t✐✈❡ ♥♦t❛t✐♦♥✳

❚❤❡♦r❡♠ ✸✳✼✳✶✷✿ ❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s ❙✉♣♣♦s❡

an → a

❙❘✿ P❘✿

❛♥❞

bn → b✳

an + bn → a + b an · bn → ab

❚❤❡♥

❈▼❘✿ ◗❘✿

c · an → ca an /bn → a/b

❢♦r ❛♥② ❝♦♠♣❧❡① ♣r♦✈✐❞❡❞

c

b 6= 0

❊❛❝❤ ♣r♦♣❡rt② ✐s ♠❛t❝❤❡❞ ❜② ✐ts ❛♥❛❧♦❣ ❢♦r ❢✉♥❝t✐♦♥s✳

❚❤❡♦r❡♠ ✸✳✼✳✶✸✿ ❆❧❣❡❜r❛ ♦❢ ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s ❙✉♣♣♦s❡

❙❘✿ P❘✿

f (x) → F

❛♥❞

g(x) → G

f (x) + g(x) → F + G f (x) · g(x) → F G

❏✉st ❛s ❜❡❢♦r❡✱ t❤❡ ♥❡①t ❝♦♥❝❡♣t ✐s ❝♦♥t✐♥✉✐t②✿

❛s

❈▼❘✿ ◗❘✿

x → a✳

❚❤❡♥

c · f (x) → cF f (x)/g(x) → F/G

❢♦r ❛♥② ❝♦♠♣❧❡① ♣r♦✈✐❞❡❞

G 6= 0

c

✸✳✽✳ ❈♦♠♣❧❡① ❧✐♥❡❛r ♦♣❡r❛t♦rs

✷✸✸

❉❡✜♥✐t✐♦♥ ✸✳✼✳✶✹✿ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ❆ ❢✉♥❝t✐♦♥ f ✐s ❝❛❧❧❡❞ ❝♦♥t✐♥✉♦✉s ❛t ♣♦✐♥t a ✐❢✿ • ❚❤❡ ❢✉♥❝t✐♦♥ f (x) ✐s ❞❡✜♥❡❞ ❛t x = a✳ • ❚❤❡ ❧✐♠✐t ♦❢ f ❡①✐sts ❛t a✳ • ❚❤❡ t✇♦ ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✿

lim f (x) = f (a) .

x→a

❚❤✉s✱ t❤❡ ❧✐♠✐ts ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ❝❛♥ ❜❡ ❢♦✉♥❞ ❜② s✉❜st✐t✉t✐♦♥✳ ❊q✉✐✈❛❧❡♥t❧②✱ ❛ ❢✉♥❝t✐♦♥ f ✐s ❝♦♥t✐♥✉♦✉s ❛t a ✐❢

lim f (xn ) = f (a) ,

n→∞

❢♦r ❛♥② s❡q✉❡♥❝❡ xn → a✳

❆ t②♣✐❝❛❧ ❢✉♥❝t✐♦♥ ✇❡ ❡♥❝♦✉♥t❡r ✐s ❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r② ♣♦✐♥t ♦❢ ✐ts ❞♦♠❛✐♥✳ ❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❝❧❛ss ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳

❚❤❡♦r❡♠ ✸✳✼✳✶✺✿ ❈♦♥t✐♥✉✐t② ♦❢ P♦❧②♥♦♠✐❛❧s ❊✈❡r② ♣♦❧②♥♦♠✐❛❧ ✐s ❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r② ♣♦✐♥t✳

❯♥❧✐❦❡ ✈❡❝t♦r ❢✉♥❝t✐♦♥s✱ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥s ❤❛✈❡ ♠♦r❡ ♦♣❡r❛t✐♦♥s t♦ ✇♦rr② ❛❜♦✉t✳ ❚❤❡ t❤❡♦r❡♠ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❧❣❡❜r❛✐❝ r❡s✉❧t✳

❚❤❡♦r❡♠ ✸✳✼✳✶✻✿ ❆❧❣❡❜r❛ ♦❢ ❈♦♥t✐♥✉✐t② ❙✉♣♣♦s❡

f

❛♥❞

g

❛r❡ ❝♦♥t✐♥✉♦✉s ❛t

❙❘✿ P❘✿

f +g f ·g

x = a✳

❈▼❘✿ ◗❘✿

c·f f /g

❚❤❡♥ s♦ ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥s✿

c g(a) 6= 0

❢♦r ❛♥② ❝♦♠♣❧❡① ♣r♦✈✐❞❡❞

✸✳✽✳ ❈♦♠♣❧❡① ❧✐♥❡❛r ♦♣❡r❛t♦rs

▲❡t✬s ❝♦♥s✐❞❡r ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❣❛✐♥✳ ❙✉♣♣♦s❡ A ✐s ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ✇✐t❤ ❛ 2 × 2 ♠❛tr✐① ✇✐t❤ r❡❛❧ ❡♥tr✐❡s✳ ❙✉♣♣♦s❡ D ✐s t❤❡ ❞✐s❝r✐♠✐♥❛♥t ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ♦❢ A✳ ❲❤❡♥ D > 0✱ ✇❡ ❤❛✈❡ t✇♦ ❞✐st✐♥❝t r❡❛❧ r♦♦ts ❝♦✈❡r❡❞ ❜② t❤❡ ❈❧❛ss✐✜❝❛t✐♦♥ ❚❤❡♦r❡♠ ♦❢ ▲✐♥❡❛r ❖♣❡r❛t♦rs ✇✐t❤ r❡❛❧ ❡✐❣❡♥✈❛❧✉❡s✳ ❲❡ ❛❧s♦ s❛✇ t❤❡ tr❛♥s✐t✐♦♥❛❧ ❝❛s❡ ✇❤❡♥ D = 0✳ ❚❤✉s✱ ✇❡ tr❛♥s✐t✐♦♥ t♦ t❤❡ ❝❛s❡ D < 0 ✇❤❡♥ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ✕ ❛s r♦♦ts ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ✕ ❛r❡ ❝♦♠♣❧❡①✳ ❲❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛r❡ ❥✉st ❛s ❣♦♦❞ ✭♦r ❜❡tt❡r✦✮ t❤❛♥ t❤❡ r❡❛❧✱ s♦ ✇❤② ♥♦t ✐♥❝❧✉❞❡ t❤✐s ♣♦ss✐❜✐❧✐t②❄

❊①❛♠♣❧❡ ✸✳✽✳✶✿ r♦t❛t✐♦♥ ❚❤✐s ✐s ❤♦✇ ✇❡ ✜♥❞ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ❢♦r t❤❡ r♦t❛t✐♦♥ ❛♥❞ ✜♥❞ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❜② s♦❧✈✐♥❣ ✐t✿         −λ −1 1 0 0 −1 0 −1 = λ2 + 1 = 0 =⇒ λ = ±i . = det −λ =⇒ det A= 1 −λ 0 1 1 0 1 0

❚❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ✐♠❛❣✐♥❛r②✦ ▲❡t✬s ♥♦t✐❝❡ t❤♦✉❣❤ t❤❛t ♥♦ r❡❛❧ ✈❡❝t♦r ♠✉❧t✐♣❧✐❡❞ ❜② ❛♥ ✐♠❛❣✐♥❛r② ♥✉♠❜❡r ❝❛♥ ♣r♦❞✉❝❡ ❛ r❡❛❧ ✈❡❝t♦r✳✳✳ ■♥❞❡❡❞✱ ❧❡t✬s tr② t♦ ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs❀ s♦❧✈❡ t❤❡ ♠❛tr✐① ❡q✉❛t✐♦♥

✸✳✽✳

✷✸✹

❈♦♠♣❧❡① ❧✐♥❡❛r ♦♣❡r❛t♦rs

AV = λV ❢♦r V ✐♥ R2 ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ♥❡❡❞ t♦ ✜♥❞ r❡❛❧✭✦✮ x, y t❤❛t s❛t✐s❢②✿       −y = ix x 0 −1 x =i =⇒ x = iy y y 1 0

❯♥❧❡ss ❜♦t❤ ③❡r♦✱ x ❛♥❞ y ❝❛♥✬t ❜❡ ❜♦t❤ r❡❛❧✳✳✳ ❙♦✱ ✇❤❡♥ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡♥✬t r❡❛❧✱ t❤❡r❡ ❛r❡ ♥♦ r❡❛❧ ❡✐❣❡♥✈❡❝t♦rs✳ ❇✉t ✇❤② st♦♣ ❤❡r❡❄ ❲❤② ♥♦t ❤❛✈❡ ❛❧❧ t❤❡ ♥✉♠❜❡rs ❛♥❞ ✈❡❝t♦rs ❛♥❞ ♠❛tr✐❝❡s ❝♦♠♣❧❡①❄ ❏✉st ❛s ❛ r❡❛❧ 2✲✈❡❝t♦r ✐s ❛ ♣❛✐r ♦❢ r❡❛❧ ♥✉♠❜❡rs✱ ❛ ❡①❛♠♣❧❡✱ 

❝♦♠♣❧❡①

2✲✈❡❝t♦r ✐s ❛ ♣❛✐r ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs❀ ❢♦r

 2 + i V = . −1 + 2i

❚❤✐s r❡♣r❡s❡♥t❛t✐♦♥ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇ ✈✐❛ t❤❡ r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ❝♦♠♣♦♥❡♥ts✿

❍❡r❡✱ ✇❡ s❡❡ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ❢♦r t❤❡ ✜rst ❛♥❞ t❤❡♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ❢♦r t❤❡ s❡❝♦♥❞ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ✈❡❝t♦r V ✳ ❚❤✐s ✐s ✇❤② ✇❡ ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ❛❧❧ ❝♦♠♣❧❡① 2✲✈❡❝t♦rs ❜② C2 ✳ ❋✉rt❤❡r♠♦r❡✱ t❤✐s ✈❡❝t♦r V ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ✐♥ t❡r♠s ♦❢ ✐ts r❡❛❧

❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts ✿

     1 2 2 + i V = . +i = 2 −1 −1 + 2i 

❊❛❝❤ ♦❢ t❤❡s❡ ✐s ❛ r❡❛❧ ✈❡❝t♦r ❛♥❞ t❤❡② ❛r❡ ✐❧❧✉str❛t❡❞ ❛❝❝♦r❞✐♥❣❧②✿

❚❤❡ ❢♦r♠❡r ✐s ♦❢ t❤❡ ♠❛✐♥ ✐♥t❡r❡st ❛♥❞ ✐t ✐s ❧♦❝❛t❡❞ ✐♥ t❤❡ ❢❛♠✐❧✐❛r r❡❛❧ ♣❧❛♥❡ R2 ✳ ◆❡①t✱ ❛ ❝♦♠♣❧❡① 2 × 2

♠❛tr✐①

A ✐s s✐♠♣❧② ❛ 2 × 2 t❛❜❧❡ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs❀ ❢♦r ❡①❛♠♣❧❡✿   0 1+i . A= i 2 − 3i

✸✳✽✳ ❈♦♠♣❧❡① ❧✐♥❡❛r ♦♣❡r❛t♦rs

✷✸✺

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✕ ❛❞❞✐t✐♦♥ ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✕ ♣r❡s❡♥t❡❞ ❛❜♦✈❡ ❛❧❧♦✇s ✉s t♦ ❝❛rr② ♦✉t t❤❡ ♦♣❡r❛t✐♦♥s ✕ ❛❞❞✐t✐♦♥✱ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❛♥❞ ♠❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✕ ♦♥ t❤❡s❡ ✈❡❝t♦rs ❛♥❞ t❤❡s❡ ♠❛tr✐❝❡s✦ ❚❤❡♥✱ ❛ ❝♦♠♣❧❡① ♠❛tr✐① A ❞❡✜♥❡s ❛ ❢✉♥❝t✐♦♥✿

A : C2 → C2 , t❤r♦✉❣❤ ♠❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ A(X) = AX ✳ ▼♦r❡♦✈❡r✱ t❤✐s ❢✉♥❝t✐♦♥ ✐s ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡✿ A(αX + βY ) = αA(X) + βA(Y ) , ❢♦r ❛♥② ❝♦♠♣❧❡① ♥✉♠❜❡rs α ❛♥❞ β ❛♥❞ ❛♥② ❝♦♠♣❧❡① ✈❡❝t♦rs X ❛♥❞ Y ✳ ❇✉t ❤♦✇ ❝❛♥ ✇❡ ✉s❡ t❤❡s❡ ❢✉♥❝t✐♦♥s t♦ ✉♥❞❡rst❛♥❞ r❡❛❧ ❧✐♥❡❛r ♦♣❡r❛t♦rs❄ ■ts ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts ❛r❡ ❝♦♠♣❧❡① ✈❡❝t♦rs ❛♥❞ t❤❡② ❤❛✈❡ r❡❛❧ ♣❛rts ❛s ❞✐s❝✉ss❡❞ ❛❜♦✈❡✳ ❙♦✱ ✇❡ ❝❛♥ r❡str✐❝t t❤❡ ❞♦♠❛✐♥ ♦❢ ❛ ❝♦♠♣❧❡① ❧✐♥❡❛r ♦♣❡r❛t♦r t♦ t❤❡ r❡❛❧ ♣❧❛♥❡ ✜rst ❛♥❞ t❤❡♥ t❛❦❡ t❤❡ r❡❛❧ ♣❛rt ♦❢ t❤❡ ♦✉t♣✉t✳ ❚❤❡ r❡s✉❧t ✐s ❛ ❢❛♠✐❧✐❛r r❡❛❧ ❧✐♥❡❛r ♦♣❡r❛t♦r✿ B : R2 → R2 , t❤❡ r❡❛❧ ♣❛rt ♦❢ t❤❡ ❝♦♠♣❧❡① ❧✐♥❡❛r ♦♣❡r❛t♦r A✳

▲❡t✬s r❡✈✐❡✇ ♦✉r t❤❡♦r② ❣❡♥❡r❛❧✐③❡❞ t❤✐s ✇❛②✳ ❚❤❡r❡ ✐s ♥♦ ❞✐✛❡r❡♥❝❡✿

❉❡✜♥✐t✐♦♥ ✸✳✽✳✷✿ ❞❡t❡r♠✐♥❛♥t ❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ❛ ❝♦♠♣❧❡① 2×2 ♠❛tr✐① ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣❧❡① ♥✉♠❜❡r✿   a b = ad − bc det c d

❚❤❡♦r❡♠ ✸✳✽✳✸✿ ◆♦♥✲③❡r♦ ❉❡t❡r♠✐♥❛♥t ❙✉♣♣♦s❡

A ✐s ❛ ❝♦♠♣❧❡① 2 × 2 ♠❛tr✐①✳ ❚❤❡♥✱ det A 6= 0 ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ s♦❧✉t✐♦♥ AX = 0 ❝♦♥s✐sts ♦❢ ♦♥❧② 0✳

s❡t ♦❢ t❤❡ ♠❛tr✐① ❡q✉❛t✐♦♥

❉❡✜♥✐t✐♦♥ ✸✳✽✳✹✿ ❡✐❣❡♥✈❛❧✉❡ ●✐✈❡♥ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r A : C2 → C2 ✱ ❛ ✭❝♦♠♣❧❡①✮ ♥✉♠❜❡r λ ✐s ❝❛❧❧❡❞ ❛♥ ❡✐❣❡♥✈❛❧✉❡ ♦❢ A ✐❢

A(V ) = λV ❢♦r s♦♠❡ ♥♦♥✲③❡r♦ ✈❡❝t♦r V ✐♥ C2 ✳ ❚❤❡♥✱ V ✐s ❝❛❧❧❡❞ ❛♥ ❡✐❣❡♥✈❡❝t♦r ♦❢ A ❝♦rr❡✲ s♣♦♥❞✐♥❣ t♦ λ✳

❉❡✜♥✐t✐♦♥ ✸✳✽✳✺✿ ❡✐❣❡♥s♣❛❝❡ ❋♦r ❛ ✭❝♦♠♣❧❡①✮ ❡✐❣❡♥✈❛❧✉❡ λ ♦❢ A✱ t❤❡ ❡✐❣❡♥s♣❛❝❡ ♦❢ ❛ ❝♦♠♣❧❡① ❧✐♥❡❛r ♦♣❡r❛t♦r A ❝♦rr❡s♣♦♥❞✐♥❣ t♦ λ ✐s ❞❡✜♥❡❞ ❛♥❞ ❞❡♥♦t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t ✐♥ C2 ✿

E(λ) = {V : A(V ) = λV } ■t ✐s ❛ ✈❡r② ✐♠♣♦rt❛♥t ❢❛❝t t❤❛t ❛❧❧ t❤❡ ❝♦♠♣✉t❛t✐♦♥s t❤❛t ✇❡ ❤❛✈❡ ♣❡r❢♦r♠❡❞ ♦♥ r❡❛❧ ♠❛tr✐❝❡s ❛♥❞ ✈❡❝t♦rs r❡♠❛✐♥ ✈❛❧✐❞✦ ❆♠♦♥❣ t❤❡ r❡s✉❧ts t❤❛t r❡♠❛✐♥ ✈❛❧✐❞ ✐s t❤❡ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r ❖♣❡r❛t♦rs ✇✐t❤ r❡❛❧ ❡✐❣❡♥✈❛❧✉❡s✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs r❡♣r❡s❡♥t❡❞ ❜② r❡❛❧ ♠❛tr✐❝❡s ❤♦✇❡✈❡r ✇✐❧❧ r❡♠❛✐♥ ♦✉r ❡①❝❧✉s✐✈❡ ✐♥t❡r❡st✳

✸✳✾✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs ✇✐t❤ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s

✷✸✻

❊✈❡r② ❧✐♥❡❛r ♦♣❡r❛t♦r r❡♣r❡s❡♥t❡❞ ❜② ❛ r❡❛❧ ♠❛tr✐① A ✐s st✐❧❧ ❥✉st ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ ❛ ❝♦♠♣❧❡① ♦♣❡r❛t♦r A : C2 → C2 ✳ ■♥ ❢❛❝t✱ ✐t ✇✐❧❧ ❛❧✇❛②s ❤❛✈❡ s♦♠❡ ❝♦♠♣❧❡① ✭♥♦♥✲r❡❛❧✮ ✈❡❝t♦rs ❛♠♦♥❣ ✐ts ✈❛❧✉❡s✱ ✉♥❧❡ss A = 0✳ ■ts ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ❤❛s r❡❛❧ ❝♦❡✣❝✐❡♥ts ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ❈❧❛ss✐✜❝❛t✐♦♥ ❚❤❡♦r❡♠ ♦❢ ◗✉❛❞r❛t✐❝ P♦❧②♥♦♠✐❛❧s ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r ❛♣♣❧✐❡s✱ ❛s ❢♦❧❧♦✇s✳

❚❤❡♦r❡♠ ✸✳✽✳✻✿ ❈❧❛ss✐✜❝❛t✐♦♥ ❚❤❡♦r❡♠ ♦❢ ❊✐❣❡♥✈❛❧✉❡s ❙✉♣♣♦s❡

A

✐s ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r r❡♣r❡s❡♥t❡❞ ❜② ❛ r❡❛❧

2×2

t❤❡ ❞✐s❝r✐♠✐♥❛♥t ♦❢ ✐ts ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✳ ❚❤❡♥ t❤❡

A

D ✐s ❡✐❣❡♥✈❛❧✉❡s λ1 , λ2 ♦❢ ♠❛tr✐①

A

❛♥❞

❢❛❧❧ ✐♥t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ❝❛t❡❣♦r✐❡s✿ ✶✳ ■❢ ✷✳ ■❢ ✸✳ ■❢

D > 0✱ D = 0✱ D < 0✱

t❤❡♥ t❤❡♥ t❤❡♥

λ1 , λ 2 λ1 , λ 2 λ1 , λ 2

❛r❡ ❞✐st✐♥❝t r❡❛❧✳ ❛r❡ ❡q✉❛❧ r❡❛❧✳ ❛r❡ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡✳

❲❡ ♦♥❧② ♥❡❡❞ t♦ ❛❞❞r❡ss t❤❡ ❧❛st ❝❛s❡✳

✸✳✾✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs ✇✐t❤ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s

❆❧❧ ♥✉♠❜❡rs ❜❡❧♦✇ ❛r❡ ❝♦♠♣❧❡① ✉♥❧❡ss st❛t❡❞ ♦t❤❡r✇✐s❡✳ ❚❤❡ ✜rst t❤✐♥❣ ✇❡ ♥♦t✐❝❡ ❛❜♦✉t t❤❡ ❝❛s❡ ✇❤❡♥ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ❛r❡♥✬t r❡❛❧ ✐s t❤❛t t❤❡r❡ ❛r❡ ♥♦ r❡❛❧ ❡✐❣❡♥✈❡❝t♦rs✳ ❚❤❡r❡ ✇✐❧❧ ❜❡ ♥♦ ❡✐❣❡♥s♣❛❝❡s s❤♦✇♥ ✐♥ t❤❡ r❡❛❧ ♣❧❛♥❡ s❤♦✇♥ ✐♥ t❤❡ ❡①❛♠♣❧❡s ❜❡❧♦✇✳

❊①❛♠♣❧❡ ✸✳✾✳✶✿ r♦t❛t✐♦♥s ❈♦♥s✐❞❡r ❛ r♦t❛t✐♦♥ t❤r♦✉❣❤ 90 ❞❡❣r❡❡s ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ ❛❣❛✐♥✿      0 −1 x u =⇒ λ1,2 = ±i . = y 1 0 v

❚♦ ✜♥❞ t❤❡ ✜rst ❡✐❣❡♥✈❡❝t♦r✱ ✇❡ s♦❧✈❡✿ 

0 −1 F V1 = 1 0

    x x . =i y y

❚❤✐s ❣✐✈❡s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ 

❲❡ ❝❤♦♦s❡ ❛ ❝♦♠♣❧❡① ❡✐❣❡♥✈❡❝t♦r✿

x

−y = ix, ❆◆❉ = iy

=⇒ y = −ix .

 1 , V1 = −i 

✸✳✾✳

▲✐♥❡❛r ♦♣❡r❛t♦rs ✇✐t❤ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s

✷✸✼

❛♥❞ s✐♠✐❧❛r❧② ❛♥ ❡✐❣❡♥✈❡❝t♦r ❢♦r t❤❡ s❡❝♦♥❞ ❡✐❣❡♥✈❛❧✉❡✿   1 . V2 = i ❚❤❡ r❡st ❛r❡ ✭❝♦♠♣❧❡①✮ ♠✉❧t✐♣❧❡s ♦❢ t❤❡s❡✳ ❙♦✱ ✉♥❞❡r F ✱ ❝♦♠♣❧❡① ✈❡❝t♦r Vk ✐s ♠✉❧t✐♣❧✐❡❞ ❜② λk ✱ ❛♥❞ s♦ ✐s ❡✈❡r② ♦❢ ✐ts ♠✉❧t✐♣❧❡s✱ k = 1, 2✳ ❙✐♥❝❡ t❤❡s❡ ♠✉❧t✐♣❧❡s ❛r❡ ❝♦♠♣❧❡①✱ t❤✐s ♠✉❧t✐♣❧✐❝❛t✐♦♥ r♦t❛t❡s ✭t❤❡ ✈❡❝t♦r ♦❢ t❤❡ ❣❡♦♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢✮ ❡✐t❤❡r ❝♦♠♣♦♥❡♥t ♦❢ t❤✐s ✈❡❝t♦r✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ r❡❛❧ ♣❛rt ♦❢ t❤✐s ✈❡❝t♦r ✐s ❛❧s♦ r♦t❛t❡❞ ✕ ♦♥ t❤❡ r❡❛❧ ♣❧❛♥❡✳ ■t ✐s t❤✐s r♦t❛t✐♦♥ t❤❛t ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥✳ ■t ✐s s❤♦✇♥ ✐♥ t❤❡ s❡❝♦♥❞ r♦✇ ❜❡❧♦✇✿

❚❤❡ ❣❡♥❡r❛❧ ✈❛❧✉❡ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ✕ ♦✈❡r t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✕ ♦❢ ♦✉r t✇♦ ❡✐❣❡♥✈❡❝t♦rs✿     1 1 . + β(−i) X = α V1 + β V2 =⇒ F X = αi i −i ▲❡t✬s ❝♦♥s✐❞❡r ❛

r♦t❛t✐♦♥ t❤r♦✉❣❤ ❛♥ ❛r❜✐tr❛r② ❛♥❣❧❡ θ✿      cos θ − sin θ x u . = y sin θ cos θ v

❚❤❡♥✱ ✇❡ ❤❛✈❡ ❚❤❡r❡❢♦r❡✱

χA (λ) = (cos θ − λ)2 + sin2 θ = cos2 θ − 2 cos θ λ + λ2 + sin2 θ = λ2 − 2 cos θ λ + 1 .

λ1,2 =

2 cos θ ±

p √ p (2 cos θ)2 − 4 2 cos θ ± 2 cos2 θ − 1 = = cos θ ± − sin2 θ = cosθ ± i sin θ . 2 2

✸✳✾✳

❙♦✱

▲✐♥❡❛r ♦♣❡r❛t♦rs ✇✐t❤ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s

✷✸✽

t❤❡ ❛r❣✉♠❡♥t ♦❢ t❤❡ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s ✐s ❡q✉❛❧ t♦ t❤❡ ❛♥❣❧❡ ♦❢ r♦t❛t✐♦♥

✭✉♣ t♦ ❛ s✐❣♥✮✿

| arg λ1,2 | = |θ| . ❚❤❡ ❡✐❣❡♥✈❡❝t♦rs ❛r❡ t❤❡ s❛♠❡ ❛s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✿

 1 V1 = −i 

❛♥❞

  1 . V2 = i

■♥❞❡❡❞✱ t❤❡② ❛r❡ r♦t❛t❡❞ t❤r♦✉❣❤ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② t❤❡ ❡✐❣❡♥✈❛❧✉❡s✿

      1 cos θ − sin θ cos θ ± i sin θ 1 . = = λk Vk = (cosθ ± i sin θ) ∓i sin θ cos θ sin θ ∓ i cos θ ∓i 

❖♥❝❡ ❛❣❛✐♥✱ ✇❡ s❡❡ ❤♦✇ t❤❡ r❡❛❧ ♣❛rt ♦❢ ❛ ❝♦♠♣❧❡① ❡✐❣❡♥✈❡❝t♦r ✐s r♦t❛t❡❞ ✈✐❛ ❝♦♠♣❧❡① ♠✉❧t✐♣❧✐❝❛t✐♦♥✿

❲❛r♥✐♥❣✦ ❚❤✐s ❞♦❡s ♥♦t ❛♣♣❧② t♦ ✈❡❝t♦rs ✐♥ ❡✐❣❡♥✈❡❝t♦rs✳

❊①❛♠♣❧❡ ✸✳✾✳✷✿ r♦t❛t✐♦♥ ✇✐t❤ str❡t❝❤✲s❤r✐♥❦ ▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ❧✐♥❡❛r ♦♣❡r❛t♦r✿



u = 3x −13y v = 5x +y

❛♥❞

 3 −13 . F = 5 1 

C2

t❤❛t ❛r❡♥✬t

✸✳✾✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs ✇✐t❤ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s

✷✸✾

❖✉r ❛♥❛❧②s✐s st❛rts ✇✐t❤ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✿  3 − λ −13 = λ2 − 4λ + 68 . χ(λ) = det(F − λI) = det 5 1−λ 

❲❡ ✜♥❞ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❢r♦♠ t❤❡ ◗✉❛❞r❛t✐❝ ❋♦r♠✉❧❛ ✿

λ1,2 = 2 ± 8i .

◆♦✇ ✇❡ ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✳ ❲❡ s♦❧✈❡ t❤❡ t✇♦ ❡q✉❛t✐♦♥s✿ F Vk = λk Vk , k = 1, 2 .

❚❤❡ ✜rst✿



3 −13 F V1 = 5 1

    x x . = (2 + 8i) y y

❚❤✐s ❣✐✈❡s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ 

3x −13y = (2 + 8i)x, ❆◆❉ 5x +y = (2 + 8i)y

=⇒

❲❡ ❝❤♦♦s❡ t❤❡ ✜rst ❡✐❣❡♥✈❡❝t♦r t♦ ❜❡✿



(1 − 8i)x −13y =0 5x +(−1 − 8i)y = 0

 1 + 8i . V1 = 5 

3 −13 F V2 = 5 1 

3x −13y = (2 − 8i)x 5x +y = (2 − 8i)y

=⇒

❲❡ ❝❤♦♦s❡ t❤❡ s❡❝♦♥❞ ❡✐❣❡♥✈❡❝t♦r t♦ ❜❡✿



    x x . = (2 − 8i) y y

(1 + 8i)x −13y =0 5x +(−1 + 8i)y = 0

=⇒ x =

(1 − 8i) y. 5

 1 − 8i . V2 = 5 

❚❤❡ ❣❡♥❡r❛❧ ❝♦♠♣❧❡① ✈❛❧✉❡ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦✿ X = α V1 + β V2

(1 + 8i) y. 5



❚❤❡ s❡❝♦♥❞ ❡✐❣❡♥✈❛❧✉❡ s❛t✐s✜❡s✿

❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠✿

=⇒ x =

   1 − 8i 1 + 8i . + β(2 − 8i) =⇒ F X = α(2 + 8i) 5 5 

❲❡ ❦♥♦✇ t❤❡ ❡✛❡❝t ♦❢ F ♦♥ t❤❡s❡ t✇♦ ✈❡❝t♦rs✿ t❤❡② ❛r❡ r♦t❛t❡❞ ❛♥❞ str❡t❝❤❡❞✳

✸✳✾✳

▲✐♥❡❛r ♦♣❡r❛t♦rs ✇✐t❤ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s

✷✹✵

❆s ②♦✉ ❝❛♥ s❡❡✱ str❡t❝❤✐♥❣ ✐s t❤❡ s❛♠❡ ❢♦r ❜♦t❤ ❝♦♠♣♦♥❡♥ts ❛♥❞ ❜♦t❤ ❜❛s✐s ✈❡❝t♦rs✳ ❚❤❛t ✐s ✇❤② ✇❡ ❤❛✈❡ ✕ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ r♦t❛t✐♦♥ ✕ ❛ ✉♥✐❢♦r♠ str❡t❝❤ ✭r❡✲s❝❛❧✐♥❣✮ ❢♦r t❤❡ r❡❛❧ ♣❧❛♥❡✳ ❚❤✐s ✐s s❤♦✇♥ ✐♥ t❤❡ s❡❝♦♥❞ r♦✇✳ ❚❤✐s ✐s t❤❡ ❛❧❣❡❜r❛ ❢♦r t❤❡ ❛❜♦✈❡ ✐❧❧✉str❛t✐♦♥✿

   −62 + 24i 1 + 8i = λ1 V1 = (2 + 8i) 10 + 40i 5 

❆❝❝♦r❞✐♥❣ t♦ t❤❡

   −62 − 24i 1 − 8i . = λ2 V2 = (2 − 8i) 10 − 40i 5 

❛♥❞

❈❧❛ss✐✜❝❛t✐♦♥ ❚❤❡♦r❡♠ ♦❢ ◗✉❛❞r❛t✐❝ P♦❧②♥♦♠✐❛❧s✱

✇❤❡♥ t❤❡ ❞✐s❝r✐♠✐♥❛♥t

D < 0✱

t❤❡ t✇♦

r♦♦ts ❛r❡ ❝♦♥❥✉❣❛t❡✿

λ1,2 = a ± bi . ❚❤❡② ❤❛✈❡ t❤❡ s❛♠❡ ♠♦❞✉❧✉s✿

||λ1,2 || =



a2 + b2 .

❚❤✐s ✐s ✇❤② ♠✉❧t✐♣❧②✐♥❣ ❛♥② ❝♦♠♣❧❡① ♥✉♠❜❡r ❜② ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ♥✉♠❜❡rs ✇✐❧❧ ♣r♦❞✉❝❡ t❤❡ s❛♠❡ r❛t❡ 2 2 ♦❢ str❡t❝❤✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t C ❛♥❞✱ ❡s♣❡❝✐❛❧❧②✱ t❤❡ r❡❛❧ ♣❧❛♥❡ R ❛r❡ str❡t❝❤❡❞ ✦ ❚❤✐s ✐s t❤❡

✉♥✐❢♦r♠❧②

s✉♠♠❛r② ♦❢ ♦✉r ❛♥❛❧②s✐s✳

❚❤❡♦r❡♠ ✸✳✾✳✸✿ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r ❖♣❡r❛t♦rs ✇✐t❤ ❈♦♠♣❧❡① ❊✐❣❡♥✈❛❧✉❡s ❙✉♣♣♦s❡ ❛ r❡❛❧ ♠❛tr✐① F ❤❛s t✇♦ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ❡✐❣❡♥✈❛❧✉❡s λ1 ❛♥❞ λ2 ✳ ❚❤❡♥✱ t❤❡ ♦♣❡r❛t♦r U = F X ❞♦❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ■t r♦t❛t❡s t❤❡ r❡❛❧ ♣❧❛♥❡ t❤r♦✉❣❤ t❤❡ ❛♥❣❧❡ θ t❤❛t s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿ | sin θ| = | arg λ1 | = | arg λ2 | . • ■t str❡t❝❤❡s✲s❤r✐♥❦s t❤❡ ♣❧❛♥❡ ✉♥✐❢♦r♠❧② ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝t♦r✿ s = ||λ1 || = ||λ2 || . ❲❡ ❝❛♥ ❛❧s♦ r❡❝❛st t❤✐s t❤❡♦r❡♠ ✐♥ ❡①❝❧✉s✐✈❡❧② ✏r❡❛❧✑ t❡r♠s✳ ❲❡ ✇✐❧❧ ♥❡❡❞ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t✿

❚❤❡♦r❡♠ ✸✳✾✳✹✿ ◆♦♥✲♣♦s✐t✐✈❡ ❉✐s❝r✐♠✐♥❛♥t ■❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✱

x2 + px + q ,

❤❛s ❛ ♥♦♥✲♣♦s✐t✐✈❡ ❞✐s❝r✐♠✐♥❛♥t✱ D = p2 − 4q ≤ 0 ,

t❤❡♥ ❡✐t❤❡r ♦❢ ✐ts r♦♦ts✱

√ 1 x = (−p ± D) , 2

❤❛s ✐ts ❛r❣✉♠❡♥t s❛t✐s❢②✿ 

sin arg x =

❛♥❞ ✐ts ♠♦❞✉❧❡ s❛t✐s❢②✿

1 2

s

||x||2 = q .

4−

p2 , q

✸✳✾✳ ▲✐♥❡❛r ♦♣❡r❛t♦rs ✇✐t❤ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s

✷✹✶

Pr♦♦❢✳

❚❤❡ ♠♦❞✉❧✉s ♦❢ t❤❡ r♦♦ts ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ||x||2 = Re x

2

+ Im x

2

 1 1 1 = (p2 + |D|) = (p2 − D) = p2 − (p2 − 4q) = q . 4 4 4

❆♥❞ t❤❡✐r ❛r❣✉♠❡♥t s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿

s s s √ 1 2 − 4q D p p2 Im x −D 1 1 1 − 4− . sin arg x = = 2√ = = = ||x|| q 2 q 2 q 2 q 

❲❡ ❞❡✜♥❡❞ t❤❡ tr❛❝❡ ♦❢ ❛ ♠❛tr✐① ❛s t❤❡ s✉♠ ♦❢ ✐ts ❞✐❛❣♦♥❛❧ ❡❧❡♠❡♥ts✿  a b = a + d. tr c d 

❚❤❡♥ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ t❛❦❡s t❤✐s ❢♦r♠✿  a−λ b χ(λ) = det c d−λ = ad − aλ − λd + λ2 − bc = λ2 − (a + d)λ + (ad − bc) = λ2 − tr F λ + det F. 

❲❡ ♠❛t❝❤ t❤✐s t♦ t❤❡ t❤❡♦r❡♠ ❛❜♦✈❡✿ p = − tr F, q = det F, D = (tr F )2 − 4 det F .

❚❤❡ r♦♦ts ❛r❡ λ1,2 =

√  1 tr F ± D . 2

❲❤❡♥ t❤❡ r♦♦ts ❛r❡ ❝♦♠♣❧❡①✱ t❤❡ ♠♦❞✉❧✉s ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ||λ1,2 || =



det F .

❆♥❞ t❤❡✐r ❛r❣✉♠❡♥t s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿ sin arg λ1,2

❚❤✐s ✐s t❤❡ ✜♥❛❧ r❡s✉❧t✳



1 = 2

r

(tr F )2 − 4. det F

❈♦r♦❧❧❛r② ✸✳✾✳✺✿ ◆♦♥✲♣♦s✐t✐✈❡ ❉✐s❝r✐♠✐♥❛♥t

❙✉♣♣♦s❡ ❛ r❡❛❧ ♠❛tr✐① F s❛t✐s✜❡s✿ D = (tr F )2 − 4 det F ≤ 0 .

❚❤❡♥✱ t❤❡ ♦♣❡r❛t♦r U = F X ❞♦❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ■t r♦t❛t❡s t❤❡ r❡❛❧ ♣❧❛♥❡ t❤r♦✉❣❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛♥❣❧❡✿ θ = sin−1

1 2

r

4 − (tr F )2 det F

!

.

• ■t str❡t❝❤❡s✲s❤r✐♥❦s t❤❡ ♣❧❛♥❡ ✉♥✐❢♦r♠❧② ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝t♦r✿ √ s = det F .

✸✳✶✵✳

❈♦♠♣❧❡① ❝❛❧❝✉❧✉s

✷✹✷

❨❡s✱ ✇❡ ❤❛✈❡ ✐♥❝❧✉❞❡❞ t❤❡ tr❛♥s✐t✐♦♥❛❧ ❝❛s❡ D = 0✦ ■♥❞❡❡❞✱ ✐t ❤❛s t❤❡ s❛♠❡ str❡t❝❤ ❜✉t ♥♦ r♦t❛t✐♦♥✳ ❲❡ ✜♥❛❧❧② ♣✉t t♦❣❡t❤❡r ♦✉r t✇♦ ❝❧❛ss✐✜❝❛t✐♦♥ t❤❡♦r❡♠s✿ t❤❡ ❜❡❤❛✈✐♦r ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✐♥ t❡r♠s ♦❢ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ✕ r❡❛❧ ♦r ❝♦♠♣❧❡① ✕ ♦❢ t❤❡✐r ♠❛tr✐❝❡s✳ ❲❡ ✐❧❧✉str❛t❡ t❤✐s ❝❧❛ss✐✜❝❛t✐♦♥ ❜❡❧♦✇ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ t❤❡ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❘♦♦ts ♦❢ ◗✉❛❞r❛t✐❝ P♦❧②♥♦♠✐❛❧s ✿

✸✳✶✵✳ ❈♦♠♣❧❡① ❝❛❧❝✉❧✉s

❊✈❡♥ t❤♦✉❣❤ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② ♣❧❛♥❡ ✈❡❝t♦rs✱ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥ ❞♦❡s♥✬t ❢♦❧❧♦✇ t❤❡ ✐❞❡❛ ♦❢ t❤❡ ❣r❛❞✐❡♥t✳ ■t r❛t❤❡r ❢♦❧❧♦✇s✱ ❛♥❞ ✐s ✐❞❡♥t✐❝❛❧ t♦✱ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ✉s✉❛❧ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✳ ❲❡ r❡❧② ♦♥ t❤❡ ❢❛❝t t❤❛t t❤❡ ❛❧❣❡❜r❛ ✐s t❤❡ s❛♠❡ ❡✈❡♥ t❤♦✉❣❤ t❤❡ ♥❛t✉r❡ ♦❢ t❤❡ ♥✉♠❜❡rs ✐s ❞✐✛❡r❡♥t✿

❉❡✜♥✐t✐♦♥ ✸✳✶✵✳✶✿ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥ u = f (x) ❛t x = a ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛t x = a ❛s t❤❡ ✐♥❝r❡♠❡♥t ∆x ✐s ❛♣♣r♦❛❝❤✐♥❣ 0✱ ❞❡♥♦t❡❞ ❜②✿

f ′ (x) = lim

x→a

f (x) − f (a) f (x) − f (a) = lim ||x−a||→0 x−a x−a

✐♥ t❤❛t ❝❛s❡ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❝❛❧❧❡❞

❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a✳

❚❤✐s ❢♦r♠✉❧❛ ✐s ♠❛❞❡ ♣♦ss✐❜❧❡ ❜② t❤❡ ❛✈❛✐❧❛❜✐❧✐t② ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥✳

✸✳✶✵✳

❈♦♠♣❧❡① ❝❛❧❝✉❧✉s

✷✹✸

❲❛r♥✐♥❣✦ ❚❤❡ ❢♦r♠✉❧❛ ✐s ♥♦t t❤❡ s❛♠❡ ❛s t❤✐s✿

f (x) − f (a) . x→a ||x − a|| lim

❲❛r♥✐♥❣✦ ❚❤❡ ✐❞❡❛ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s♥✬t ❛❜♦✉t t❤❡ s❧♦♣❡✱ r✐s❡ ♦✈❡r t❤❡ r✉♥✱ ❛♥②♠♦r❡✳

❚❤❡ ❞❡r✐✈❛t✐✈❡ ✐s st✐❧❧ t❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ ♦✉t♣✉t r❡❧❛t✐✈❡ t♦ t❤❡ ✐♥♣✉t✳ ▼❛♥② r❡s✉❧ts ❛r❡ ❢❛♠✐❧✐❛r s✉❝❤ ❛s t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✸✳✶✵✳✷✿ ❉✐✛ ❂❃ ❈♦♥t ■❢ ❛ ❢✉♥❝t✐♦♥ ✐s ❞✐✛❡r❡♥t✐❛❜❧❡✱ ✐t ✐s ❛❧s♦ ❝♦♥t✐♥✉♦✉s✳

❊①❛♠♣❧❡ ✸✳✶✵✳✸✿ ❞❡r✐✈❛t✐✈❡ ❈♦♠♣✉t❛t✐♦♥s ✇♦r❦ ♦✉t ✐♥ t❤❡ ❡①❛❝t❧② t❤❡ s❛♠❡ ♠❛♥♥❡r✳ ▲❡t✬s ❝♦♠♣✉t❡ f ′ (2) ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ❢♦r f (x) = −x2 − x.

❉❡✜♥✐t✐♦♥✿

f (2 + h) − f (2) . h→0 h

f ′ (2) = lim

❚♦ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ ✇❡ ♥❡❡❞ t♦ s✉❜st✐t✉t❡ t✇✐❝❡✿ f (2 + h) = −(2 + h)2 − (2 + h), f (2) = −22 − 2 .

◆♦✇✱ ✇❡ s✉❜st✐t✉t❡ ✐♥t♦ t❤❡ ❞❡✜♥✐t✐♦♥✿ [−(2 − h)2 − (2 − h)] − [−22 − 2] h→0 h 2 −4 − 4h − h − 2 − h + 4 + 2 = lim h→0 h −5h − h2 = lim h→0 h = lim (−5 − h)

f ′ (2) = lim

h→0

= −5 − 0 = 5.

❚❤❡♦r❡♠ ✸✳✶✵✳✹✿ ■♥t❡❣❡r P♦✇❡r ❋♦r♠✉❧❛ (xn )′ = nxn−1

Pr♦♦❢✳ ❚❤❡ ♣r♦♦❢ r❡❧✐❡s ❡♥t✐r❡❧② ♦♥ t❤❡ ❢♦r♠✉❧❛✿ an − bn = (a − b)(an−1 + an−2 b + ... + abn−2 + bn−1 ) .

✸✳✶✶✳ ❙❡r✐❡s ❛♥❞ ♣♦✇❡r s❡r✐❡s

✷✹✹

❊①❡r❝✐s❡ ✸✳✶✵✳✺

❋✐♥✐s❤ t❤❡ ♣r♦♦❢✳

❚❤❡r❡ ✐s ❛ ❝♦✉♥t❡r♣❛rt ❢♦r ❡❛❝❤ r✉❧❡ ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥✦ ❚❤❡♦r❡♠ ✸✳✶✵✳✻✿ ❆❧❣❡❜r❛ ♦❢ ❉❡r✐✈❛t✐✈❡s

❲❤❡r❡✈❡r ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥s f ❛♥❞ g ❛r❡ ❞✐✛❡r❡♥t✐❛❜❧❡✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❙❘✿

(f + g)′ = f ′ + g ′ ❈▼❘✿

P❘✿

(f g)′ = f ′ g + f g ′

◗❘✿

(cf )′ = cf ′ f ′g − f g′ (f /g)′ = g2

❢♦r ❛♥② ❝♦♠♣❧❡① c ♣r♦✈✐❞❡❞ g 6= 0

❲❡ ❝❛♥ ❞✐✛❡r❡♥t✐❛t❡ ❛♥② ♣♦❧②♥♦♠✐❛❧ ❡❛s✐❧② ♥♦✇✳ ❚❤❡♦r❡♠ ✸✳✶✵✳✼✿ ❉❡r✐✈❛t✐✈❡ ♦❢ P♦❧②♥♦♠✐❛❧

❋♦r ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n ❛♥❞ ❛♥② ❝♦♠♣❧❡① ♥✉♠❜❡rs a0 , ..., an ✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ an x n + an−1 xn−1 +an−2 xn−2 +... +a2 x2 +a1 x +a0 = an nxn−1 + an−1 (n − 1)xn−2 +an−2 (n − 2)xn−3 +... +a2 2x +a1

′

❆s ❢❛r ❛s ❛♣♣❧✐❝❛t✐♦♥s ❛r❡ ❝♦♥❝❡r♥❡❞✱ t❤❡r❡ ✐s ♥♦ s✉❝❤ ❛ r❡❧❛t✐♦♥ ❛s ✏❧❡ss✑ ♦r ✏♠♦r❡✑ ❛♠♦♥❣ ❝♦♠♣❧❡① ♥✉♠❜❡rs✦ ❚❤❛t✬s ✇❤② ✇❡ ❞♦♥✬t ❤❛✈❡ t♦ ✇♦rr② ❛❜♦✉t✿ ✶✳ ♠♦♥♦t♦♥✐❝✐t② ✷✳ ❡①tr❡♠❡ ♣♦✐♥ts ✸✳ ❝♦♥❝❛✈✐t②✱ ❡t❝✳ ❏✉st ❛s ❜❡❢♦r❡✱ r❡✈❡rs✐♥❣ ❞✐✛❡r❡♥t✐❛t✐♦♥ ✐s ❝❛❧❧❡❞ ✐♥t❡❣r❛t✐♦♥ ❛♥❞ t❤❡ r❡s✉❧t✐♥❣ ❢✉♥❝t✐♦♥s ❛r❡ ❝❛❧❧❡❞ ❛♥t✐❞❡r✐✈❛✲ ′ t✐✈❡s ❀ F ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ f ✐❢ F = f ✳ ❚❤❡r❡ ✐s ❛ ❝♦✉♥t❡r♣❛rt ❢♦r ❡❛❝❤ r✉❧❡ ♦❢ ✐♥t❡❣r❛t✐♦♥✦ ❚❤❡♦r❡♠ ✸✳✶✵✳✽✿ ❆❧❣❡❜r❛ ♦❢ ❆♥t✐❞❡r✐✈❛t✐✈❡s

❲❤❡r❡✈❡r f ❛♥❞ g ❛r❡ ✐♥t❡❣r❛❜❧❡✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❙❘✿ P❘✿

Z

Z

(f + g) dx = f dg = f g −

Z

Z

f dx + g df

Z

Z

Z

g dx ❈▼❘✿ (cf ) dx = c f dx Z Z 1 ▲❈❘✿ f (mx + b) dx = m f (t) dt

❢ t=mx+b

✸✳✶✶✳ ❙❡r✐❡s ❛♥❞ ♣♦✇❡r s❡r✐❡s

❆❧❧ t❤❡ ❞❡✜♥✐t✐♦♥s ❛♥❞ t❤❡♦r❡♠s ❝♦♥t✐♥✉❡ t♦ ❜❡ ✐❞❡♥t✐❝❛❧ ♦r ✈✐rt✉❛❧❧② ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ♦♥❡s ✐♥ ❱♦❧✉♠❡ ✸ ✭❈❤❛♣t❡r ✸■❈✲✺✮✳



✸✳✶✶✳

✷✹✺

❙❡r✐❡s ❛♥❞ ♣♦✇❡r s❡r✐❡s

❉❡✜♥✐t✐♦♥ ✸✳✶✶✳✶✿ s❡q✉❡♥❝❡ ♦❢ s✉♠s✱ ♣❛rt✐❛❧ s✉♠s ❋♦r ❛ ❣✐✈❡♥ s❡q✉❡♥❝❡ {zn } = {zn : n = s, s + 1, s + 2, ...} ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✱ ✐ts s❡q✉❡♥❝❡ ♦❢ s✉♠s✱ ♦r ♣❛rt✐❛❧ s✉♠s✱ {pn : n = s, s + 1, s + 2, ...} ✐s ❛ s❡q✉❡♥❝❡ ❞❡✜♥❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿ ps = z s ,

pn+1 = pn + zn , n = s, s + 1, s + 2, ...

❉❡✜♥✐t✐♦♥ ✸✳✶✶✳✷✿ s✉♠ ♦❢ s❡r✐❡s ❋♦r ❛ s❡q✉❡♥❝❡ {zn }✱ t❤❡ ❧✐♠✐t S ♦❢ ✐ts s❡q✉❡♥❝❡ ♦❢ ♣❛rt✐❛❧ s✉♠s {pn } ✐s ❝❛❧❧❡❞ ❜② t❤❡ s✉♠ ♦❢ t❤❡ s❡q✉❡♥❝❡ ♦r✱ ♠♦r❡ ❝♦♠♠♦♥❧②✱ t❤❡ s✉♠ ♦❢ t❤❡ s❡r✐❡s✱ ❞❡♥♦t❡❞ ❜②✿ S=

∞ X

zi = lim

n→∞

i=s

n X

zi

i=s

❚❤✐s ❧✐♠✐t ♠✐❣❤t ❛❧s♦ ❜❡ ✐♥✜♥✐t❡✳ ❋r♦♠ t❤❡ ❯♥✐q✉❡♥❡ss

♦❢ ▲✐♠✐t

✇❡ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳

❚❤❡♦r❡♠ ✸✳✶✶✳✸✿ ❯♥✐q✉❡♥❡ss ♦❢ ❙✉♠ ❆ s❡r✐❡s ❝❛♥ ❤❛✈❡ ♦♥❧② ♦♥❡ ❧✐♠✐t ✭✜♥✐t❡ ♦r ✐♥✜♥✐t❡✮❀ ✐✳❡✳✱ ✐❢ a ❛♥❞ b ❛r❡ s✉♠s ♦❢ t❤❡ s❛♠❡ s❡r✐❡s✱ t❤❡♥ a = b✳ ❋r♦♠ t❤❡ ❈♦♠♣♦♥❡♥t✲✇✐s❡

❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❙❡q✉❡♥❝❡s

✇❡ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳

❚❤❡♦r❡♠ ✸✳✶✶✳✹✿ ❈♦♠♣♦♥❡♥t✲✇✐s❡ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❙❡r✐❡s ❆ s❡r✐❡s ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs

∞ X

zi ✐ ❝♦♥✈❡r❣❡s t♦ ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r z ✐❢ ❛♥❞

i=s

♦♥❧② ✐❢ ❜♦t❤ t❤❡ r❡❛❧ ❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ t❤❡ s❡q✉❡♥❝❡ ♦❢ ✭♣❛rt✐❛❧✮ s✉♠s ♦❢ zk ❝♦♥✈❡r❣❡ t♦ t❤❡ r❡❛❧ ❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ z r❡s♣❡❝t✐✈❡❧②❀ ✐✳❡✳✱ ∞ X i=s

zi = z ⇐⇒ Re

∞ X

zi = Re(z) ❛♥❞ Im

i=s

i=s

❲❡ ❝♦♥t✐♥✉❡ ✐♥ t❤❡ s❛♠❡ ♦r❞❡r ❛s ✇✐t❤ r❡❛❧✲✈❛❧✉❡s s❡r✐❡s✿

❉❡✜♥✐t✐♦♥ ✸✳✶✶✳✺✿ ❝♦♥✈❡r❣❡s ❛❜s♦❧✉t❡❧② ❋♦r ❛ s❡q✉❡♥❝❡ {zn }✱ t❤❡ s❡r✐❡s✿

n X

zi ,

i=∞ ❝♦♥✈❡r❣❡s ❛❜s♦❧✉t❡❧②

✐❢ t❤❡ s❡r✐❡s ♦❢ ✐ts ♠♦❞✉❧✐✱ n X

i=∞

❝♦♥✈❡r❣❡s✳ ◆♦✇✱ t❤❡ ❛❧❣❡❜r❛ ♦❢ s❡r✐❡s✳

∞ X

||zi || ,

zi = Im(z) .

✸✳✶✶✳

❙❡r✐❡s ❛♥❞ ♣♦✇❡r s❡r✐❡s

❏✉st ❛s ❜❡❢♦r❡✱ ✇❡ ❝❛♥

✷✹✻

♠✉❧t✐♣❧② ❛ ❝♦♥✈❡r❣❡♥t s❡r✐❡s ❜② ❛ ♥✉♠❜❡r t❡r♠ ❜② t❡r♠✳

❚❤❡♦r❡♠ ✸✳✶✶✳✻✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❙❡r✐❡s ❙✉♣♣♦s❡

{sn }

✐s ❛ s❡q✉❡♥❝❡✳ ❋♦r ❛♥② ✐♥t❡❣❡r

∞ X a

(c · sn ) = c ·

a

❛♥❞ ❛♥② ❝♦♠♣❧❡①

∞ X

c✱

✇❡ ❤❛✈❡✿

sn

a

♣r♦✈✐❞❡❞ t❤❡ s❡r✐❡s ❝♦♥✈❡r❣❡s✳

❏✉st ❛s ❜❡❢♦r❡✱ ✇❡ ❝❛♥

❛❞❞ t✇♦ ❝♦♥✈❡r❣❡♥t s❡r✐❡s t❡r♠ ❜② t❡r♠✳

❚❤❡♦r❡♠ ✸✳✶✶✳✼✿ ❙✉♠ ❘✉❧❡ ❢♦r ❙❡r✐❡s ❙✉♣♣♦s❡

{sn }

❛♥❞

{tn }

❛r❡ s❡q✉❡♥❝❡s✳ ❋♦r ❛♥② ✐♥t❡❣❡r

∞ X

(sn + tn ) =

a

∞ X a

sn +

∞ X

a✱

✇❡ ❤❛✈❡✿

tn

a

♣r♦✈✐❞❡❞ t❤❡ t✇♦ s❡r✐❡s ❝♦♥✈❡r❣❡✳

❊①❡r❝✐s❡ ✸✳✶✶✳✽ Pr♦✈❡ t❤❡s❡ t❤❡♦r❡♠s✳

❉❡✜♥✐t✐♦♥ ✸✳✶✶✳✾✿ ❣❡♦♠❡tr✐❝ s❡r✐❡s ❚❤❡ s❡r✐❡s ♣r♦❞✉❝❡❞ ❜② t❤❡ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ❝♦♠♣❧❡① ♥✉♠❜❡r✮ ∞ X

an = a · rn

✇✐t❤ r❛t✐♦

r

✭❛

arn = ar1 + ar2 + ar3 + ar4 + ... + arn + ...

n=s

✐s ❝❛❧❧❡❞ t❤❡

❣❡♦♠❡tr✐❝ s❡r✐❡s

✇✐t❤ r❛t✐♦

r✳

❊①❛♠♣❧❡ ✸✳✶✶✳✶✵✿ ❝♦♠♣❧❡① ♣♦✇❡r s❡r✐❡s ❚❤✐s ✐s ❛ ✈✐s✉❛❧✐③❛t✐♦♥ ♦❢ ❛ ♣♦✇❡r s❡r✐❡s ♦✈❡r ❝♦♠♣❧❡① ♥✉♠❜❡rs✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ s❡r✐❡s✿

z + z 2 + z 3 + ... ❋♦r s❡✈❡r❛❧ ✈❛❧✉❡s ♦❢ z ✱ t❤❡ s❡q✉❡♥❝❡ ❛♥❞ t❤❡♥ t❤❡ ♣❛rt✐❛❧ s✉♠s ♦❢ t❤❡ s❡r✐❡s ❛r❡ ♣❧♦tt❡❞✳ ❲❡ ❝❛♥ s❡❡ ❤♦✇ st❛rt✐♥❣ ❢r♦♠ ❛ ♣♦✐♥t ✐♥s✐❞❡ t❤❡ ❞✐s❦

||z|| ≤ 1

♣r♦❞✉❝❡s ❞✐✈❡r❣❡♥❝❡ ❛♥❞ ♦✉ts✐❞❡ ♣r♦❞✉❝❡s ❞✐✈❡r❣❡♥❝❡✿

✸✳✶✶✳ ❙❡r✐❡s ❛♥❞ ♣♦✇❡r s❡r✐❡s

✷✹✼

■♥❞❡❡❞✱ t❤❡ s❡r✐❡s

1 + z + z 2 + z 3 + ...

✐s ❛ ❣❡♦♠❡tr✐❝ s❡r✐❡s ✇✐t❤ t❤❡ r❛t✐♦ r = z ✳ ❚❤❡r❡❢♦r❡✱ ✐t ❝♦♥✈❡r❣❡s ❢♦r ❛❧❧ ||z|| < 1✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠✱ ❛♥❞ ❞✐✈❡r❣❡s ❢♦r ❛❧❧ ||z|| > 1✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐t ❝♦♥✈❡r❣❡s ♦♥ t❤❡ ❞✐s❦ ♦❢ r❛❞✐✉s 1 ❝❡♥t❡r❡❞ ❛t 0 ✐♥ C ❛♥❞ ❞✐✈❡r❣❡s ♦✉ts✐❞❡ ♦❢ ✐t✳ ❚❤✐s ❝✐r❝❧❡ ✐s t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❜② t❤❡ s❡r✐❡s✳ ❲❡ ❡✈❡♥ ❤❛✈❡ ❛ ❢♦r♠✉❧❛ ❢♦r t❤✐s ❢✉♥❝t✐♦♥✿ 1 + z + z 2 + z 3 + .. =

❚❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ✐s ✐♥ t❤❡ ❞♦♠❛✐♥✳

1 . 1−z

■ts s✉♠ ✐s ❢♦✉♥❞ t❤❡ s❛♠❡ ✇❛② ❛s ❢♦r r❡❛❧ ✈❛r✐❛❜❧❡✳ ❚❤❡ ♦♥❧② ❞✐✛❡r❡♥❝❡ ✐s t❤❛t t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ✐s r❡♣❧❛❝❡❞ ✇✐t❤ t❤❡ ♠♦❞✉❧✉s✿ ❚❤❡♦r❡♠ ✸✳✶✶✳✶✶✿ ❙✉♠ ♦❢ ●❡♦♠❡tr✐❝ ❙❡r✐❡s ❚❤❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s ✇✐t❤ r❛t✐♦ ✇❤❡♥

||r|| > 1✳

r ❝♦♥✈❡r❣❡s ❛❜s♦❧✉t❡❧② ✇❤❡♥ ||r|| < 1 ❛♥❞ ❞✐✈❡r❣❡s

■♥ t❤❡ ❢♦r♠❡r ❝❛s❡✱ t❤❡ s✉♠ ✐s✿

∞ X

arn =

n=0

a . 1−r

❊①❡r❝✐s❡ ✸✳✶✶✳✶✷

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❊①❛♠♣❧❡ ✸✳✶✶✳✶✸✿ tr✐❣ r❡♣r❡s❡♥t❛t✐♦♥

❚❤❡r❡ ✐s ❛ ❤✐♥t ❛t t❤❡ ✐❞❡❛ ♦❢ ❝♦♠♣❧❡① ♣♦✇❡r s❡r✐❡s ✐♥ ❈❤❛♣t❡r ✹❍❉✲✶✳ ▲❡t✬s ❝♦♠♣❛r❡ t❤❡ ❚❛②❧♦r s❡r✐❡s ♦❢ t❤❡ s✐♥❡✱ t❤❡ ❝♦s✐♥❡✱ ❛♥❞ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✳ ❚❤❡ s✐♥❡ ✐s ♦❞❞✱ ❛♥❞ ✐ts ❚❛②❧♦r s❡r✐❡s ♦♥❧② ✐♥❝❧✉❞❡s ♦❞❞ t❡r♠s✿ ∞ sin x =

X (−1)k x2k+1 . (2k + 1)! k=0

❚❤❡ ❝♦s✐♥❡ ✐s ❡✈❡♥✱ ❛♥❞ ✐ts ❚❛②❧♦r s❡r✐❡s ♦♥❧② ✐♥❝❧✉❞❡s ❡✈❡♥ t❡r♠s✿ cos x =

∞ X (−1)k k=0

(2k)!

x2k .

■❢ ✇❡ ✇r✐t❡ t❤❡♠ ♦♥❡ ✉♥❞❡r t❤❡ ♦t❤❡r✱ ✇❡ s❡❡ ❤♦✇ t❤❡② ❝♦♠♣❧❡♠❡♥t ❡❛❝❤ ♦t❤❡r✿ n 0 1 sin x

2

x

cos x 1

x2 − 2!

3 −

x3 3!

4

x4 4!

5

...

x5 ... 5! ...

❲❤❛t ✐❢ ✇❡ ❛❞❞ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ t♦ t❤✐s❄ n 0 1

2

sin x 0 x

0 x2 2! x2 2!

cos x 1 0 − ex 1 x

3

4

5

x3 3!

0

x5 ... 5!



0 x3 3!

x4 4! x4 4!

0

...

...

x5 ... 5!

✸✳✶✶✳ ❙❡r✐❡s ❛♥❞ ♣♦✇❡r s❡r✐❡s

✷✹✽

❚❤✐s ❧♦♦❦s ❛❧♠♦st ❧✐❦❡ ❛❞❞✐t✐♦♥✦✳✳ ❡①❝❡♣t ❢♦r t❤♦s❡ ♠✐♥✉s s✐❣♥s✳ ❚❤❛t✬s ✇❤❡r❡ i ❝♦♠❡s ✐♥✳ ▲❡t✬s s✉❜st✐t✉t❡ x = it✿ 3 4 5 2 eit

(it) (it) (it) (it) + + + + ··· 2! 3! 4! 5! t2 it3 t4 it5 + + − ··· = 1 + it − − 2! 3! 4!  5!   t 3 x5 t2 t4 = 1 − + − ··· + i t − + − ··· 2! 4! 3! 5! = cos t + i sin t .

= 1 + it +

■t ✐s ❝❛❧❧❡❞ ❊✉❧❡r✬s ❢♦r♠✉❧❛✳ ▼♦r❡ ❣❡♥❡r❛❧ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ea+bi = ea (cos b + i sin b) .

■♥ ❝♦♠♣❧❡① ❝❛❧❝✉❧✉s✱ ❢✉♥❝t✐♦♥s ❛r❡ ♠♦r❡ ✐♥t❡rr❡❧❛t❡❞✦

❊①❡r❝✐s❡ ✸✳✶✶✳✶✹ ❙❤♦✇ t❤❛t sin2 x + cos2 x = 1✳

❉❡✜♥✐t✐♦♥ ✸✳✶✶✳✶✺✿ ♣♦✇❡r s❡r✐❡s ❆ s❡q✉❡♥❝❡ {qn } ♦❢ ♣♦❧②♥♦♠✐❛❧s ❣✐✈❡♥ ❜② ❛ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿ qk+1 (x) = qk (x) + ck+1 (x − a)k+1 , k = 0, 1, 2, ... ,

❢♦r s♦♠❡ ✜①❡❞ ✭❝♦♠♣❧❡①✮ ♥✉♠❜❡r a ❛♥❞ ❛ s❡q✉❡♥❝❡ ♦❢ ✭❝♦♠♣❧❡①✮ ♥✉♠❜❡rs {ck }✱ ✐s ❝❛❧❧❡❞ ❛ ♣♦✇❡r s❡r✐❡s ❝❡♥t❡r❡❞ ❛t a✳ ❚❤❡ ❢✉♥❝t✐♦♥ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❧✐♠✐t ♦❢ qn ✐s ❝❛❧❧❡❞ t❤❡ s✉♠ ♦❢ t❤❡ s❡r✐❡s✱ ✇r✐tt❡♥ ❛s✿ f (x) = c0 + c1 (x − a) + c2 (x − a)2 + ... =

X k

ck (x − a)k = lim qk (x) k→∞

❢♦r ❛❧❧ x ❢♦r ✇❤✐❝❤ t❤❡ ❧✐♠✐t ❡①✐sts✳ ❚❤❡♥ t❤❡ t❤r❡❡ ♣♦✇❡r s❡r✐❡s ✐♥ t❤❡ ❡①❛♠♣❧❡ ❛❜♦✈❡ ♠❛② s❡r✈❡ ❛s t❤❡ ❞❡✜♥✐t✐♦♥s ♦❢ t❤❡s❡ t❤r❡❡ ❢✉♥❝t✐♦♥s ♦❢ ❝♦♠♣❧❡① ✈❛r✐❛❜❧❡✿ n 0 sin x =

1

2

x

3 −

4

x3 3!

5 +

...

x5 + ... 5!

x2 x4 + − ... 2! 4! x2 x3 x4 x5 + + + + ..., ex = 1 +x + 2! 3! 4! 5!

cos x = 1



♣❡♥❞✐♥❣ t❤❡ ♣r♦♦❢ ♦❢ t❤❡✐r ❝♦♥✈❡r❣❡♥❝❡✳ ❲❡ ✇✐❧❧ ❛❝❝❡♣t t❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐t❤♦✉t ♣r♦♦❢✳

❚❤❡♦r❡♠ ✸✳✶✶✳✶✻✿ ❲❡✐❡rstr❛ss ▼✲❚❡st

❈♦♥s✐❞❡r t❤❡ ♣♦✇❡r s❡r✐❡s

∞ X n=0

cn (z − a)n .

✸✳✶✶✳

❙❡r✐❡s ❛♥❞ ♣♦✇❡r s❡r✐❡s

✷✹✾

❙✉♣♣♦s❡ t❤❡r❡ ❡①✐sts s✉❝❤ ❛ s❡q✉❡♥❝❡ ♦❢ ♥♦♥✲♥❡❣❛t✐✈❡ r❡❛❧ ♥✉♠❜❡rs

{Mn }

t❤❛t

|cn (z − a)n | ≤ Mn , n = 0, 1, 2, ... , ❢♦r ❛❧❧

z

✐♥ ❛♥ ♦♣❡♥ ❞✐s❦

D

❛r♦✉♥❞

a

❛♥❞ t❤❡ s❡r✐❡s

∞ X n=0

❝♦♥✈❡r❣❡s✳ ❚❤❡♥ t❤❡ ♣♦✇❡r s❡r✐❡s ❝♦♥✈❡r❣❡s ✉♥✐❢♦r♠❧② ♦♥

Mn

✭♦❢ r❡❛❧ ♥✉♠❜❡rs✮

D✳

❈♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ t❤r❡❡ s❡r✐❡s ❛❜♦✈❡ ❢♦❧❧♦✇s✳

❉❡✜♥✐t✐♦♥ ✸✳✶✶✳✶✼✿ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❚❤❡ ❣r❡❛t❡st ❧♦✇❡r ❜♦✉♥❞

r

✭t❤❛t ❝♦✉❧❞ ❜❡ ✐♥✜♥✐t❡✮ ♦❢ t❤❡ ❞✐st❛♥❝❡s ❢r♦♠

❛ ♣♦✐♥t ❢♦r ✇❤✐❝❤ ❛ ♣♦✇❡r s❡r✐❡s ❝❡♥t❡r❡❞ ❛t

❝♦♥✈❡r❣❡♥❝❡

a

❞✐✈❡r❣❡s ✐s ❝❛❧❧❡❞ t❤❡

a

t♦

r❛❞✐✉s ♦❢

♦❢ t❤❡ s❡r✐❡s✳

❚❤✐s ❞❡✜♥✐t✐♦♥ ✐s ❧❡❣✐t✐♠❛t❡ ❛❝❝♦r❞✐♥❣ t♦ t❤❡

❊①✐st❡♥❝❡ ♦❢ sup ❚❤❡♦r❡♠

❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✶✳

❲❡ ✜♥❛❧❧② ❝❛♥ s❡❡ ✇❤❡r❡ t❤❡ ✇♦r❞ ✏r❛❞✐✉s✑ ❝♦♠❡s ❢r♦♠✳

❚❤❡♦r❡♠ ✸✳✶✶✳✶✽✿ ❘❛❞✐✉s ♦❢ ❈♦♥✈❡r❣❡♥❝❡ ❙✉♣♣♦s❡

r

✐s t❤❡ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛ ♣♦✇❡r s❡r✐❡s ∞ X n=0

cn (x − a)n .

❚❤❡♥ ✇❡ ❤❛✈❡✿ ✶✳ ❲❤❡♥

r < ∞✱ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ s❡r✐❡s ✐s ❛ ❞✐s❦ B(a, R) ✐♥ C ♦❢ r❛❞✐✉s r a ✇✐t❤ s♦♠❡ ♣♦✐♥ts ♦♥ ✐ts ❜♦✉♥❞❛r② ♣♦ss✐❜❧② ✐♥❝❧✉❞❡❞✳ r = ∞✱ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ s❡r✐❡s ✐s t❤❡ ✇❤♦❧❡ C✳

❝❡♥t❡r❡❞ ❛t ✷✳ ❲❤❡♥

✸✳✶✷✳

❙♦❧✈✐♥❣ ❖❉❊s ✇✐t❤ ♣♦✇❡r s❡r✐❡s

✷✺✵

❊①❛♠♣❧❡ ✸✳✶✶✳✶✾✿ ❞♦♠❛✐♥s ❆ ❢❡✇ ❜❛s✐❝ s❡r✐❡s ❛s ❢✉♥❝t✐♦♥s✿

s❡r✐❡s

∞ X

s✉♠

❞♦♠❛✐♥

1 1−x

xk

=

(−1)k k x k!

= ex

k=0 ∞ X k=0

B(0, 1) C

∞ X (−1)k 2k+1 x = sin x (2k + 1)! k=0 ∞ X (−1)k 2k x = cos x (2k)! k=0

C C

✸✳✶✷✳ ❙♦❧✈✐♥❣ ❖❉❊s ✇✐t❤ ♣♦✇❡r s❡r✐❡s

❘❡❝❛❧❧ ❢r♦♠ ❈❤❛♣t❡r ✸■❈✲✺ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t r❡s✉❧ts✳

❚❤❡♦r❡♠ ✸✳✶✷✳✶✿ ❯♥✐q✉❡♥❡ss ♦❢ P♦✇❡r ❙❡r✐❡s ■❢ t✇♦ ♣♦✇❡r s❡r✐❡s ❛r❡ ❡q✉❛❧✱ ❛s ❢✉♥❝t✐♦♥s✱ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ (a−r, a+r), r > 0✱ t❤❡♥ t❤❡✐r ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦❡✣❝✐❡♥ts ❛r❡ ❡q✉❛❧✱ ✐✳❡✳✱ ∞ X n=0

cn (x − a)

n

=

∞ X n=0

dn (x − a)n ❢♦r ❛❧❧ a − r < x < a + r

cn = dn ❢♦r ❛❧❧ n = 0, 1, 2, 3, ...

=⇒

❚❤❡♦r❡♠ ✸✳✶✷✳✷✿ ❚❡r♠✲❜②✲❚❡r♠ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ■♥t❡❣r❛t✐♦♥ ❙✉♣♣♦s❡ R > 0 ✐s t❤❡ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛ ♣♦✇❡r s❡r✐❡s f (x) =

∞ X n=0

an (x − a)n .

❚❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ f r❡♣r❡s❡♥t❡❞ ❜② t❤✐s ♣♦✇❡r s❡r✐❡s ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ✭❛♥❞✱ t❤❡r❡✲ ❢♦r❡✱ ✐♥t❡❣r❛❜❧❡✮ ♦♥ t❤❡ ♦♣❡♥ ❞✐s❦ |x−a| < R ❛♥❞ t❤❡ ♣♦✇❡r s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ✐ts ❞❡r✐✈❛t✐✈❡ ❛♥❞ ✐ts ❛♥t✐❞❡r✐✈❛t✐✈❡ ❝♦♥✈❡r❣❡ ♦♥ t❤✐s ❞✐s❦ ❛♥❞ ❛r❡ ❢♦✉♥❞ ❜② t❡r♠ ❜② t❡r♠ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ✐♥t❡❣r❛t✐♦♥ ♦❢ t❤❡ ♣♦✇❡r s❡r✐❡s ♦❢ f ✱ ✐✳❡✳✱ ′

f (x) =

∞ X n=0

❛♥❞ Z

f (x) dx =

Z

cn (x − a)

∞ X n=0

n

!′

cn (x − a)

n

=

!

∞ X n=0

n ′

(cn (x − a) ) =

dx =

∞ Z X n=0

∞ X n=1

ncn (x − a)n−1 ,

∞ X cn (x−a)n+1 . cn (x−a) dx = n+1 n=0 n

✸✳✶✷✳

❙♦❧✈✐♥❣ ❖❉❊s ✇✐t❤ ♣♦✇❡r s❡r✐❡s

✷✺✶

❊①❛♠♣❧❡ ✸✳✶✷✳✸✿ ❖❉❊s ♦❢ ✜rst ♦r❞❡r ❙✉♣♣♦s❡ ✇❡ ♥❡❡❞ t♦ s♦❧✈❡ t❤✐s ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ ✭✇❡ ♣r❡t❡♥❞ ✇❡ ❞♦♥✬t ❦♥♦✇ t❤❡ ❛♥s✇❡r✮✿ y ′ = ky, y(0) = y0 .

❚❤❡ s♦❧✉t✐♦♥ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ♦♥❡ ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✸■❈✲✺✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥ y = y(x) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ t❡r♠✲❜②✲t❡r♠ ❞✐✛❡r❡♥t✐❛❜❧❡ ♣♦✇❡r s❡r✐❡s✳ ❲❡ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ s❡r✐❡s ❛♥❞ t❤❡♥ ♠❛t❝❤ t❤❡ t❡r♠s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❡q✉❛t✐♦♥✿ y y′ =⇒ y′ || k·y

= =

c0

=

c1 || = kc0

+ c2 x2 + 2c2 x ւ ւ + 2c2 x + 3c3 x2 || || + kc1 x + kc2 x2 +

c1 x c1

+ c3 x3 + 3c3 x2 ւ + ... +

...

+ + ... +

... ...

ncn xn−1 || + kcn−1 xn−1

+ cn xn + ... n−1 + ncn x + ... ւ + (n + 1)cn+1 xn +... || + kcn xn +...

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❯♥✐q✉❡♥❡ss ♦❢ P♦✇❡r ❙❡r✐❡s✱ t❤❡ ❝♦❡✣❝✐❡♥ts ❤❛✈❡ t♦ ♠❛t❝❤✦ ❚❤✉s✱ ✇❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡ ♦❢ ❡q✉❛t✐♦♥s✿ c1 2c2 3c3 ... (n + 1)cn+1 +... || || || || kc0 kc1 kc2 ... kcn +...

❲❡ ❝❛♥ st❛rt s♦❧✈✐♥❣ t❤❡s❡ ❡q✉❛t✐♦♥s ❢r♦♠ ❧❡❢t t♦ r✐❣❤t✿ c1 =⇒ c1 = kc0 2c2 =⇒ c2 = k 2 c0 /2 3c3 ... || || || kc0 =⇒ kc1 = k 2 c0 =⇒ kc2 = k 3 c0 /2 ...

❚❤❡ ❝♦♥❞✐t✐♦♥ y(0) = y0 ♠❡❛♥s t❤❛t c0 = y0 ✳ ❚❤❡r❡❢♦r❡✱ cn = y0

kn . n!

❲❡ r❡❝♦❣♥✐③❡ t❤❡ r❡s✉❧t✐♥❣ s❡r✐❡s✿ ∞ X 1 kn n (kx)n = y0 ekx . y= y0 x = y0 n! n! n=0 n=0 ∞ X

◆♦t❡ t♦ ❣❡t t❤❡ nt❤ t❡r♠ ✇❡ s♦❧✈❡ ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡ ❛✉❣♠❡♥t❡❞ ♠❛tr✐①✿   k −1 0 0 0 ...  0 k −2 0 0 ...   0 0 k −3 0 ...   ... ... ... ... ... ... 0 0 0 0 0 ...

0 0 0 0 0 0 ... ... k −n

0 0 0 ... 0

  .  

❲❛r♥✐♥❣✦ ❘❡❝♦❣♥✐③✐♥❣ ♣❡❝t❡❞✳

❊①❡r❝✐s❡ ✸✳✶✷✳✹ ❙♦❧✈❡ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠✿

y ′ = ky + 1, y(0) = y0 .

t❤❡

r❡s✉❧t✐♥❣

s❡r✐❡s

✐s♥✬t

t♦

❜❡

❡①✲

✸✳✶✷✳

❙♦❧✈✐♥❣ ❖❉❊s ✇✐t❤ ♣♦✇❡r s❡r✐❡s

✷✺✷

❊①❛♠♣❧❡ ✸✳✶✷✳✺✿ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r

❙✉♣♣♦s❡ ✇❡ ♥❡❡❞ t♦ s♦❧✈❡ t❤✐s ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠✿ y ′′ = −y, y(0) = y0 , y ′ (0) = v0 .

❆❣❛✐♥✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥ y = y(x) ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ t❡r♠✲❜②✲t❡r♠ ❞✐✛❡r❡♥t✐❛❜❧❡ ♣♦✇❡r s❡r✐❡s✳ ❲❡ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ s❡r✐❡s t✇✐❝❡ ❛♥❞ t❤❡♥ ♠❛t❝❤ t❤❡ t❡r♠s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❡q✉❛t✐♦♥✿ y y′ y ′′ =⇒ y ′′ || −y

= = =

c0

+

c1 x c1

+ + +

c2 x2 2c2 x 2c2

c3 x3 3c3 x2 3 · 2c3 x

+ + +

+ c4 x4 + c5 x5 + ... 3 + 4cn x + 5cn x4 + ... + 4 · 3cn x2 + 5 · 4cn x3 + ...

= 2c2 + 3 · 2c3 x + 4 · 3c4 x2 + 5 · 4cn x3 + || || || = −c0 + −c1 x + −c2 x2 + −c3 x3 +

... ...

❚❤❡ ❝♦❡✣❝✐❡♥ts ❤❛✈❡ t♦ ♠❛t❝❤✿ 2c2 3 · 2c3 4 · 3c4 5 · 4c5 ... n(n − 1)cn ... || || || || −c0 −c1 −c2 −c3 ... −cn−2 ...

❲❡ ❝❛♥ st❛rt s♦❧✈✐♥❣ t❤❡s❡ ❡q✉❛t✐♦♥s ❢r♦♠ ❧❡❢t t♦ r✐❣❤t✱ ♦❞❞ s❡♣❛r❛t❡ ❢r♦♠ ❡✈❡♥✳ ❋✐rst✱ ❢♦r ❡✈❡♥ n✿ 2c2 =⇒ c2 = c0 /2 4 · 3c4 =⇒ c4 = −c0 /(4 · 3 · 2) ... =⇒ cn = ±c0 /n! ... || || −c0 =⇒ −c2 = −c0 /2

❚❤❡ ❝♦♥❞✐t✐♦♥ y(0) = y0 ♠❡❛♥s t❤❛t c0 = y0 ✳ ❚❤❡r❡❢♦r❡✱ n ❡✈❡♥ =⇒ cn = (−1)n/2+1

y0 . n!

❲❡ r❡❝♦❣♥✐③❡ t❤❡ r❡s✉❧t✐♥❣ s❡r✐❡s✿ ∞ X

❡✈❡♥✱

(−1)

n/2+1 y0

n=0

n!

n

x = y0

∞ X

❡✈❡♥

(−1)n/2+1

n=0

1 n x = y0 cos x . n!

❙❡❝♦♥❞✱ ❢♦r ♦❞❞ n✿ 3 · 2c3 ⇒ c3 = c1 /(3 · 2) 5 · 4c5 ⇒ c5 = −c1 /(5 · 4 · 3 · 2) ... ⇒ cn = ±c1 /n! || || −c1 ⇒ −c3 = −c1 /(3 · 2)

❚❤❡ ❝♦♥❞✐t✐♦♥ y ′ (0) = v0 ♠❡❛♥s t❤❛t c1 = v0 ✳ ❚❤❡r❡❢♦r❡✱ n ♦❞❞ =⇒ cn = (−1)(n+1)/2

v0 . n!

❲❡ r❡❝♦❣♥✐③❡ t❤❡ r❡s✉❧t✐♥❣ s❡r✐❡s✿ y=

∞ X

♦❞❞

n=1

(−1)

(n+1)/2 v0

n!

n

x = v0

∞ X

♦❞❞✱

(−1)(n+1)/2

n=1

❚❤✐s ✐s t❤❡ r❡s✉❧t✿ y = y0 cos x + v0 sin x .

1 n x = v0 sin x . n!

✸✳✶✷✳

❙♦❧✈✐♥❣ ❖❉❊s ✇✐t❤ ♣♦✇❡r s❡r✐❡s

✷✺✸

❊①❡r❝✐s❡ ✸✳✶✷✳✻ Pr♦✈✐❞❡ t❤❡ ❛✉❣♠❡♥t❡❞ ♠❛tr✐① ❢♦r t❤✐s s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✳

❊①❡r❝✐s❡ ✸✳✶✷✳✼ ❙♦❧✈❡ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠✿

y ′′ + xy ′ + y = 0, y(0) = 1, y ′ (0) = 1 . ❲❤❡♥ t❤❡ r❡s✉❧t✐♥❣ s❡r✐❡s ✐s♥✬t r❡❝♦❣♥✐③❛❜❧❡✱ ✐t ✐s ✉s❡❞ t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❛♥s✇❡r ✈✐❛ ✐ts ♣❛rt✐❛❧ s✉♠s✳ ❚❤❡ ❛❝❝✉r❛❝② ♦❢ t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❣✐✈❡♥ ❜② t❤❡ ❡rr♦r ❜♦✉♥❞ ❢♦r ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s ❣✐✈❡♥ ✐♥ ❈❤❛♣t❡r ✹❍❉✲✶✱ ❛s ❢♦❧❧♦✇s✳

❚❤❡♦r❡♠ ✸✳✶✷✳✽✿ ❊rr♦r ❇♦✉♥❞

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ y = y(x) ✐s (n + 1) t✐♠❡s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t ❢♦r ❡❛❝❤ i = 0, 1, 2, ..., n + 1✱ ✇❡ ❤❛✈❡ |y (i) (t)| < Ki ❢♦r ❛❧❧ t ❜❡t✇❡❡♥ a ❛♥❞ x ,

❛♥❞ s♦♠❡ r❡❛❧ ♥✉♠❜❡r Ki ✳ ❚❤❡♥ en (x) = |y(x) − Tn (x)| ≤ Kn+1

✇❤❡r❡ Tn ✐s t❤❡ nt❤ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧ ♦❢ y ✳

|x − a|n+1 , (n + 1)!

❈❤❛♣t❡r ✹✿ ❙②st❡♠s ♦❢ ❖❉❊s

❈♦♥t❡♥ts

✹✳✶ P❛r❛♠❡tr✐❝ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺✹ ✹✳✷ ❚❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻✸ ✹✳✸ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻✼ ✹✳✹ ❙♦❧✈✐♥❣ t❤❡ ▲♦t❦❛✕❱♦❧t❡rr❛ ❡q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼✵ ✹✳✺ ❱❡❝t♦r ✜❡❧❞s ❛♥❞ s②st❡♠s ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼✸ ✹✳✻ ❉✐s❝r❡t❡ s②st❡♠s ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼✾ ✹✳✼ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ s②st❡♠s ♦❢ ❖❉❊s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽✸ ✹✳✽ ❚❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥ ❛♥❞ ❧✐♥❡❛r s②st❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽✼ ✹✳✾ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾✸ ✹✳✶✵ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾✽

✹✳✶✳ P❛r❛♠❡tr✐❝ ❝✉r✈❡s

❋✉♥❝t✐♦♥s ♣r♦❝❡ss ❛♥ ✐♥♣✉t ♦❢ ❛♥② ♥❛t✉r❡ ❛♥❞ ♣r♦❞✉❝❡ ❛♥ ♦✉t♣✉t ♦❢ ❛♥② ♥❛t✉r❡✳ ■♥ ❣❡♥❡r❛❧✱ ✇❡ r❡♣r❡s❡♥t ❛ ❢✉♥❝t✐♦♥ ❞✐❛❣r❛♠♠❛t✐❝❛❧❧② ❛s ❛ ❜❧❛❝❦ ❜♦① t❤❛t ♣r♦❝❡ss❡s t❤❡ ✐♥♣✉t ❛♥❞ ♣r♦❞✉❝❡s t❤❡ ♦✉t♣✉t✿ ✐♥♣✉t ❢✉♥❝t✐♦♥ ♦✉t♣✉t x

7→

f

7→

y

❋✉♥❝t✐♦♥s ✐♥ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡s t❛❦❡ ♣♦✐♥ts ♦r ✈❡❝t♦rs ❛s t❤❡ ✐♥♣✉t ❛♥❞ ♣r♦❞✉❝❡ ♣♦✐♥ts ♦r ✈❡❝t♦rs ♦❢ ✈❛r✐♦✉s ❞✐♠❡♥s✐♦♥s ❛s t❤❡ ♦✉t♣✉t✳ ❲❡ ❝❛♥ s❛② t❤❛t t❤❡ ✐♥♣✉t X ✐s ✐♥ Rn ❛♥❞ t❤❡ ♦✉t♣✉t U = F (X) ♦❢ X ✐s ✐♥ Rm ✿ F : P

✐♥ R

n

7→ U

✐♥ Rm

❚❤❡♥✱ t❤❡ ❞♦♠❛✐♥ ♦❢ s✉❝❤ ❛ ❢✉♥❝t✐♦♥ ✐s ✐♥ Rn ❛♥❞ t❤❡ r❛♥❣❡ ✭✐♠❛❣❡✮ ✐s ✐♥ Rm ✳ ❚❤❡ ❞♦♠❛✐♥ ❝❛♥ ❜❡ ❧❡ss t❤❛♥ t❤❡ ✇❤♦❧❡ s♣❛❝❡✳ ■♥ ❢❛❝t✱ t❤❡ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ❞❡✜♥❡❞ ♦♥ t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ s✉❜s❡t ♦❢ Rn ✳ ❇❡❧♦✇ ✇❡ ✐❧❧✉str❛t❡ t❤❡ ❢♦✉r ♣♦ss✐❜✐❧✐t✐❡s ❢♦r n = 1, 2 ❛♥❞ m = 1, 2✿

✹✳✶✳

✷✺✺

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ✉s✉❛❧ ❢✉♥❝t✐♦♥s✱ ✇❡ s❡❡ • ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ P : t 7→ (x, y) ♦r P : t 7→< x, y >✱

• ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s F : (x, y) 7→ z ♦r F :< x, y >7→ z ✱ ❛♥❞

• ❛ ✈❡❝t♦r ✜❡❧❞ ♦♥ t❤❡ ♣❧❛♥❡ V : (x, y) 7→< u, v >✳

❲❡ ♥❡❡❞ t♦ ❧❡❛r♥ ❤♦✇✱ ✐♥st❡❛❞ ♦❢ tr❡❛t✐♥❣ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ♦♥❡ ❛①✐s ❛t ❛ t✐♠❡✱ t♦ st✉❞② t❤❡♠ ❛s ❢✉♥❝t✐♦♥s ✇✐t❤ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts✳ ❱❡❝t♦r ❛❧❣❡❜r❛ ✇✐❧❧ ❜❡ ❡s♣❡❝✐❛❧❧② ✉s❡❢✉❧✳ ❲❡ ✇✐❧❧ r❡❢❡r t♦ ❛s ❛

♣❛r❛♠❡tr✐❝ ❝✉r✈❡

t♦

• ❛♥② ❢✉♥❝t✐♦♥ ♦❢ t❤❡ r❡❛❧ ✈❛r✐❛❜❧❡✱ ✐✳❡✳✱ t❤❡ ❞♦♠❛✐♥ ❧✐❡s ✐♥s✐❞❡ R✱ ❛♥❞

• ✇✐t❤ ✐ts ✈❛❧✉❡s ✐♥ Rm ❢♦r s♦♠❡ m = 1, 2, 3...✳

■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✇✐❧❧ ❧✐♠✐t ♦✉rs❡❧✈❡s t♦ t❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ✈✐❛ ♠♦t✐♦♥✳ ❚❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ✐s t❤❡♥ t✐♠❡ ❛♥❞ t❤❡ ✈❛❧✉❡ ✐s t❤❡ ❧♦❝❛t✐♦♥✳ ❆ ♣♦✐♥t ✐s t❤❡ s✐♠♣❧❡st ❝✉r✈❡✳ ❙✉❝❤ ❛ ❝✉r✈❡ ✇✐t❤ ♥♦ ♠♦t✐♦♥ ✐s ♣r♦✈✐❞❡❞ ❜② ❛ ❝♦♥st❛♥t ❆ str❛✐❣❤t

❧✐♥❡

❢✉♥❝t✐♦♥✳

✐s t❤❡ s❡❝♦♥❞ s✐♠♣❧❡st ❝✉r✈❡✳

❲❡ st❛rt ✇✐t❤ ❧✐♥❡s ✐♥ R2 ✳ ❲❡ ❛❧r❡❛❞② ❦♥♦✇ ❤♦✇ t♦ r❡♣r❡s❡♥t str❛✐❣❤t ❧✐♥❡s ♦♥ t❤❡ ♣❧❛♥❡❀ t❤❡ ✜rst ♠❡t❤♦❞ ✐s t❤❡ s❧♦♣❡✲✐♥t❡r❝❡♣t ❢♦r♠ ✿ y = mx + b .

❚❤✐s ♠❡t❤♦❞ ❞♦❡s ♥♦t ✐♥❝❧✉❞❡ ✈❡rt✐❝❛❧ ❧✐♥❡s✿ t❤❡ s❧♦♣❡ ✐s ✐♥✜♥✐t❡✦ ■♥ ♦✉r st✉❞② ♦❢ ❝✉r✈❡s ✭s♣❡❝✐✜❝❛❧❧② t♦ r❡♣r❡s❡♥t ♠♦t✐♦♥✮ ♦♥ t❤❡ ♣❧❛♥❡✱ t❤❡r❡ ❛r❡ ♥♦ ♣r❡❢❡rr❡❞ ❞✐r❡❝t✐♦♥s ❛♥❞ t❤❡♥ ✐t ✐s ✉♥❛❝❝❡♣t❛❜❧❡ t♦ ❡①❝❧✉❞❡ ❛♥② str❛✐❣❤t ❧✐♥❡s✳ ❚❤❡ s❡❝♦♥❞ ♠❡t❤♦❞ ✐s ✐♠♣❧✐❝✐t ✿ px + qy = r .

❚❤❡ ❝❛s❡ ♦❢ p 6= 0, q = 0 ❣✐✈❡s ✉s ❛ ✈❡rt✐❝❛❧ ❧✐♥❡✳ ❚❤❡ t❤✐r❞ ♠❡t❤♦❞ ✐s ✐♥t❡r♣r❡t❛t✐♦♥✳

♣❛r❛♠❡tr✐❝✳

■t ❤❛s ❛ ❞②♥❛♠✐❝

✹✳✶✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✷✺✻

❊①❛♠♣❧❡ ✹✳✶✳✶✿ str❛✐❣❤t ♠♦t✐♦♥

❙✉♣♣♦s❡ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ tr❛❝❡ t❤❡ ❧✐♥❡ t❤❛t st❛rts ❛t t❤❡ ♣♦✐♥t (1, 3) ❛♥❞ ♣r♦❝❡❡❞s ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r < 2, 3 >✳

❲❡ ✉s❡ ♠♦t✐♦♥ ❛s ❛ st❛rt✐♥❣ ♣♦✐♥t ❛♥❞ ❛s ✇❡❧❧ ❛s ❛ ♠❡t❛♣❤♦r ❢♦r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✱ ❛s ❢♦❧❧♦✇s✳ ❲❡ st❛rt ♠♦✈✐♥❣ ✭t = 0✮ • ❢r♦♠ t❤❡ ♣♦✐♥t P0 = (1, 3)✱ • ✉♥❞❡r ❛ ❝♦♥st❛♥t ✈❡❧♦❝✐t② ♦❢ V =< 2, 3 >✳ ❲❡ ♠♦✈❡ • 2 ❢❡❡t ♣❡r s❡❝♦♥❞ ❤♦r✐③♦♥t❛❧❧②✱ ❛♥❞ • 3 ❢❡❡t ♣❡r s❡❝♦♥❞ ✈❡rt✐❝❛❧❧②✳

■♥ t❡r♠s ♦❢ ✈❡❝t♦rs✱ ✐❢ ✇❡ ❛r❡ ❛t ♣♦✐♥t P ♥♦✇✱ ✇❡ ✇✐❧❧ ❜❡ ❛t ♣♦✐♥t P + V ❛❢t❡r ♦♥❡ s❡❝♦♥❞✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❛r❡ ❛t P1 = P0 + V = (1, 3)+ < 2, 3 >= (3, 6) ❛t t✐♠❡ t = 1✳ ❲❡ t❤❡♥ ❤❛✈❡ ❛❧r❡❛❞② t✇♦ ♣♦✐♥ts ♦♥ ♦✉r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ P ✿ P (0) = P0 = (1, 3) ❛♥❞ P (1) = P1 = (3, 6) .

❖❢ ❝♦✉rs❡✱ t❤❡s❡ ❛r❡ ❛❧s♦ t❤❡ ✈❛❧✉❡s ♦❢ ♦✉r ❢✉♥❝t✐♦♥✳ ▲❡t✬s ✜♥❞ t❤❡ ❢♦r♠✉❧❛s ❢♦r t❤✐s ❢✉♥❝t✐♦♥✳ ❊❛r❧② ♦♥✱ ✐t✬s ❖❑ t♦ ❞♦ t❤✐s ❝♦♠♣♦♥❡♥t✲✇✐s❡✳ ❚❤❡♥ P (t) = (x(t), y(t)) ,

❛♥❞

x(0) = 1, x(1) = 3 ❛♥❞ y(0) = 3, y(1) = 6 .

❚❤❡s❡ ❢✉♥❝t✐♦♥s ♠✉st ❜❡ ❧✐♥❡❛r❀ t❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿ x(t) = 1 + 2t ❛♥❞ y(t) = 3 + 3t .

❚❤✐s ✐s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳✳✳ ❜✉t ♥♦t ❛♥ ❛❝❝❡♣t❛❜❧❡ ❛♥s✇❡r ✐❢ ✇❡ ❛r❡ t♦ ❧❡❛r♥ ❤♦✇ t♦ ✉s❡ ✈❡❝t♦rs✦ ❚❤❡ ❢♦✉r ❝♦❡✣❝✐❡♥ts✱ ♦❢ ❝♦✉rs❡✱ ❝♦♠❡ ❢r♦♠ t❤❡ s♣❡❝✐✜❝ ♥✉♠❜❡rs t❤❛t ❣✐✈❡ ✉s P0 ❛♥❞ V ✳ ▲❡t✬s ❛ss❡♠❜❧❡ t❤❡ t✇♦ ❝♦♦r❞✐♥❛t❡ ❢✉♥❝t✐♦♥ ✐♥t♦ ♦♥❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿ P (t) = (x(t), y(t)) = (1 + 2t, 3 + 3t) .

✹✳✶✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✷✺✼

❚❤✐s ✐s st✐❧❧ ♥♦t ❣♦♦❞ ❡♥♦✉❣❤❀ ✇❡ st✐❧❧ ❝❛♥✬t s❡❡ t❤❡ P0 ❛♥❞ V ❞✐r❡❝t❧②✦ ❲❡ ❝♦♥t✐♥✉❡ ❜② ✉s✐♥❣ ✈❡❝t♦r ❛❧❣❡❜r❛✿ ❲❡ ✉s❡ ✈❡❝t♦r ❛❞❞✐t✐♦♥✳ P (t) = (1 + 2t, 3 + 3t) = (1, 3)+ < 2t, 3t > ❚❤❡♥ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ = (1, 3) + t < 2, 3 > ❆♥❞ ✜♥❛❧❧② s✉❜st✐t✉t❡✳ = P0 + tV.

❲❡ ❤❛✈❡ ❞✐s❝♦✈❡r❡❞ ❛ ✈❡❝t♦r r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ str❛✐❣❤t ♠♦t✐♦♥✿ P (t) = P0 + tV ,

✇❤❡r❡ P0 ✐s t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ ❛♥❞ V ✐s t❤❡ ✭❝♦♥st❛♥t✮ ✈❡❧♦❝✐t②✳

❲❛r♥✐♥❣✦ ❖♥❡ ❝❛♥✱ ♦❢ ❝♦✉rs❡✱ ♠♦✈❡ ❛❧♦♥❣ ❛ str❛✐❣❤t ❧✐♥❡ ❛t ❛

✈❛r✐❛❜❧❡

✈❡❧♦❝✐t②✳

❙♦✱ ♣♦s✐t✐♦♥ ❛t t✐♠❡ t = ✐♥✐t✐❛❧ ♣♦s✐t✐♦♥ + t · ✈❡❧♦❝✐t② ❲❡ st❛t❡❞ t❤✐s ❝♦♥❝❧✉s✐♦♥ ❜❡❢♦r❡✦ ❚❤✐s t✐♠❡ ♦♥❧② t❤❡ ❝♦♥t❡①t ❤❛s ❝❤❛♥❣❡❞✳ ❚❤❡ ♣❛tt❡r♥ ✐s ❝❧❡❛r✿ t❤❡ ❧✐♥❡ st❛rt✐♥❣ ❛t t❤❡ ♣♦✐♥t (a, b) ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r < u, v > ✐s r❡♣r❡s❡♥t❡❞ ♣❛r❛♠❡tr✐❝❛❧❧② ❛s✿ P (t) = (a, b) + t < u, v > .

❙✐♠✐❧❛r ❢♦r ❞✐♠❡♥s✐♦♥ 3✿ t❤❡ ❧✐♥❡ st❛rt✐♥❣ ❛t t❤❡ ♣♦✐♥t (a, b, c) ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r < u, v, w > ✐s r❡♣r❡s❡♥t❡❞ ❛s✿ P (t) = (a, b, c) + t < u, v, w > .

❆♥❞ s♦ ♦♥✳ ❆t t❤❡ ♥❡①t ❧❡✈❡❧✱ ✇❡✬❞ r❛t❤❡r ❤❛✈❡ ♥♦ r❡❢❡r❡♥❝❡s t♦ ♥❡✐t❤❡r t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ s♣❛❝❡ ♥♦r t❤❡ s♣❡❝✐✜❝ ❝♦♦r❞✐♥❛t❡s✳

❉❡✜♥✐t✐♦♥ ✹✳✶✳✷✿ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ t❤❡ ✉♥✐❢♦r♠ ♠♦t✐♦♥ ❙✉♣♣♦s❡ P0 ✐s ❛ ♣♦✐♥t ✐♥ Rm ❛♥❞ V ✐s ❛ ✈❡❝t♦r✳ ❚❤❡♥ t❤❡

♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢

t❤❡ ✉♥✐❢♦r♠ ♠♦t✐♦♥ t❤r♦✉❣❤ P0 ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t② ♦❢ V ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ P (t) = P0 + tV

❚❤❡♥✱ t❤❡ ❧✐♥❡ t❤r♦✉❣❤ P0 ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳

✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢

V ✐s t❤❡ ♣❛t❤ ✭✐♠❛❣❡✮ ♦❢ t❤✐s

✹✳✶✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

❙t❛t❡❞ ❢♦r

m = 1✱

✷✺✽

t❤❡ ❞❡✜♥✐t✐♦♥ ♣r♦❞✉❝❡s t❤❡ ❢❛♠✐❧✐❛r ♣♦✐♥t✲s❧♦♣❡ ❢♦r♠✦

❚❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ✐s ❛ s✐♥❣❧❡

♥✉♠❜❡r ✭t❤❡ s❧♦♣❡✮ ❜❡❝❛✉s❡ t❤❡ ❝❤❛♥❣❡ ✐s ❡♥t✐r❡❧② ✇✐t❤✐♥ t❤❡ y ✲❛①✐s✳ ❲❤❛t ❤❛s ❝❤❛♥❣❡❞ ✐s t❤❡ ❝♦♥t❡①t✿ 2 t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ❞✐r❡❝t✐♦♥s ✐♥ R ❢♦r ❝❤❛♥❣❡✳ ❚❤❛t ✐s ✇❤② t❤❡ ❝❤❛♥❣❡ ❛♥❞ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ✐s ❛ ✈❡❝t♦r✳ ❇✉t t❤❡ ❡q✉❛t✐♦♥ ❧♦♦❦s ❡①❛❝t❧② t❤❡ s❛♠❡✳✳✳ ❡✈❡♥ t❤♦✉❣❤ ❡❛❝❤ ❧❡tt❡r ♠❛② ❝♦♥t❛✐♥ ✉♥❧✐♠✐t❡❞ ❛♠♦✉♥t ♦❢ ✐♥❢♦r♠❛t✐♦♥✦ ❊①❛♠♣❧❡ ✹✳✶✳✸✿ ♣r✐❝❡s

❚❤❡ ❞❡✜♥✐t✐♦♥ ❛♣♣❧✐❡s t♦ t❤❡ ❛❜str❛❝t s♣❛❝❡s✳ ■❢ ✇❡ ♠✐❣❤t ❤❛✈❡

m = 10, 000✳

Rm

✐s t❤❡ s♣❛❝❡ ♦❢ ♣r✐❝❡s ✭♦❢ st♦❝❦s ♦r ❝♦♠♠♦❞✐t✐❡s✮✱

❚❤❡ ♣r✐❝❡s r❡❝♦r❞❡❞ ❝♦♥t✐♥✉♦✉s❧② ✇✐❧❧ ♣r♦❞✉❝❡ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❛♥❞ t❤✐s

❝✉r✈❡ ♠✐❣❤t ❜❡ ❛ str❛✐❣❤t ❧✐♥❡✳ ❚❤✐s ❤❛♣♣❡♥s ✇❤❡♥ t❤❡ ♣r✐❝❡s ❛r❡ ❣r♦✇✐♥❣ ✭♦r ❞❡❝❧✐♥✐♥❣✮

♣r♦♣♦rt✐♦♥❛❧❧②

❜✉t✱ ♣♦ss✐❜❧②✱ ❛t ❞✐✛❡r❡♥t r❛t❡s✳ ❆❧s♦✱ ✐♥ t❤❡ s❤♦rt t❡r♠ t❤✐s ❝✉r✈❡ ✐s ❧✐❦❡❧② t♦ ❧♦♦❦ ❧✐❦❡ ❛ str❛✐❣❤t ❧✐♥❡✿ t❤❡ ♠♦st r❡❝❡♥t ❝❤❛♥❣❡ ♦❢ ❡❛❝❤ ♣r✐❝❡ ✐s r❡❝♦r❞❡❞ ✐s t❤❡♥ t❤❡ s❛♠❡ ❝❤❛♥❣❡ ✐s ♣r❡❞✐❝t❡❞ ❢♦r t❤❡ ♥❡①t t✐♠❡ ♣❡r✐♦❞✳ ■♥ ❡❛❝❤ ❝♦❧✉♠♥ ✇❡ ✉s❡ t❤❡ s❛♠❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❢♦r t❤❡

k t❤

♣r✐❝❡✿

xk (t + ∆t) = xk (t) + vk ∆t , ✇❤❡r❡

vk

✐s t❤❡

k t❤

❚❤❡ t❛❜❧❡ ✐s ♦✉r

r❛t❡ ♦❢ ❝❤❛♥❣❡✳

10, 000✲❞✐♠❡♥s✐♦♥❛❧

❝✉r✈❡✦ ❈❛♥ ✇❡ ✈✐s✉❛❧✐③❡ s✉❝❤ ❛ ❝✉r✈❡ ✐♥ ❛♥② ✇❛②❄ ❲❡ ♣✐❝❦ t✇♦

❝♦❧✉♠♥s ❛t ❛ t✐♠❡ ❛♥❞ ♣❧♦t t❤❛t ❝✉r✈❡ ♦♥ t❤❡ ♣❧❛♥❡✳ ❙✐♥❝❡ t❤❡s❡ ❝♦❧✉♠♥s ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❛①❡s✱ ✇❡ ❛r❡ ♣❧♦tt✐♥❣ ❛ ✏s❤❛❞♦✇✑ ♦❢ ♦✉r ❝✉r✈❡ ❝❛st ♦♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡✳ ❚❤❡② ❛r❡ ❛❧❧ str❛✐❣❤t ❧✐♥❡s✳ ❊①❡r❝✐s❡ ✹✳✶✳✹

❋✐♥❞ ❛ ♣❛r❛♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ t❤r♦✉❣❤ t✇♦ ❞✐st✐♥❝t ♣♦✐♥ts

P

❛♥❞

Q✳

■♥ t❤❡ ♣❤②s✐❝❛❧ s♣❛❝❡✱ ❛ str❛✐❣❤t ❧✐♥❡ ✐s ❢♦❧❧♦✇❡❞ ❜② ❛♥ ♦❜❥❡❝t ✇❤❡♥ t❤❡r❡ ❛r❡ ♥♦t ❢♦r❝❡s ❛t ♣❧❛②✳

❊✈❡♥ ❛

❝♦♥st❛♥t ❢♦r❝❡ ❧❡❛❞s t♦ ❛❝❝❡❧❡r❛t✐♦♥ ✇❤✐❝❤ ♠❛② ❝❤❛♥❣❡ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ♠♦t✐♦♥✳ ❊①❛♠♣❧❡ ✹✳✶✳✺✿ ❝♦♥st❛♥t ✈❡❧♦❝✐t②

❘❡❝❛❧❧ t❤❡s❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s t❤❛t ❣✐✈❡s t❤❡ ❧♦❝❛t✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ✇❤❡♥ t❤❡ ✈❡❧♦❝✐t② ✐s ❝♦♥st❛♥t ✭k

= 0, 1, ...✮✿ x : pk+1 = pk +v∆t y : qk+1 = qk +u∆t

✹✳✶✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✷✺✾

❚❤❡s❡ q✉❛♥t✐t✐❡s ❛r❡ ♥♦✇ ❝♦♠❜✐♥❡❞ ✐♥t♦ ♣♦✐♥ts ♦♥ t❤❡ ♣❧❛♥❡✿

Pk = (pk , qk ) , ❛♥❞ t❤❡ ❡q✉❛t✐♦♥s t❛❦❡ ❛ ✈❡❝t♦r ❢♦r♠ t♦♦✿

Pk+1 = Pk + Vk ∆t . ❊①❛♠♣❧❡ ✹✳✶✳✻✿ t❤r♦✇♥ ❜❛❧❧

▲❡t✬s r❡✈✐❡✇ t❤❡ ❞②♥❛♠✐❝s ♦❢ ❛ t❤r♦✇♥ ❜❛❧❧✳ ❆ ❝♦♥st❛♥t ❢♦r❝❡ ❝❛✉s❡s t❤❡ ✈❡❧♦❝✐t② t♦ ❝❤❛♥❣❡ ❧✐♥❡❛r❧②✱ ❥✉st ❛s t❤❡ ❧♦❝❛t✐♦♥ ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✳ ❍♦✇ ❞♦❡s t❤❡ ❧♦❝❛t✐♦♥ ❝❤❛♥❣❡ t❤✐s t✐♠❡❄ ■♥ t❤❡ ❤♦r✐③♦♥t❛❧ ❞✐r❡❝t✐♦♥✱ ❛s t❤❡r❡ ✐s ♥♦ ❢♦r❝❡ ❝❤❛♥❣✐♥❣ t❤❡ ✈❡❧♦❝✐t②✱ t❤❡ ❧❛tt❡r r❡♠❛✐♥s ❝♦♥st❛♥t✳ ▼❡❛♥✇❤✐❧❡✱ t❤❡ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t② ✐s ❝♦♥st❛♥t❧② ❝❤❛♥❣❡❞ ❜② t❤❡ ❣r❛✈✐t②✳ ❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ❤❡✐❣❤t ♦♥ t❤❡ t✐♠❡ ✐s q✉❛❞r❛t✐❝✳ ❚❤❡ ♣❛t❤ ♦❢ t❤❡ ❜❛❧❧ ✇✐❧❧ ❛♣♣❡❛r t♦ ❛♥ ♦❜s❡r✈❡r ✕ ❢r♦♠ t❤❡ r✐❣❤t ❛♥❣❧❡ ✕ ❛s ❛ ❝✉r✈❡✿

❆ ❢❛❧❧✐♥❣ ❜❛❧❧ ✐s s✉❜❥❡❝t t♦ t❤❡s❡ ❛❝❝❡❧❡r❛t✐♦♥s✱ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧✿

x : ak+1 = 0;

y : ak+1 = −g .

◆♦✇ r❡❝❛❧❧ t❤❡ s❡t✉♣ ❝♦♥s✐❞❡r❡❞ ♣r❡✈✐♦✉s❧②✿ ❢r♦♠ ❛ 200 ❢❡❡t ❡❧❡✈❛t✐♦♥✱ ❛ ❝❛♥♥♦♥ ✐s ✜r❡❞ ❤♦r✐③♦♥t❛❧❧② ❛t 200 ❢❡❡t ♣❡r s❡❝♦♥❞✳

❚❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛r❡✿ • ❚❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥✱ x : p0 = 0 ❛♥❞ y : p0 = 200✳ • ❚❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②✱ x : v0 = 200 ❛♥❞ y : v0 = 0✳

✹✳✶✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✷✻✵

❚❤❡♥ ✇❡ ❤❛✈❡ t✇♦ ♣❛✐rs ♦❢ r❡❝✉rs✐✈❡ ❡q✉❛t✐♦♥s ✕ ❢♦r t❤❡ ❧♦❝❛t✐♦♥ ✐♥ t❡r♠s ♦❢ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ✈❡❧♦❝✐t② ✐♥ t❡r♠s ♦❢ ❛❝❝❡❧❡r❛t✐♦♥ ✕ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❡❛❝❤ ♦t❤❡r✿

x : vk+1 pk+1 y : uk+1 qk+1

= v0 = pk +vk ∆t = vk −g∆t = qk +uk ∆t

❚❤❡s❡ ❛r❡ t❤❡ ❢♦r♠✉❧❛s ✐♥ t❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥✿

Vk+1 = Vk +A ·∆t Pk+1 = Pk +Vk+1 ·∆t ❚❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ✈❡❝t♦r ❛♣♣r♦❛❝❤ ✐s t❤❛t t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐s ♥♦ ❧♦♥❣❡r ❛ ❝♦♥❝❡r♥✦ ❊①❛♠♣❧❡ ✹✳✶✳✼✿ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s

■♥ ❞✐♠❡♥s✐♦♥ 2✱ ❢♦r ❡①❛♠♣❧❡✱ ✇❡ ❞♦♥✬t ❤❛✈❡ t♦ ❛❧✐❣♥ t❤❡ x✲❛①✐s ✇✐t❤ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ t❤r♦✇ ❛♥❞ ✐♥ ❞✐♠❡♥s✐♦♥ 3 ✇❡ ❞♦♥✬t ❤❛✈❡ t♦ ❛❧✐❣♥ t❤❡ z ✲❛①✐s ✇✐t❤ t❤❡ ✈❡rt✐❝❛❧ ❞✐r❡❝t✐♦♥✳ ◆♦♥❡t❤❡❧❡ss✱ ❧❡t✬s st❛rt ✇✐t❤ ❢♦r♠❡r ❝❛s❡✳ ❆ 6✲❢♦♦t ♠❛♥ t❤r♦✇s ✕ str❛✐❣❤t ❢♦r✇❛r❞ ✕ ❛ ❜❛❧❧ ✇✐t❤ t❤❡ s♣❡❡❞ ♦❢ 100 ❢❡❡t ♣❡r s❡❝♦♥❞✳ ■❢ t❤❡ t❤r♦✇ ✐s ❛❧♦♥❣ t❤❡ x✲❛①✐s ❛♥❞ t❤❡ y ✲❛①✐s ✐s ✈❡rt✐❝❛❧✱ ✇❡ ❤❛✈❡✿

A =< 0, −32 >, V0 =< 0, 100 >, P0 = (6, 0) . ❚❤✐s ❞❛t❛ ❣♦❡s ✐♥t♦ t❤❡ ✜rst r♦✇ ♦❢ ♦✉r t❛❜❧❡ ❢♦r t❤❡ ❝♦❧✉♠♥s ♠❛r❦❡❞ x′′ , y ′′ ✱ x′ , y ′ ✱ ❛♥❞ x, y r❡s♣❡❝t✐✈❡❧②✳

❲❡ ❛♣♣❧② t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❣✐✈❡♥ ❛❜♦✈❡✳ ■♥ t❤❡ s♣r❡❛❞s❤❡❡t✱ • ❚❤❡ ✈❡❧♦❝✐t② ✐s ❝♦♠♣✉t❡❞ ❢r♦♠ t❤❡ ✈❡❧♦❝✐t②✳ • ❚❤❡ ❧♦❝❛t✐♦♥ ✐s ❝♦♠♣✉t❡❞ ❢r♦♠ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✳ ❲❤❛t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ♦✉r ✈❡❝t♦rs ❛♣♣r♦❛❝❤ ❢r♦♠ ♦✉r ♣r❡✈✐♦✉s tr❡❛t♠❡♥t ♦❢ t❤❡ ✢✐❣❤t ♦❢ ❛ ❜❛❧❧❄ ■♥st❡❛❞ ♦❢ t❤r❡❡ ❝♦❧✉♠♥s ❢♦r x′′ , x′ , x ❛♥❞ t❤❡♥ t❤r❡❡ ❝♦❧✉♠♥s ❢♦r y ′′ , y ′ , y ✱ ♦♥❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡ t✇♦ ❝♦♠♣♦♥❡♥ts ♦❢ ❛❝❝❡❧❡r❛t✐♦♥✱ ✈❡❧♦❝✐t②✱ ❛♥❞ ❧♦❝❛t✐♦♥ ❛r❡ ❝♦♠❜✐♥❡❞ ✐♥t♦ ✈❡❝t♦rs ❝♦♥t❛✐♥❡❞ ✐♥ t✇♦ ❝♦❧✉♠♥s ❡❛❝❤✿ x′′ , y ′′ ✱ t❤❡♥ x′ , y ′ ✱ t❤❡♥ x, y ✳ ❚❤❡ ❢♦r♠✉❧❛ ✐s ❛❧♠♦st t❤❡ s❛♠❡ ❛s ❜❡❢♦r❡✿

❂❘❬✲✶❪❈✰✭❘❈❬✲✷❪✲❘❬✲✶❪❈❬✲✷❪✮✯❘✷❈✶ ◆❡①t ❛♥ ❛♥❣❧❡❞ t❤r♦✇✳✳✳ ❚❤❡ ♦♥❧② ❝❤❛♥❣❡ ✐s t❤❡ ✈❡❝t♦r ♦❢ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②✿

A =< 0, −32 >, V0 =< 100 cos α, 100 sin α >, P0 = (6, 0) , ✇❤❡r❡ α ✐s t❤❡ ❛♥❣❧❡ ♦❢ t❤❡ t❤r♦✇✳

✹✳✶✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✷✻✶

❊①❛♠♣❧❡ ✹✳✶✳✽✿ ❝♦♥t✐♥✉♦✉s ♠♦t✐♦♥ ◆♦✇ t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡✳✳✳ ❙t❛rt✐♥❣ ✇✐t❤ t❤❡ ♣❤②s✐❝s✱  ′′ x = 0, y ′′ = −g, ✇❡ ✐♥t❡❣r❛t❡ ✕ ❝♦♦r❞✐♥❛t❡✲✇✐s❡ ✕ ♦♥❝❡✿  ′ x = vx , y ′ = −gt + vy , ❛♥❞ t✇✐❝❡✿ 

x′ (0) = vx ✐s t❤❡ ✐♥✐t✐❛❧ ❤♦r✐③♦♥t❛❧ ✈❡❧♦❝✐t②, y ′ (0) = vy ✐s t❤❡ ✐♥✐t✐❛❧ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t②;

x = vx t + px , y = − 21 gt2 + vy t + py ,

❚❤✉s✱ ✇❡ ❤❛✈❡✿  ❞❡♣t❤ = ❤❡✐❣❤t =

✐♥✐t✐❛❧ ❞❡♣t❤ ✐♥✐t✐❛❧ ❤❡✐❣❤t

+ +

x(0) = px ✐s t❤❡ ✐♥✐t✐❛❧ ❤♦r✐③♦♥t❛❧ ♣♦s✐t✐♦♥, y(0) = py ✐s t❤❡ ✐♥✐t✐❛❧ ✈❡rt✐❝❛❧ ♣♦s✐t✐♦♥.

✐♥✐t✐❛❧ ❤♦r✐③♦♥t❛❧ ✈❡❧♦❝✐t② ✐♥✐t✐❛❧ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t②

· t✐♠❡ , · t✐♠❡ − 21 g · t✐♠❡ 2 .

❲❡ t❛❦❡ t❤✐s s♦❧✉t✐♦♥ t♦ t❤❡ ♥❡①t ❧❡✈❡❧ ❜② ❛ss❡♠❜❧✐♥❣ t❤❡s❡ ❝♦♠♣♦♥❡♥ts ✐♥t♦ ✈❡❝t♦rs ❥✉st ❛s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✳ ❧♦❝❛t✐♦♥ =

✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ +

✐♥✐t✐❛❧ ✈❡❧♦❝✐t②

· t✐♠❡

+ < 0, − 21 g · t✐♠❡ 2 > .

❚❤❡ ❧❛st t❡r♠ ♥❡❡❞s ✇♦r❦✳ ❚❤❡ ③❡r♦ r❡♣r❡s❡♥ts t❤❡ ③❡r♦ ❤♦r✐③♦♥t❛❧ ❛❝❝❡❧❡r❛t✐♦♥ ✇❤✐❧❡ −g ✐s t❤❡ ✈❡rt✐❝❛❧ t2 ❛❝❝❡❧❡r❛t✐♦♥✳ ❚❤❡♥ t❤❡ ❧❛st t❡r♠ ✐s t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ t✐♠❡s ✳ ❆❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ ❤❛✈❡✿ 2         t2 px x vx 0 = + · . ·t + y py vy −g 2 ❚❤❡ ♥❛t✉r❡ ♦❢ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ✐rr❡❧❡✈❛♥t❀ ✇❡ ♦♥❧② ♥❡❡❞ ✐t t♦ ❜❡ ❝♦♥st❛♥t✳

❉❡✜♥✐t✐♦♥ ✹✳✶✳✾✿ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ ✉♥✐❢♦r♠❧② ❛❝❝❡❧❡r❛t❡❞ ♠♦t✐♦♥ ❙✉♣♣♦s❡ P0 ✐s ❛ ♣♦✐♥t ✐♥ Rm ❛♥❞ V0 , A ❛r❡ ✈❡❝t♦rs✳ ❚❤❡♥ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ ✉♥✐❢♦r♠❧② ❛❝❝❡❧❡r❛t❡❞ ♠♦t✐♦♥ t❤r♦✉❣❤ P0 ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t② ♦❢ V ❛♥❞ ❛❝❝❡❧❡r❛t✐♦♥ A ✐s✿

P (t) = P0 + V0 · t + A ·

t2 2

✹✳✶✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s

✷✻✷

❲❡ ❤❛✈❡ ❛♥ ❡①tr❛ t❡r♠✱ t❤❛t ❞✐s❛♣♣❡❛rs ✇❤❡♥

1✲❞✐♠❡♥s✐♦♥❛❧

A = 0✱

✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ✉♥✐❢♦r♠ ♠♦t✐♦♥✳ ❏✉st ❛s ✐♥ t❤❡

❝❛s❡✱ ❛ ❝♦♥st❛♥t ❛❝❝❡❧❡r❛t✐♦♥ ♣r♦❞✉❝❡s ❛ q✉❛❞r❛t✐❝ ♠♦t✐♦♥✦

❊①❡r❝✐s❡ ✹✳✶✳✶✵ ❙❤♦✇ t❤❛t t❤❡ ♣❛t❤ ♦❢ t❤✐s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s ❛ ♣❛r❛❜♦❧❛✳

❚❤❡ ✈❛❧✉❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ r❡♣r❡s❡♥t❡❞ ❜② ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❧✐❡ ✐♥

✈❡❝t♦rs✳

Rm

❛s

♣♦✐♥ts

❜✉t ❝❛♥ ❛❧s♦ ❜❡ s❡❡♥ ❛s

❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ r❡✲✇r✐t❡ t❤❡ ❢❛♠✐❧✐❛r ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ ♣♦✐♥ts✿

P (t) = P0 + V0 · t + A ·

t2 , 2

R(t) = R0 + V0 · t + A ·

t2 . 2

❛s ♦♥❡ ♦❢ ✈❡❝t♦rs✿

■♥st❡❛❞ ♦❢ ♣❛ss✐♥❣ t❤r♦✉❣❤ ♣♦✐♥t

P0

✐t ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ❡♥❞ ♣♦✐♥t ♦❢ ✈❡❝t♦r

t❤✐♥❣✳ ❆♥❞✱ ♦❢ ❝♦✉rs❡✱ t❤❡ ❡♥❞ ♦❢ ✈❡❝t♦r

R(t) ✐s t❤❡ ♣♦✐♥t P (t)✳

R0 = OP0 ✱

✇❤✐❝❤ ✐s t❤❡ s❛♠❡

❚❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ❧❛tt❡r ❛♣♣r♦❛❝❤ ✐s t❤❛t

✐t ❛❧❧♦✇s ✉s t♦ ❛♣♣❧② ✈❡❝t♦r ♦♣❡r❛t✐♦♥s t♦ t❤❡ ❝✉r✈❡s✳

❊①❛♠♣❧❡ ✹✳✶✳✶✶✿ ❝✐r❝❧❡ tr❛♥s❢♦r♠❡❞ ❘❡❝❛❧❧ ❤♦✇ ✇❡ ♣❛r❛♠❡tr✐③❡❞ t❤❡ ✉♥✐t ❝✐r❝❧❡ ✉s✐♥❣ t❤❡ ❛♥❣❧❡ ❛s t❤❡ ♣❛r❛♠❡t❡r✳ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛ ♣♦✐♥t ❛t ❛♥❣❧❡

t

✐s

cos t

❛♥❞

sin t

❍❡r❡✱ t❤❡

x✲

❛♥❞

y✲

r❡s♣❡❝t✐✈❡❧②✿

x = cos t, y = sin t . ❚❤❡ ✈❛❧✉❡s ♦❢

t

♠❛② ❜❡ t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ s✉❝❤ ❛s

[0, 2π]

♦r r✉♥ t❤r♦✉❣❤ t❤❡ ✇❤♦❧❡

✐♥t❡r✈❛❧✳

❲❡ ❝❛♥ ❛❧s♦ ❧♦♦❦ ❛t t❤✐s ❢♦r♠✉❧❛ ❛s ❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡✳ ❚❤❡♥ t❤✐s ✐s ❛ r❡❝♦r❞ ♦❢ ♠♦t✐♦♥ ✇✐t❤ ❛ ❝♦♥st❛♥t s♣❡❡❞ ♦r✱ ✐♥ ♦t❤❡r ✇♦r❞s✱ ❛ ❝♦♥st❛♥t

❛♥❣✉❧❛r

✈❡❧♦❝✐t②✳ ◆♦✇✱ t❤✐s ✐s t❤❡ ✈❡❝t♦r

r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤✐s ❝✉r✈❡✿

R(t) =< cos t, sin t > . ❙♦✱ ❛♣♣❧②✐♥❣ ✈❡❝t♦r ♦♣❡r❛t✐♦♥s t♦ t❤✐s ❝✉r✈❡ ✇✐❧❧ ❣✐✈❡ ❛s ♥❡✇ ❝✉r✈❡s✱ ❥✉st ❛s ✐♥ t❤❡ ✭❈❤❛♣t❡r ✶P❈✲✹✮✳ ❋♦r ❡①❛♠♣❧❡✱ ✉s✐♥❣ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜②

Q(t) = 2R(t) = 2 < cos t, sin t > , 2✳

❝❛s❡

2 ♦♥ ❛❧❧ ✈❡❝t♦rs ♠❡❛♥s str❡t❝❤✐♥❣ r❛❞✐❛❧❧②

t❤❡ ✇❤♦❧❡ s♣❛❝❡✳ ❲❡ t❤❡♥ ❞✐s❝♦✈❡r t❤❛t t❤❡ ❝✉r✈❡ ✐♥ t❤❡ ♣❧❛♥❡ ❣✐✈❡♥ ❜②✿

✐s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s

1✲❞✐♠❡♥s✐♦♥❛❧

✹✳✷✳

❚❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧

✷✻✸

❙✐♠✐❧❛r❧②✱ ✉s✐♥❣ ✈❡❝t♦r ❛❞❞✐t✐♦♥ ✇✐t❤

W =< 3, 1 >

♦♥ ❛❧❧ ✈❡❝t♦rs ♠❡❛♥s

t❤✐s ✈❡❝t♦r✳ ❲❡ t❤❡♥ ❞✐s❝♦✈❡r t❤❛t t❤❡ ❝✉r✈❡ ✐♥ t❤❡ ♣❧❛♥❡ ❣✐✈❡♥ ❜②✿

S(t) = W + 2R(t) = (1, 2) + 2 < cos t, sin t > , ✐s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦❢ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s

❆♥❞ s♦ ♦♥ ✇✐t❤ ♦t❤❡r

tr❛♥s❢♦r♠❛t✐♦♥s

2

❝❡♥t❡r❡❞ ❛t

(1, 2)✳

♦❢ t❤❡ ♣❧❛♥❡✳

✹✳✷✳ ❚❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧

❚❤✐s ✐s t❤❡ ■❱P ✇❡ ❤❛✈❡ ❝♦♥s✐❞❡r❡❞ s♦ ❢❛r✿

∆y = f (t, y), y(t0 ) = y0 , ∆t ❛♥❞

y ′ = f (t, y), y(t0 ) = y0 . ❚❤❡ ❡q✉❛t✐♦♥s s❤♦✇ ❤♦✇ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ ❲❤❛t ✐❢ ✇❡ ❤❛✈❡

t✇♦

y

❞❡♣❡♥❞s ♦♥

✈❛r✐❛❜❧❡ q✉❛♥t✐t✐❡s ❞❡♣❡♥❞❡♥t ♦♥

❚❤❡ s✐♠♣❧❡st ❡①❛♠♣❧❡ ✐s ❛s ❢♦❧❧♦✇s✿

• x

✐s t❤❡ ❤♦r✐③♦♥t❛❧ ❧♦❝❛t✐♦♥✱ ❛♥❞

• y

✐s t❤❡ ✈❡rt✐❝❛❧ ❧♦❝❛t✐♦♥✳

t❄

t

❛♥❞

y✳

s❤✐❢t✐♥❣

t❤❡ ✇❤♦❧❡ s♣❛❝❡ ❜②

✹✳✷✳ ❚❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧

✷✻✹

❲❡ ❤❛✈❡ ❛❧r❡❛❞② s❡❡♥ t❤✐s s✐♠♣❧❡st s❡tt✐♥❣ ♦❢ ❢r❡❡ ❢❛❧❧✿

   ∆x = vx , ∆t  ∆y  = vy − gt , ∆t

❛♥❞

   x′ = vx ,

  y ′ = v − gt . y

■t ✐s ❥✉st ❛s s✐♠♣❧❡ ✇❤❡♥ ❛r❜✐tr❛r② ❢✉♥❝t✐♦♥s ❛r❡ ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥s ✭t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡ ✐s s♦❧✈❡❞ ❜② ✐♥t❡❣r❛t✐♦♥✮✳ ❍❡r❡ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ ❧♦❝❛t✐♦♥ ❞❡♣❡♥❞s ♦♥ t❤❡ t✐♠❡

t

♦♥❧②✳

▼♦r❡ ❝♦♠♣❧❡① ✐s t❤❡ s✐t✉❛t✐♦♥ ✇❤❡♥ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❤❡ ❧♦❝❛t✐♦♥ ❞❡♣❡♥❞s ♦♥ t❤❡ ❧♦❝❛t✐♦♥✳ ❲❤❡♥ t❤❡ ❢♦r♠❡r ❞❡♣❡♥❞s ♦♥❧② ♦♥ ✐ts ♦✇♥ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❧♦❝❛t✐♦♥✱ t❤❡ ♠♦t✐♦♥ ✐s ❞❡s❝r✐❜❡❞ ❜② t❤✐s ♣❛✐r ♦❢ ❖❉❊s✿

   ∆x = g(x) , ∆t  ∆y  = h(y) , ∆t

❛♥❞

   x′ = g(x) ,   y ′ = h(y) .

❚❤❡ s♦❧✉t✐♦♥ ❝♦♥s✐sts ♦❢ t✇♦ s♦❧✉t✐♦♥s t♦ t❤❡ t✇♦✱ ✉♥r❡❧❛t❡❞✱ ❖❉❊s✳

❲❡ ❝❛♥ t❤❡♥ ❛♣♣❧② t❤❡ ♠❡t❤♦❞s ♦❢

❈❤❛♣t❡r ✸✳ ❆s ❛♥ ❡①❛♠♣❧❡ ♦❢ q✉❛♥t✐t✐❡s t❤❛t ❞♦ ✐♥t❡r❛❝t✱ ❧❡t✬s ❝♦♥s✐❞❡r t❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧✳ ▲❡t

• x

❜❡ t❤❡ ♥✉♠❜❡r ♦❢ r❛❜❜✐ts ❛♥❞

• y

❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❢♦①❡s ✐♥ t❤❡ ❢♦r❡st✳

▲❡t

• ∆t

❜❡ t❤❡ ✜①❡❞ ✐♥❝r❡♠❡♥t ♦❢ t✐♠❡✳

❊✈❡♥ t❤♦✉❣❤ t✐♠❡

t

✐s ♥♦✇ ❞✐s❝r❡t❡✱ t❤❡ ✏♥✉♠❜❡r✑ ♦❢ r❛❜❜✐ts

♦✉r ♠♦❞❡❧✳ ❖♥❡ ❝❛♥ t❤✐♥❦ ♦❢ ✐s ♦♥❡ s♦♠❡✇❤❡r❡ ✐s

0.1

x

♦r ❢♦①❡s

y

✐s♥✬t✳ ❚❤♦s❡ ❛r❡ r❡❛❧ ♥✉♠❜❡rs ✐♥

r❛❜❜✐ts ❛s ✐❢ t❤❡ ❛❝t✉❛❧ ♥✉♠❜❡r ✐s ✉♥❦♥♦✇♥ ❜✉t t❤❡ ❧✐❦❡❧✐❤♦♦❞ t❤❛t t❤❡r❡

10%✳

❲❡ ❜❡❣✐♥ ✇✐t❤ t❤❡ r❛❜❜✐ts✳ ❚❤❡r❡ ❛r❡ t✇♦ ❢❛❝t♦rs ❛✛❡❝t✐♥❣ t❤❡✐r ♣♦♣✉❧❛t✐♦♥✳ ❋✐rst✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡② ❤❛✈❡ ❛♥ ✉♥❧✐♠✐t❡❞ ❢♦♦❞ s✉♣♣❧② ❛♥❞ r❡♣r♦❞✉❝❡ ✐♥ ❛ ♠❛♥♥❡r ❞❡s❝r✐❜❡❞ ♣r❡✈✐♦✉s❧② ✕ ✇❤❡♥ t❤❡r❡ ✐s ♥♦ ♣r❡❞❛t♦r✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❣❛✐♥ ♦❢ t❤❡ r❛❜❜✐t ♣♦♣✉❧❛t✐♦♥ ♣❡r ✉♥✐t ♦❢ t✐♠❡ t❤r♦✉❣❤ t❤❡✐r ♥❛t✉r❛❧ r❡♣r♦❞✉❝t✐♦♥ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ s✐③❡ ♦❢ t❤❡✐r ❝✉rr❡♥t ♣♦♣✉❧❛t✐♦♥✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿ ❘❛❜❜✐ts✬ ❣❛✐♥ ❢♦r s♦♠❡

= α · x · ∆t ,

α > 0✳

❙❡❝♦♥❞✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ r❛t❡ ♦❢ ♣r❡❞❛t✐♦♥ ✉♣♦♥ t❤❡ r❛❜❜✐ts t♦ ❜❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ r❛t❡ ❛t ✇❤✐❝❤ t❤❡ r❛❜❜✐ts ❛♥❞ t❤❡ ❢♦①❡s ♠❡❡t✱ ✇❤✐❝❤✱ ✐♥ t✉r♥✱ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ s✐③❡s ♦❢ t❤❡✐r ❝✉rr❡♥t ♣♦♣✉❧❛t✐♦♥s✱

y✳

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿ ❘❛❜❜✐ts✬ ❧♦ss

= β · x · y · ∆t ,

x

❛♥❞

✹✳✷✳

✷✻✺

❚❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧

❢♦r s♦♠❡ β > 0✳ ❈♦♠❜✐♥❡❞✱ t❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ r❛❜❜✐t ♣♦♣✉❧❛t✐♦♥ ♦✈❡r t❤❡ ♣❡r✐♦❞ ♦❢ t✐♠❡ ♦❢ ❧❡♥❣t❤ ∆t ✐s✿

∆x = αx∆t − βxy∆t . ❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤ t❤❡ ❢♦①❡s✳ ❚❤❡r❡ ❛r❡ t✇♦ ❢❛❝t♦rs ❛✛❡❝t✐♥❣ t❤❡✐r ♣♦♣✉❧❛t✐♦♥✳ ❋✐rst✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ❢♦①❡s ❤❛✈❡ ♦♥❧② ❛ ❧✐♠✐t❡❞ ❢♦♦❞ s✉♣♣❧②✱ ✐✳❡✳✱ t❤❡ r❛❜❜✐ts✳ ❚❤❡ ❢♦①❡s ❞✐❡ ♦✉t ❣❡♦♠❡tr✐❝❛❧❧② ✐♥ ❛ ♠❛♥♥❡r ❞❡s❝r✐❜❡❞ ♣r❡✈✐♦✉s❧② ✕ ✇❤❡♥ t❤❡r❡ ✐s ♥♦ ♣r❡②✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❧♦ss ♦❢ t❤❡ ❢♦① ♣♦♣✉❧❛t✐♦♥ ♣❡r ✉♥✐t ♦❢ t✐♠❡ t❤r♦✉❣❤ t❤❡✐r ♥❛t✉r❛❧ ❞❡❛t❤ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ s✐③❡ ♦❢ t❤❡✐r ❝✉rr❡♥t ♣♦♣✉❧❛t✐♦♥✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿ ❋♦①❡s✬ ❧♦ss = γ · y · ∆t , ❢♦r s♦♠❡ γ > 0✳

❙❡❝♦♥❞✱ ✇❡ ❛❣❛✐♥ ❛ss✉♠❡ t❤❛t t❤❡ r❛t❡ ♦❢ r❡♣r♦❞✉❝t✐♦♥ ♦❢ t❤❡ ❢♦①❡s ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ r❛t❡ ♦❢ t❤❡✐r ♣r❡❞❛t✐♦♥ ✉♣♦♥ t❤❡ r❛❜❜✐ts ✇❤✐❝❤ ✐s✱ ❛s ✇❡ ❦♥♦✇✱ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ s✐③❡s ♦❢ t❤❡✐r ❝✉rr❡♥t ♣♦♣✉❧❛t✐♦♥s✱ x ❛♥❞ y ✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿ ❋♦①❡s✬ ❣❛✐♥ = δ · x · y · ∆t , ❢♦r s♦♠❡ δ > 0✳

❈♦♠❜✐♥❡❞✱ t❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ ❢♦① ♣♦♣✉❧❛t✐♦♥ ♦✈❡r t❤❡ ♣❡r✐♦❞ ♦❢ t✐♠❡ ♦❢ ❧❡♥❣t❤ ∆t ✐s✿

∆y = δxy∆t − γy∆t . P✉tt✐♥❣ t❤❡s❡ t✇♦ t♦❣❡t❤❡r ❣✐✈❡s ✉s ❛

❞✐s❝r❡t❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧ ✿



∆x = (αx − βxy) ∆t, ∆y = (δxy − γy) ∆t.

❚❤❡♥ t❤❡ s♣r❡❛❞s❤❡❡t ❢♦r♠✉❧❛s ❛r❡ ❢♦r x ❛♥❞ y r❡s♣❡❝t✐✈❡❧②✿

❂❘❬✲✶❪❈✰❘✷❈✸✯❘❬✲✶❪❈✯❘✸❈✶✲❘✷❈✹✯❘❬✲✶❪❈✯❘❬✲✶❪❈❬✶❪✯❘✸❈✶ ❂❘❬✲✶❪❈✲❘✷❈✺✯❘❬✲✶❪❈✯❘✸❈✶✰❘✷❈✻✯❘❬✲✶❪❈✯❘❬✲✶❪❈❬✲✶❪✯❘✸❈✶ ▲❡t✬s t❛❦❡ ❛ ❧♦♦❦ ❛t ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ♣♦ss✐❜❧❡ ❞②♥❛♠✐❝s ✭α = 0.10✱ β = 0.50✱ γ = 0.20✱ δ = 0.20✱ h = 0.2✮✿

✹✳✷✳

❚❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧

✷✻✻

❚❤✐s ✐s ✇❤❛t ✇❡ s❡❡✳ ■♥✐t✐❛❧❧②✱ t❤❡r❡ ❛r❡ ♠❛♥② r❛❜❜✐ts ❛♥❞✱ ✇✐t❤ t❤✐s ♠✉❝❤ ❢♦♦❞✱ t❤❡ ♥✉♠❜❡r ♦❢ ❢♦①❡s ✇❛s

↑✳ տ✳

←✳

❣r♦✇✐♥❣✿

❚❤✐s ✇❛s ❝❛✉s✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ r❛❜❜✐ts t♦ ❞❡❝❧✐♥❡✿

s②st❡♠✿

▲❛t❡r✱ t❤❡ ♥✉♠❜❡r ♦❢ r❛❜❜✐ts ❞❡❝❧✐♥❡s s♦ ♠✉❝❤ t❤❛t✱ ✇✐t❤ s♦ ❧✐tt❧❡ ❢♦♦❞✱ t❤❡ ♥✉♠❜❡r ♦❢ ❢♦①❡s ❛❧s♦

st❛rt❡❞ t♦ ❞❡❝❧✐♥❡✿

↓✳

❈♦♠❜✐♥❡❞✱ t❤✐s ✐s t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡

❆t t❤❡ ❡♥❞✱ ❜♦t❤ ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥s s❡❡♠ t♦ ❤❛✈❡ ❞✐s❛♣♣❡❛r❡❞✳✳✳

❆♥♦t❤❡r ❡①♣❡r✐♠❡♥t s❤♦✇s t❤❛t t❤❡② ❝❛♥ r❡❝♦✈❡r ✭α

= 3✱ β = 2✱ γ = 3✱ δ = 1✱ h = 0.03✮✿

■♥ ❢❛❝t✱ ✇❡ ❝❛♥ s❡❡ ❛ r❡♣❡❛t✐♥❣ ♣❛tt❡r♥✳ ❋✉rt❤❡r♠♦r❡✱ ✇✐t❤

∆t → 0✱

✇❡ ❤❛✈❡ t✇♦✱ r❡❧❛t❡❞✱ ❖❉❊s✱ ❛

❝♦♥t✐♥✉♦✉s ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧ ✿

   dx = αx − βxy , dt  dy  = δxy − γy . dt

■t ❛♣♣r♦①✐♠❛t❡s t❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧✳ ❚❤❡ ❡q✉❛t✐♦♥s ❛r❡ ❦♥♦✇♥ ❛s t❤❡

▲♦t❦❛✕❱♦❧t❡rr❛ ❡q✉❛t✐♦♥s✳

✹✳✸✳

✷✻✼

◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧

✹✳✸✳ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧

❚♦ ❝♦♥✜r♠ ♦✉r ♦❜s❡r✈❛t✐♦♥s✱ ✇❡ ✇✐❧❧ ❝❛rr② ♦✉t q✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s✳ ■t ✐s ❡q✉❛❧❧② ❛♣♣❧✐❝❛❜❧❡ t♦ ❜♦t❤ t❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧ ❛♥❞ t❤❡ s②st❡♠ ♦❢ ❖❉❊s✳ ■♥❞❡❡❞✱ t❤❡② ❜♦t❤ ❤❛✈❡ t❤❡ s❛♠❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✿    ∆x = αx − βxy , ∆t   ∆y = δxy − γy , ∆t

❛♥❞

   dx = αx − βxy , dt   dy = δxy − γy . dt

❲❡ ✇✐❧❧ ✐♥✈❡st✐❣❛t❡ t❤❡ ❞②♥❛♠✐❝s ❛t ❛❧❧ ❧♦❝❛t✐♦♥s ✐♥ t❤❡ ✜rst q✉❛❞r❛♥t ♦❢ t❤❡ tx✲♣❧❛♥❡✳ ❋✐rst✱ ✇❡ ✜♥❞ t❤❡ ❧♦❝❛t✐♦♥s ✇❤❡r❡ x ❤❛s ❛ ③❡r♦ ❞❡r✐✈❛t✐✈❡✱ ✐✳❡✳✱ x′ = 0✱ ✇❤✐❝❤ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ❧♦❝❛t✐♦♥s ✇❤❡r❡ t❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧ ❧❡❛❞s t♦ ♥♦ ❝❤❛♥❣❡ ✐♥ x✱ ✐✳❡✳✱ ∆x = 0✳ ❚❤❡ ❝♦♥❞✐t✐♦♥ ✐s✿ αx − βxy = 0 .

❲❡ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿

x = 0 ♦r y = α/β .

❲❡ ❞✐s❝♦✈❡r t❤❛t✱ ✜rst✱ • x = 0 ✐s ❛ s♦❧✉t✐♦♥✱ ❛♥❞✱ s❡❝♦♥❞✱

• t❤❡ ❤♦r✐③♦♥t❛❧ ❧✐♥❡ y = α/β ✐s ❝r♦ss❡❞ ✈❡rt✐❝❛❧❧② ❜② t❤❡ s♦❧✉t✐♦♥s✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✜rst✱ t❤❡ ❢♦①❡s ❛r❡ ❞②✐♥❣ ♦✉t ✇✐t❤ ♥♦ r❛❜❜✐ts ❛♥❞✱ s❡❝♦♥❞✱ t❤❡r❡ ♠❛② ❜❡ ❛ r❡✈❡rs❛❧ ✐♥ t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ r❛❜❜✐ts ❛t ❛ ❝❡rt❛✐♥ ♥✉♠❜❡r ♦❢ ❢♦①❡s✳ ❚♦ ✜♥❞ ♦✉t✱ s♦❧✈❡ t❤❡ ✐♥❡q✉❛❧✐t②✿ x′ > 0 ♦r ∆x > 0 =⇒ αx − βxy > 0 =⇒ y < α/β .

■t ❢♦❧❧♦✇s t❤❛t✱ ✐♥❞❡❡❞✱ t❤❡ ♥✉♠❜❡r ♦❢ r❛❜❜✐ts ✐♥❝r❡❛s❡s ✇❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ ❢♦①❡s ✐s ❜❡❧♦✇ α/β ✱ ♦t❤❡r✇✐s❡ ✐t ❞❡❝r❡❛s❡s✳

✹✳✸✳

◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧

✷✻✽

❙❡❝♦♥❞✱ ✇❡ ✜♥❞ t❤❡ ❧♦❝❛t✐♦♥s ✇❤❡r❡ y ❤❛s ❛ ③❡r♦ ❞❡r✐✈❛t✐✈❡✱ ✐✳❡✳✱ y ′ = 0✱ ✇❤✐❝❤ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ❧♦❝❛t✐♦♥s ✇❤❡r❡ t❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧ ❧❡❛❞s t♦ ♥♦ ❝❤❛♥❣❡ ✐♥ y ✱ ✐✳❡✳✱ ∆y = 0✳ ❚❤❡ ❝♦♥❞✐t✐♦♥ ✐s✿ δxy − γy = 0 .

❲❡ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿

y = 0 ♦r x = γ/δ .

❲❡ ❞✐s❝♦✈❡r t❤❛t✱ ✜rst✱ • y = 0 ✐s ❛ s♦❧✉t✐♦♥✱ ❛♥❞✱ s❡❝♦♥❞✱

• t❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡ x = γ/δ ✐s ❝r♦ss❡❞ ❤♦r✐③♦♥t❛❧❧② ❜② t❤❡ s♦❧✉t✐♦♥s✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✜rst✱ t❤❡ r❛❜❜✐ts t❤r✐✈❡ ✇✐t❤ ♥♦ ❢♦①❡s ❛♥❞✱ s❡❝♦♥❞✱ t❤❡r❡ ♠❛② ❜❡ ❛ r❡✈❡rs❛❧ ✐♥ t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ ❢♦①❡s ❛t ❛ ❝❡rt❛✐♥ ♥✉♠❜❡r ♦❢ r❛❜❜✐ts✳ ❚♦ ✜♥❞ ♦✉t✱ s♦❧✈❡ t❤❡ ✐♥❡q✉❛❧✐t②✿ y ′ > 0 ♦r ∆y > 0 =⇒ δxy − γy > 0 =⇒ x > γ/δ .

■t ❢♦❧❧♦✇s t❤❛t✱ ✐♥❞❡❡❞✱ t❤❡ ♥✉♠❜❡r ♦❢ ❢♦①❡s ✐♥❝r❡❛s❡s ✇❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ r❛❜❜✐ts ✐s ❛❜♦✈❡ γ/δ ✱ ♦t❤❡r✇✐s❡ ✐t ❞❡❝r❡❛s❡s✳

◆♦✇ ✇❡ ♣✉t t❤❡s❡ t✇♦ ♣❛rts t♦❣❡t❤❡r✳ ❲❡ ❤❛✈❡ ❢♦✉r s❡❝t♦rs ✐♥ t❤❡ ✜rst q✉❛❞r❛♥t ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❢♦✉r ❞✐✛❡r❡♥t ❝❤♦✐❝❡s ♦❢ t❤❡ s✐❣♥s ♦❢ x′ ❛♥❞ y ′ ✱ ♦r ∆x ❛♥❞ ∆y ✿

✹✳✸✳ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧

✷✻✾

■♥ ❡✐t❤❡r ❝❛s❡✱ t❤❡ r❡s✉❧t ✐s ❛ r♦✉❣❤ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❞②♥❛♠✐❝s ♦♥ t❤❡ ❧♦❝❛❧ ❧❡✈❡❧✿ ✐❢ t❤✐s ✐s t❤❡ ❝✉rr❡♥t st❛t❡✱ t❤❡♥ t❤✐s ✐s t❤❡ ❞✐r❡❝t✐♦♥ ✐t ✐s ❣♦✐♥❣✳ ■t ✐s ❛ ✈❡❝t♦r ✜❡❧❞ ✦ ❲❡ ✈✐s✉❛❧✐③❡ t❤✐s ✈❡❝t♦r ✜❡❧❞ ✇✐t❤ t❤❡ s❛♠❡ ♣❛r❛♠❡t❡rs ❛s ❜❡❢♦r❡✿

❚❤❡ ❛rr♦✇s ❛r❡♥✬t ♠❡❛♥t ②❡t t♦ ❜❡ ❝♦♥♥❡❝t❡❞ ✐♥t♦ ❝✉r✈❡s t♦ ♣r♦❞✉❝❡ s♦❧✉t✐♦♥s✳ ❚❤❡ ♦♥❧② ❢♦✉r ❞✐st✐♥❝t s♦❧✉t✐♦♥s ✇❡ ❦♥♦✇ ❢♦r s✉r❡ ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

• t❤❡ ❞❡❝❧✐♥❡ ♦❢ t❤❡ ❢♦①❡s ✐♥ t❤❡ ❛❜s❡♥❝❡ r❛❜❜✐ts ✕ ♦♥ t❤❡ y ✲❛①✐s❀

• t❤❡ ❡①♣❧♦s✐♦♥ ♦❢ t❤❡ r❛❜❜✐ts ✐♥ t❤❡ ❛❜s❡♥❝❡ ♦❢ ❢♦①❡s ✕ ♦♥ t❤❡ x✲❛①✐s❀

• t❤❡ ❢r❡❡③✐♥❣ ♦❢ ❜♦t❤ r❛❜❜✐ts ❛♥❞ ❢♦①❡s ❛t t❤❡ s♣❡❝✐❛❧ ❧❡✈❡❧s ✕ ✐♥ t❤❡ ♠✐❞❞❧❡❀ ❛♥❞ ❛❧s♦ • t❤❡ ❢r❡❡③✐♥❣ ♦❢ ❜♦t❤ r❛❜❜✐ts ❛♥❞ ❢♦①❡s ❛t t❤❡ ③❡r♦ ❧❡✈❡❧✳

❊✐t❤❡r ♦❢ t❤❡ ❧❛st t✇♦ ✐s ❛ ❝♦♥st❛♥t s♦❧✉t✐♦♥ ❝❛❧❧❡❞ ❛♥ ❡q✉✐❧✐❜r✐✉♠✳ ❚❤❡ ♠❛✐♥✱ ♥♦♥✲③❡r♦✱ ❡q✉✐❧✐❜r✐✉♠ ✐s✿

x(t) = γ/δ, y(t) = α/β . ❲❤❛t ❛❜♦✉t t❤❡ r❡st ♦❢ t❤❡ s♦❧✉t✐♦♥s❄ ■♥ ♦r❞❡r t♦ ❝♦♥✜r♠ t❤❛t t❤❡ s♦❧✉t✐♦♥s ❝✐r❝❧❡ t❤❡ ♠❛✐♥ ❡q✉✐❧✐❜r✐✉♠✱ ✇❡ ♥❡❡❞ ❛ ♠♦r❡ ♣r❡❝✐s❡ ❛♥❛❧②s✐s✳ ■♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦✉r s❡❝t♦rs✱ t❤❡ ♠♦♥♦t♦♥✐❝✐t② ♦❢ t❤❡ s♦❧✉t✐♦♥s ❤❛s ❜❡❡♥ ♣r♦✈❡♥✳ ❍♦✇❡✈❡r✱ ✐t ✐s st✐❧❧ ♣♦ss✐❜❧❡ t❤❛t s♦♠❡ s♦❧✉t✐♦♥s ✇✐❧❧ st❛② ✇✐t❤✐♥ t❤❡ s❡❝t♦r ✇❤❡♥ ♦♥❡ ♦r ❜♦t❤ ♦❢ x ❛♥❞ y ❜❡❤❛✈❡ ❛s②♠♣t♦t✐❝❛❧❧②✿

x(t) → a ❛♥❞✴♦r y(t) → b ❛s t → ∞ . ❙✐♥❝❡ ❜♦t❤ ❢✉♥❝t✐♦♥s ❛r❡ ♠♦♥♦t♦♥✐❝✱ t❤✐s ✐♠♣❧✐❡s t❤❛t

x′ (t) → 0 ❛♥❞✴♦r y ′ (t) → 0 ❛s t → ∞ , ❛♥❞ t❤❡ s❛♠❡ ❢♦r ∆x ❛♥❞ ∆y ✳ ❲❡ ❝❛♥ s❤♦✇ t❤❛t t❤✐s ✐s ✐♠♣♦ss✐❜❧❡✳ ❋♦r ❡①❛♠♣❧❡✱ s✉♣♣♦s❡ ✇❡ st❛rt ✐♥ t❤❡ ❜♦tt♦♠ r✐❣❤t s❡❝t♦r✱ ✐✳❡✳✱ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛r❡✿

• x(0) = p > γ/δ ❀

• y(0) = q < α/β ✳

✹✳✹✳

❙♦❧✈✐♥❣ t❤❡ ▲♦t❦❛✕❱♦❧t❡rr❛ ❡q✉❛t✐♦♥s

✷✼✵

❚❤❡♥✱ ❢♦r ❛s ❧♦♥❣ ❛s t❤❡ s♦❧✉t✐♦♥ ✐s ✐♥ t❤✐s s❡❝t♦r✱ ✇❡ ❤❛✈❡

• x′ > 0 =⇒ x > p • y ′ > 0 =⇒ y > q

❚❤❡r❡❢♦r❡✱

❚❤✐s ♥✉♠❜❡r ✐s ❛ ❣❛♣ ❜❡t✇❡❡♥

y′

y ′ = y(δx − γ) > q(δp − γ) > 0 . ❛♥❞

0✳

❚❤❡r❡❢♦r❡✱

❢♦❧❧♦✇ t❤❛t t❤❡ s♦❧✉t✐♦♥ ✇✐❧❧ r❡❛❝❤ t❤❡ ❧✐♥❡

y = α/β

y′

❝❛♥♥♦t ❞✐♠✐♥✐s❤ t♦

0✱

❛♥❞ t❤❡ s❛♠❡ ✐s tr✉❡ ❢♦r

∆y ✳

■t

✏✐♥ ✜♥✐t❡ t✐♠❡✑✳

❊①❡r❝✐s❡ ✹✳✸✳✶

Pr♦✈❡ t❤❡ ❛♥❛❧♦❣♦✉s ❢❛❝ts ❛❜♦✉t t❤❡ t❤r❡❡ r❡♠❛✐♥✐♥❣ s❡❝t♦rs✳

❲❡ ❤❛✈❡ ❞❡♠♦♥str❛t❡❞ t❤❛t ❛ s♦❧✉t✐♦♥ ✇✐❧❧ ❣♦ ❛r♦✉♥❞ t❤❡ ♠❛✐♥ ❡q✉✐❧✐❜r✐✉♠✱ ❜✉t ✇❤❡♥ ✐t ❝♦♠❡s ❜❛❝❦✱ ✇✐❧❧ ✐t ❜❡ ❝❧♦s❡r t♦ t❤❡ ❝❡♥t❡r✱ ❢❛rt❤❡r✱ ♦r ✇✐❧❧ ✐t ❝♦♠❡ t♦ t❤❡ s❛♠❡ ❧♦❝❛t✐♦♥❄

❚❤❡ ✜rst ♦♣t✐♦♥ ✐s ✐♥❞✐❝❛t❡❞ ❜② ♦✉r s♣r❡❛❞s❤❡❡t r❡s✉❧t✳ ◆❡①t✱ ✇❡ s❡t t❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧ ❛s✐❞❡ ❛♥❞ ❝♦♥❝❡♥tr❛t❡ ♦♥ s♦❧✈✐♥❣ ❛♥❛❧②t✐❝❛❧❧② ♦✉r s②st❡♠ ♦❢ ❖❉❊s t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✱ ✐s t❤✐s

❛ ❝②❝❧❡ ♦r ❛ s♣✐r❛❧ ❄

✹✳✹✳ ❙♦❧✈✐♥❣ t❤❡ ▲♦t❦❛✕❱♦❧t❡rr❛ ❡q✉❛t✐♦♥s

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦

❡❧✐♠✐♥❛t❡ t✐♠❡

❢r♦♠ t❤❡ ❡q✉❛t✐♦♥s ✭x

❚❤✐s st❡♣ ✐s ♠❛❞❡ ♣♦ss✐❜❧❡ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣

> 0, y > 0✮✿

   dx = αx − βxy, dt   dy = δxy − γy. dt tr✐❝❦✳

❲❡ ✐♥t❡r♣r❡t t❤❡s❡ ❞❡r✐✈❛t✐✈❡s ✐♥ t❡r♠s ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧

❢♦r♠s✿

dx αx − βxy dy =⇒ dt = δxy − γy

dx = (αx − βxy)dt =⇒ dt = dy = (δxy − γy)dt ❚❤❡r❡❢♦r❡✱

dt = ❲❡ ♥❡①t

s❡♣❛r❛t❡ ✈❛r✐❛❜❧❡s ✿

dx dy = . αx − βxy δxy − γy

α − βy δx − γ dx = dy . x y

✹✳✹✳

❲❡

❙♦❧✈✐♥❣ t❤❡ ▲♦t❦❛✕❱♦❧t❡rr❛ ❡q✉❛t✐♦♥s ✐♥t❡❣r❛t❡ ✿

Z 

❛♥❞ ✇❡ ❤❛✈❡✿

❚❤❡ s②st❡♠ ✐s

✷✼✶

γ dx = δ− x

Z 

 α − β dy , y

δx − γ ln x = α ln y − βy + C .

s♦❧✈❡❞ ✦

❇✉t ✇❤❛t ❞♦❡s t❤✐s ❡q✉❛t✐♦♥ ♠❡❛♥❄ ❊✈❡r② s♦❧✉t✐♦♥

x = x(t)

❛♥❞

y = y(t)✱

✇❤❡♥ s✉❜st✐t✉t❡❞ ✐♥t♦ t❤❡ ❢✉♥❝t✐♦♥

G(x, y) = δx − γ ln x − α ln y + βy , ♣r♦❞✉❝❡s ❛ ❝♦♥st❛♥t✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤✐s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s ❛

❧❡✈❡❧ ❝✉r✈❡

♦❢

z = G(x, y)✳

❊①❡r❝✐s❡ ✹✳✹✳✶ Pr♦✈❡ t❤❡ ❧❛st st❛t❡♠❡♥t✳

❖♥❝❡ ✇❡ ❤❛✈❡ ♥♦ ❞❡r✐✈❛t✐✈❡s ❧❡❢t✱ ✇❡ ❞❡❝❧❛r❡ t❤❡ s②st❡♠ s♦❧✈❡❞ ❡✈❡♥ t❤♦✉❣❤ ♦♥❧② ✐♠♣❧✐❝✐t❧②✳ ✇❡ ❞♦♥✬t ❤❛✈❡ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛s ❢♦r

x = x(t)

♦r

y = y(t)✱

❊✈❡♥ t❤♦✉❣❤

✇❡ ❝❛♥ ✉s❡ ✇❤❛t ✇❡ ❤❛✈❡ t♦ ❢✉rt❤❡r st✉❞② t❤❡

q✉❛❧✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ s②st❡♠✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❢❛❝t t❤❛t t❤✐s ✐s ❛ ❧❡✈❡❧ ❝✉r✈❡ ❛❧r❡❛❞② s✉❣❣❡sts t❤❛t t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s

♥♦t ❛ s♣✐r❛❧ ✿

❏✉st tr② t♦ ✐♠❛❣✐♥❡ s✉❝❤ ❛ s✉r❢❛❝❡ t❤❛t ✐ts ❧❡✈❡❧ ❝✉r✈❡s ❛r❡ s♣✐r❛❧s✳ ❲❡ t✉r♥ ✐♥st❡❛❞ t♦ t❤❡ ❛❝t✉❛❧ ❢✉♥❝t✐♦♥✳

α = 3✱ β = 2✱ γ = 3✱ δ = 1✮✿

❋✐rst✱ ✇❡ ♣❧♦t ✐t ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t ✭

❚❤❡ ❧❡✈❡❧ ❝✉r✈❡s ❛r❡ ✈✐s✐❜❧❡✳

❙♦♠❡ ♦❢ t❤❡♠ ❛r❡ ❝❧❡❛r❧② ❝✐r❝✉❧❛r ❛♥❞ ♦t❤❡rs ❛r❡♥✬t✳

❛r❡♥✬t s❤♦✇♥ ❛❧❧ t❤❡ ✇❛② t♦ t❤❡ ❛①❡s ❜❡❝❛✉s❡ t❤❡ ✈❛❧✉❡ ♦❢

G

❚❤❡ r❡❛s♦♥ ✐s t❤❛t t❤❡②

r✐s❡s s♦ q✉✐❝❦❧② ✭✐♥ ❢❛❝t✱ ❛s②♠♣t♦t✐❝❛❧❧②✮✳

❆s ❡①♣❡❝t❡❞✱ t❤❡ s✉r❢❛❝❡ s❡❡♠s t♦ ❤❛✈❡ ❛ s✐♥❣❧❡ ♠✐♥✐♠✉♠ ♣♦✐♥t✳ ▲❡t✬s ♣r♦✈❡ t❤❛t ❛❧❣❡❜r❛✐❝❛❧❧②✿

∂G (x, y) = δ − γ/x, ∂x ∂G (x, y) = −α/y + β. ∂y

✹✳✹✳ ❙♦❧✈✐♥❣ t❤❡ ▲♦t❦❛✕❱♦❧t❡rr❛ ❡q✉❛t✐♦♥s

✷✼✷

❲❡ ♥❡①t ✜♥❞ t❤❡ ❡①tr❡♠❡ ♣♦✐♥ts ♦❢ G✳ ❲❡ s❡t t❤❡ t✇♦ ❞❡r✐✈❛t✐✈❡s ❡q✉❛❧ t♦ ③❡r♦ ❛♥❞ s♦❧✈❡ ❢♦r x ❛♥❞ y ✿ γ ∂G (x, y) = δ − γ/x = 0 =⇒ x = ∂x δ ∂G α (x, y) = −α/y + β = 0 =⇒ y = ∂y β ❚❤✐s ♣♦✐♥t ✐s ✐♥❞❡❡❞ ♦✉r ♠❛✐♥ ❡q✉✐❧✐❜r✐✉♠ ♣♦✐♥t✳ ❚❤❡ s✉r❢❛❝❡ ❤❡r❡ ❤❛s ❛ ❤♦r✐③♦♥t❛❧ t❛♥❣❡♥t ♣❧❛♥❡✳ ❲❡ ❤❛✈❡

❛❧s♦ ❞❡♠♦♥str❛t❡❞ t❤❛t t❤❡r❡ ❛r❡ ♥♦ ♦t❤❡rs ♣♦✐♥ts ❧✐❦❡ t❤❛t✦

❇✉t ❝♦✉❧❞ t❤✐s ❜❡ ❛ ♠❛①✐♠✉♠ ♣♦✐♥t❄ ❏✉st ❛s y = x3 ❝r♦ss❡s t❤❡ x✲❛①✐s ❛t 0 ❞❡❣r❡❡s✱ ❛ s✉r❢❛❝❡ ❝❛♥ ❝r♦ss ✐ts t❛♥❣❡♥t ♣❧❛♥❡✳ ❲❡ ❝♦♠♣✉t❡ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡s✿ ∂ 2G 1 (x, y) = γ 2 > 0, 2 ∂x x 1 ∂ 2G (x, y) = α > 0. ∂y 2 y2

❇♦t❤ ❛r❡ ♣♦s✐t✐✈❡ t❤r♦✉❣❤♦✉t✱ t❤❡r❡❢♦r❡✱ ❡✐t❤❡r ♦❢ t❤❡ ❝r♦ss✲s❡❝t✐♦♥s ♦❢ t❤❡ s✉r❢❛❝❡ ❛❧♦♥❣ t❤❡ ❛①❡s ❤❛✈❡ ❛ ♠✐♥✐♠✉♠ ♣♦✐♥t ❤❡r❡ ❛♥❞ ✐t ❤❛s t♦ st❛② ♦♥ ♦♥❡ s✐❞❡ ♦❢ t❤❡ ♣❧❛♥❡✳ ❍♦✇❡✈❡r✱ ✐t ♠✐❣❤t ❝r♦ss ✐t ❛t ♦t❤❡r ❞✐r❡❝t✐♦♥s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s st✐❧❧ ♠✐❣❤t ❜❡ ❛ s❛❞❞❧❡ ♣♦✐♥t✦ ❲❡ ✐♥✈♦❦❡ t❤❡ ❙❡❝♦♥❞ ❉❡r✐✈❛t✐✈❡ ❚❡st ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✺ t♦ r❡s♦❧✈❡ t❤✐s✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❍❡ss✐❛♥ ♠❛tr✐① ✭❞✐s❝✉ss❡❞ ✐♥ ❈❤❛♣t❡r ✹❍❉✲✸✮ ♦❢ G✳ ■t ✐s t❤❡ 2 × 2 ♠❛tr✐① ♦❢ t❤❡ ❢♦✉r ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ♦❢ G✿     ∂ 2G  ∂x2 H(x, y) =   ∂ 2G ∂y∂x

∂ 2G 1 0  γ x 2 ∂x∂y  = . 1 ∂ 2G   0 α 2 y ∂y 2

❍❡r❡✱ ✐♥ ❛❞❞✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ♠✐①❡❞ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡s✿

∂ 2G ∂ 2G (x, y) = (x, y) = 0 . ∂x∂y ∂y∂x

◆❡①t✱ ✇❡ ❧♦♦❦ ❛t t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ t❤❡ ❍❡ss✐❛♥✿ D(x, y) = det(H(x, y)) =

■t✬s ♣♦s✐t✐✈❡✦ ❚❤❡r❡❢♦r❡✱ t❤❡ ♣♦✐♥t ✐s ❛ ♠✐♥✐♠✉♠✳



1 γ 2 x

   1 αγ · α 2 = 2 2 > 0. y xy

❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ❡✈❡r② ❧❡✈❡❧ ❝✉r✈❡ ♦❢ G✱ ✐✳❡✳✱ ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ s②st❡♠✱ ❣♦❡s ❛r♦✉♥❞ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❛♥❞ ❝♦♠❡s ❜❛❝❦ t♦ ❝r❡❛t❡ ❛ ❝②❝❧❡✳ ❆♥ ❡❛s✐❡r✱ ❜✉t ♠♦r❡ ❛❞ ❤♦❝✱ ✇❛② t♦ r❡❛❝❤ t❤✐s ❝♦♥❝❧✉s✐♦♥ ✐s t♦ ✐♠❛❣✐♥❡ t❤❛t ❛ s♦❧✉t✐♦♥ st❛rts ♦♥✱ s❛②✱ t❤❡ ❧✐♥❡ y = α/β ❛t x = x0 t♦ t❤❡ r✐❣❤t ♦❢ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❛♥❞ t❤❡♥ ❝♦♠❡s ❜❛❝❦ t♦ t❤❡ ❧✐♥❡ ❛t x = x1 ✳ ❙✐♥❝❡ t❤✐s ✐s ❛ ❧❡✈❡❧ ❝✉r✈❡✱ ✇❡ ❤❛✈❡ G(x0 , α/β) = G(x1 , α/β)✳ ❆❝❝♦r❞✐♥❣ t♦ ❘♦❧❧❡✬s ❚❤❡♦r❡♠ ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✺✱ t❤✐s > 0 ❛❧♦♥❣ t❤✐s ❧✐♥❡✳ ❝♦♥tr❛❞✐❝ts ♦✉r ❝♦♥❝❧✉s✐♦♥ t❤❛t ∂G ∂x P❧♦tt❡❞ ✇✐t❤ t❤❡ s❛♠❡ ♣❛r❛♠❡t❡rs✱ t❤✐s ✐s ✇❤❛t t❤❡s❡ ❝✉r✈❡s ❧♦♦❦ ❧✐❦❡✿

✹✳✺✳ ❱❡❝t♦r ✜❡❧❞s ❛♥❞ s②st❡♠s ♦❢ ❖❉❊s

✷✼✸

❊①❡r❝✐s❡ ✹✳✹✳✷

Pr♦✈❡ t❤❛t t❤❡ ♣r❡❞❛t♦r✲♣r❡② ❞✐s❝r❡t❡ ♠♦❞❡❧✱ ✐✳❡✳✱ ❊✉❧❡r✬s ♠❡t❤♦❞ ❢♦r t❤✐s s②st❡♠ ♦❢ ❖❉❊s✱ ♣r♦❞✉❝❡s s♦❧✉t✐♦♥s t❤❛t s♣✐r❛❧ ❛✇❛② ❢r♦♠ t❤❡ ❡q✉✐❧✐❜r✐✉♠✳ ❍✐♥t✿

✹✳✺✳ ❱❡❝t♦r ✜❡❧❞s ❛♥❞ s②st❡♠s ♦❢ ❖❉❊s ◆✉♠❡r♦✉s ♣r♦❝❡ss❡s ❛r❡ ♠♦❞❡❧❡❞ ❜② s②st❡♠s ♦❢ ❖❉❊s✳ ❋♦r ❡①❛♠♣❧❡✱ ✐❢ ❧✐tt❧❡ ✢❛❣s ❛r❡ ♣❧❛❝❡❞ ♦♥ t❤❡ ❧❛✇♥✱ t❤❡♥ t❤❡✐r ❞✐r❡❝t✐♦♥s t❛❦❡♥ t♦❣❡t❤❡r r❡♣r❡s❡♥t ❛ s②st❡♠ ♦❢ ❖❉❊s✱ ✇❤✐❧❡ t❤❡ ✭✐♥✈✐s✐❜❧❡✮ ❛✐r ✢♦✇ ✐s t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤✐s s②st❡♠✳

❆ s✐♠✐❧❛r ✐❞❡❛ ✐s ✉s❡❞ t♦ ♠♦❞❡❧ ❛ ✢✉✐❞ ✢♦✇✳ ❚❤❡ ❞②♥❛♠✐❝s ♦❢ ❡❛❝❤ ♣❛rt✐❝❧❡ ✐s ❣♦✈❡r♥❡❞ ❜② t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✢♦✇✱ ❛t ❡❛❝❤ ❧♦❝❛t✐♦♥✱ t❤❡ s❛♠❡ ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡✳

❚♦ s♦❧✈❡ s✉❝❤ ❛ s②st❡♠ ✇♦✉❧❞ r❡q✉✐r❡ tr❛❝✐♥❣ t❤❡ ♣❛t❤ ♦❢ ❡✈❡r② ♣❛rt✐❝❧❡ ♦❢ t❤❡ ❧✐q✉✐❞✳ ▲❡t✬s r❡✈✐❡✇ t❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧ ♦❢ ❛ ✢♦✇✿ ❣✐✈❡♥ ❛ ✢♦✇ ♦♥ ❛ ♣❧❛♥❡✱ tr❛❝❡ ❛ s✐♥❣❧❡ ♣❛rt✐❝❧❡ ♦❢ t❤✐s str❡❛♠✳

✹✳✺✳ ❱❡❝t♦r ✜❡❧❞s ❛♥❞ s②st❡♠s ♦❢ ❖❉❊s

✷✼✹

❋♦r ❜♦t❤ ❝♦♦r❞✐♥❛t❡s✱ x ❛♥❞ y ✱ t❤❡ ❢♦❧❧♦✇✐♥❣ t❛❜❧❡ ✐s ❜❡✐♥❣ ❜✉✐❧t✳ ❚❤❡ ✐♥✐t✐❛❧ t✐♠❡ t0 ❛♥❞ t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ p0 ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ✜rst r♦✇ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t✳ ❆s ✇❡ ♣r♦❣r❡ss ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡✱ ♥❡✇ ♥✉♠❜❡rs ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ♥❡①t r♦✇ ♦❢ ♦✉r s♣r❡❛❞s❤❡❡t✿ tn , vn , pn , n = 1, 2, 3, ... ❚❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❛r❡ ✉s❡❞✿

tn+1 = tn + ∆t ❛♥❞ ❚❤❡ r❡s✉❧t ✐s ❛ ❣r♦✇✐♥❣ t❛❜❧❡ ♦❢ ✈❛❧✉❡s✿

pn+1 = pn + vn+1 · ∆t .

✐t❡r❛t✐♦♥ n t✐♠❡ tn ✐♥✐t✐❛❧✿ 0 3.5 1 3.6 ... ... 1000 103.5 ... ...

✈❡❧♦❝✐t② vn ❧♦❝❛t✐♦♥ pn −− 22 33 25.3 ... ... 4 336 ... ...

❙♦✱ ✐♥st❡❛❞ ♦❢ t✇♦ ✭✈❡❧♦❝✐t② ✕ ❧♦❝❛t✐♦♥✱ ❛s ❜❡❢♦r❡✮✱ t❤❡r❡ ✇✐❧❧ ❜❡ ❢♦✉r ♠❛✐♥ ❝♦❧✉♠♥s ✇❤❡♥ t❤❡ ♠♦t✐♦♥ ✐s t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ s✐① ✇❤❡♥ ✐t ✐s t❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧✿ t✐♠❡ ❤♦r✐③✳ ❤♦r✐③✳ ✈❡rt✳ ✈❡rt✳ t ✈❡❧✳ x′ ❧♦❝✳ x ✈❡❧✳ y ′ ❧♦❝✳ y n 0 3.5 −− 22 −− 3 3.6 33 25.3 4 3.5 1 ... ... ... ... ... ... 1000 103.5 4 336 66 4 ... ... ... ... ... ...

... ... ... ... ... ... ...

❊①❛♠♣❧❡ ✹✳✺✳✶✿ s✐♠✉❧❛t✐♦♥

❘❡❝❛❧❧ t❤❡ ❡①❛♠♣❧❡s s✉❝❤ ✢♦✇s✳ ■❢ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✢♦✇ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❧♦❝❛t✐♦♥✿

vn+1 = 0.2 · pn , ❢♦r ❜♦t❤ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧✱ t❤❡ r❡s✉❧t ✐s ♣❛rt✐❝❧❡s ✢②✐♥❣ ❛✇❛② ❢r♦♠ t❤❡ ❝❡♥t❡r✱ ❢❛st❡r ❛♥❞ ❢❛st❡r✿

■❢ t❤❡ ❤♦r✐③♦♥t❛❧ ✈❡❧♦❝✐t② ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✈❡rt✐❝❛❧ ❧♦❝❛t✐♦♥ ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t② ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ♥❡❣❛t✐✈❡ ♦❢ t❤❡ ❤♦r✐③♦♥t❛❧ ❧♦❝❛t✐♦♥✱ t❤❡ r❡s✉❧t r❡s❡♠❜❧❡s r♦t❛t✐♦♥✿

✹✳✺✳ ❱❡❝t♦r ✜❡❧❞s ❛♥❞ s②st❡♠s ♦❢ ❖❉❊s

✷✼✺

◆♦✇✱ s✉♣♣♦s❡ t❤❡ ✈❡❧♦❝✐t② ❝♦♠❡s ❢r♦♠ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛s ❛s ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ❧♦❝❛t✐♦♥✿ u = f (x, y), v = g(x, y) ,

❛r❡ t❤❡r❡ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛s ❢♦r t❤❡ ❧♦❝❛t✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✿ x = x(t), y = y(t)?

❲❡ ❛ss✉♠❡ t❤❛t t❤❡r❡ ✐s ❛ ✈❡rs✐♦♥ ♦❢ ♦✉r r❡❝✉rs✐✈❡ r❡❧❛t✐♦♥✱ pn+1 = pn + vn+1 · ∆t ,

❢♦r ❡✈❡r② ∆t > 0 s♠❛❧❧ ❡♥♦✉❣❤✳ ❚❤❡♥ ♦✉r t✇♦ ❢✉♥❝t✐♦♥s ❤❛✈❡ t♦ s❛t✐s❢② ❢♦r x✿ vn = f (pn ) ❛♥❞ pn = x(tn ) ,

❛♥❞ ❢♦r y ✿

vn = g(pn ) ❛♥❞ pn = y(tn ) ,

❲❡ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦✱ ❛s ✇❡❧❧ ❛s t = tn ✱ ✐♥t♦ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❢♦r pn+1 ❢♦r x✿ ❛♥❞ ❢♦r y ✿ ❚❤❡♥✱ ✇❡ ❤❛✈❡ ❢♦r x✿ ❛♥❞ ❢♦r y ✿

x(t + ∆t) = x(t) + f (x(t + ∆t), y(t + ∆t)) · ∆t , y(t + ∆t) = y(t) + g(x(t + ∆t), y(t + ∆t)) · ∆t . x(t + ∆t) − x(t) = f (x(t + ∆t), y(t + ∆t)) , ∆t y(t + ∆t) − y(t) = g(x(t + ∆t), y(t + ∆t)) . ∆t

❚❛❦✐♥❣ t❤❡ ❧✐♠✐t ♦✈❡r ∆t → 0 ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠✿ 

x′ (t) = f (x(t), y(t)), y ′ (t) = g(x(t), y(t)),

♣r♦✈✐❞❡❞ x = x(t), y = y(t) ❛r❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t ❛♥❞ u = f (x, y), v = g(x, y) ❛r❡ ❝♦♥t✐♥✉♦✉s ❛t (x(t), y(t))✳ ◆♦✇✱ ❧❡t✬s r❡❝❛❧❧ ❢r♦♠ ❈❤❛♣t❡r ✹❍❉✲✻ t❤❛t ❛ ✈❡❝t♦r ✜❡❧❞ s✉♣♣❧✐❡s ❛ ❞✐r❡❝t✐♦♥ t♦ ❡✈❡r② ❧♦❝❛t✐♦♥✱ ✐✳❡✳✱ t❤❡r❡ ✐s ❛ ✈❡❝t♦r ❛tt❛❝❤❡❞ t♦ ❡❛❝❤ ♣♦✐♥t ♦❢ t❤❡ ♣❧❛♥❡✿ point 7→ vector .

❆ ✈❡❝t♦r ✜❡❧❞ ✐♥ ❞✐♠❡♥s✐♦♥ 2 ✐s ✐♥ ❢❛❝t ❛♥② ❢✉♥❝t✐♦♥✿ ❣✐✈❡♥ ❜② t✇♦ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✿

F : R2 → R2 ,

F (x, y) =< f (x, y), g(x, y) > .

❚❤❡♥✱ t❤❡ ✈❡❝t♦r ✜❡❧❞ ❞❡✜♥❡s ❛ s②st❡♠ ♦❢ t✇♦ ✭t✐♠❡✲✐♥❞❡♣❡♥❞❡♥t✮ ❖❉❊s✳

✹✳✺✳

❱❡❝t♦r ✜❡❧❞s ❛♥❞ s②st❡♠s ♦❢ ❖❉❊s

✷✼✻

❉❡✜♥✐t✐♦♥ ✹✳✺✳✷✿ s♦❧✉t✐♦♥ ♦❢ s②st❡♠ ♦❢ ❖❉❊s ❆

s♦❧✉t✐♦♥

♦❢ ❛ s②st❡♠ ♦❢ ❖❉❊s ✐s ❛ ♣❛✐r ♦❢ ❢✉♥❝t✐♦♥s

x = x(t)

❛♥❞

♣❛r❛♠❡tr✐❝ ❝✉r✈❡✮ ✇✐t❤ ❡✐t❤❡r ♦♥❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ ❢♦r ❡✈❡r②

t

✐♥

I

y = y(t) ✭❛ I s✉❝❤ t❤❛t

✇❡ ❤❛✈❡✿



x′ (t) = f (x(t), y(t)) y ′ (t) = g(x(t), y(t))

♦r ❛❜❜r❡✈✐❛t❡❞✿



x′ = f (x, y) y ′ = g(x, y)

❚❤❡ ✈❡❝t♦r ✜❡❧❞ ♦❢ t❤✐s s②st❡♠ ✐s t❤❡ s❧♦♣❡ ✜❡❧❞ ♦❢ t❤❡ ❖❉❊✳ ❍♦✇ ❞♦ ✇❡ ✈✐s✉❛❧✐③❡ t❤❡ s♦❧✉t✐♦♥s ♦❢ s✉❝❤ ❛ s②st❡♠❄ ❲✐t❤ ❛ s✐♥❣❧❡ ❖❉❊✱

y ′ = f (t) =⇒ y = y(t) ,

❣r❛♣❤s✱

ty ✲♣❧❛♥❡✳ ❚❤✐s t✐♠❡✱ t❤❡ s♦❧✉t✐♦♥s ❛r❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✦ ❚❤❡✐r ❣r❛♣❤s✱ ✐✳❡✳✱ t❤❡ ❝♦❧❧❡❝t✐♦♥s ♦❢ (t, x(t), y(t)) ❧✐❡ ✐♥ t❤❡ 3✲ ❞✐♠❡♥s✐♦♥❛❧ txy ✲s♣❛❝❡✳ ❚❤❛t ✐s ✇❤②✱ ✇❡✱ ✐♥st❡❛❞✱ ♣❧♦t t❤❡✐r ✐♠❛❣❡s✱ ✐✳❡✳✱ t❤❡ ❝♦❧❧❡❝t✐♦♥s ♦❢ ♣♦✐♥ts (x(t), y(t)) ♦♥ t❤❡ xy ✲♣❧❛♥❡✳ ■♥ t❤❡ t❤❡♦r② ♦❢ ❖❉❊s✱ t❤❡② ❛r❡ ❛❧s♦ ❦♥♦✇♥ ❛s tr❛❥❡❝t♦r✐❡s✱ ♦r ♣❛t❤s✳

✇❡ s✐♠♣❧② ♣❧♦t t❤❡✐r

✐✳❡✳✱ t❤❡ ❝♦❧❧❡❝t✐♦♥s ♦❢ ♣♦✐♥ts

(t, y(t))✱

♦❢ s♦♠❡ ♦❢ t❤❡♠ ♦♥ t❤❡

❚❤❡♥ t❤❡ ✈❡❝t♦rs ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞ ❛r❡ t❛♥❣❡♥t t♦ t❤❡s❡ tr❛❥❡❝t♦r✐❡s✳ ❙✐♥❝❡ t❤❡ ✈❡❝t♦r ✜❡❧❞ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t✱ s✉❝❤ ❛ r❡♣r❡s❡♥t❛t✐♦♥ ✐s ♦❢t❡♥ s✉✣❝✐❡♥t ❛s ❡①♣❧❛✐♥❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t✳

❚❤❡♦r❡♠ ✹✳✺✳✸✿ ❙❤✐❢t❡❞ ❙♦❧✉t✐♦♥s x = x(t), y = y(t) ✐s ❛ s♦❧✉t✐♦♥ x(t + s), y = y(t + s) ❢♦r ❛♥② r❡❛❧ s✳

■❢

♦❢ t❤❡ s②st❡♠ ♦❢ ❖❉❊s✱ t❤❡♥ s♦ ✐s

x =

■t ✐s ❛❧s♦ ✐♠♣♦rt❛♥t t♦ ❜❡ ❛✇❛r❡ ♦❢ t❤❡ ❢❛❝t t❤❛t t❤❡ t❤❡♦r② ♦❢ s②st❡♠s ♦❢ ❖❉❊s ✏✐♥❝❧✉❞❡s✑ t❤❡ t❤❡♦r② ♦❢ s✐♥❣❧❡ ❖❉❊s✳ ❘❡❝❛❧❧✱ ✜rst✱ ❤♦✇ t❤❡ ❣r❛♣❤ ♦❢ ❡✈❡r② ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ tr❛❥❡❝t♦r② ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿

y = r(x) −→



x = t, y = r(x).

❙✐♠✐❧❛r❧②✱ t❤❡ s♦❧✉t✐♦♥s ♦❢ ❡✈❡r② t✐♠❡✲✐♥❞❡♣❡♥❞❡♥t ❖❉❊ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ tr❛❥❡❝t♦r✐❡s ♦❢ ❛ s②st❡♠ ♦❢ t✇♦ ❖❉❊s✿



y = g(x) −→



x′ = 1, y ′ = g(y).

✹✳✺✳ ❱❡❝t♦r ✜❡❧❞s ❛♥❞ s②st❡♠s ♦❢ ❖❉❊s

✷✼✼

❉❡✜♥✐t✐♦♥ ✹✳✺✳✹✿ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ ❋♦r ❛ ❣✐✈❡♥ s②st❡♠ ♦❢ ❖❉❊s ❛♥❞ ❛ ❣✐✈❡♥ tr✐♣❧❡ (t0 , x0 , y0 )✱ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜✲ ❧❡♠✱ ♦r ❛♥ ■❱P✱ ✐s x′ = f (x, y), y ′ = g(x, y);





x(t0 ) = x0 , y(t0 ) = y0 ;

❛♥❞ ✐ts s♦❧✉t✐♦♥ ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❖❉❊ t❤❛t s❛t✐s✜❡s t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❛❜♦✈❡✳ ❇② t❤❡ ❧❛st t❤❡♦r❡♠✱ t❤❡ ✈❛❧✉❡ ♦❢ t0 ❞♦❡s♥✬t ♠❛tt❡r❀ ✐t ❝❛♥ ❛❧✇❛②s ❜❡ ❝❤♦s❡♥ t♦ ❜❡ 0✳

❉❡✜♥✐t✐♦♥ ✹✳✺✳✺✿ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ❲❡ s❛② t❤❛t ❛ s②st❡♠ ♦❢ ❖❉❊s s❛t✐s✜❡s t❤❡ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ❛t ❛ ♣♦✐♥t (t0 , x0 , y0 ) ✇❤❡♥ t❤❡ ■❱P ❛❜♦✈❡ ❤❛s ❛ s♦❧✉t✐♦♥✳ ■❢ ②♦✉r ♠♦❞❡❧ ♦❢ ❛ r❡❛❧✲❧✐❢❡ ♣r♦❝❡ss ❞♦❡s♥✬t s❛t✐s❢② ❡①✐st❡♥❝❡✱ ✐t r❡✢❡❝ts ❧✐♠✐t❛t✐♦♥s ♦❢ ②♦✉r ♠♦❞❡❧✳ ■t ✐s ❛s ✐❢ t❤❡ ♣r♦❝❡ss st❛rts ❜✉t ♥❡✈❡r ❝♦♥t✐♥✉❡s✳

❉❡✜♥✐t✐♦♥ ✹✳✺✳✻✿ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt② ❲❡ s❛② t❤❛t ❛♥ ❖❉❊ s❛t✐s✜❡s t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt② ❛t ❛ ♣♦✐♥t (t0 , x0 , y0 ) ✐❢ ❡✈❡r② ♣❛✐r ♦❢ s♦❧✉t✐♦♥s✱ (x1 , y1 ), (x2 , y2 )✱ ♦❢ t❤❡ ■❱P ❛❜♦✈❡ ❛r❡ ❡q✉❛❧✱ x1 (t) = x2 (t), y1 (t) = y2 (t) ,

❢♦r ❡✈❡r② t ✐♥ s♦♠❡ ♦♣❡♥ ✐♥t❡r✈❛❧ t❤❛t ❝♦♥t❛✐♥s t0 ✳

■❢ ②♦✉r ♠♦❞❡❧ ♦❢ ❛ r❡❛❧✲❧✐❢❡ ♣r♦❝❡ss ❞♦❡s♥✬t s❛t✐s❢② ✉♥✐q✉❡♥❡ss✱ ✐t r❡✢❡❝ts ❧✐♠✐t❛t✐♦♥s ♦❢ ②♦✉r ♠♦❞❡❧✳ ■t✬s ❛s ✐❢ ②♦✉ ❤❛✈❡ ❛❧❧ t❤❡ ❞❛t❛ ❜✉t ❝❛♥✬t ♣r❡❞✐❝t ❡✈❡♥ t❤❡ ♥❡❛r❡st ❢✉t✉r❡✳ ❚❤✉s s②st❡♠s ♦❢ ❖❉❊s ♣r♦❞✉❝❡ ❢❛♠✐❧✐❡s ♦❢ ❝✉r✈❡s ❛s t❤❡ s❡ts ♦❢ t❤❡✐r s♦❧✉t✐♦♥s✳ ❈♦♥✈❡rs❡❧②✱ ✐❢ ❛ ❢❛♠✐❧② ♦❢ ❝✉r✈❡s ✐s ❣✐✈❡♥ ❜② ❛♥ ❡q✉❛t✐♦♥ ✇✐t❤ ❛ s✐♥❣❧❡ ♣❛r❛♠❡t❡r✱ ✇❡ ♠❛② ❜❡ ❛❜❧❡ t♦ ✜♥❞ ❛ s②st❡♠ ♦❢ ❖❉❊s ❢♦r ✐t✳

❊①❛♠♣❧❡ ✹✳✺✳✼✿ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❚❤❡ ❢❛♠✐❧② ♦❢ ✈❡rt✐❝❛❧❧② s❤✐❢t❡❞ ❣r❛♣❤s✱

y = x2 + C ,

❝r❡❛t❡s ❛♥ ❖❉❊ ✐❢ ✇❡ ❞✐✛❡r❡♥t✐❛t❡ ✭✐♠♣❧✐❝✐t❧②✮ ✇✐t❤ r❡s♣❡❝t t♦ t✿ y ′ = 2xx′ .

❙✐♥❝❡ t❤❡s❡ ❛r❡ ❥✉st ❢✉♥❝t✐♦♥s ♦❢ x✱ ✇❡ ❝❛♥ ❝❤♦♦s❡ x = t✳ ❚❤✐s ✐s ❛ ♣♦ss✐❜❧❡ ✈❡❝t♦r ✜❡❧❞ ❢♦r t❤✐s ❢❛♠✐❧②✿ 

x′ = 1, y ′ = 2x.

✹✳✺✳ ❱❡❝t♦r ✜❡❧❞s ❛♥❞ s②st❡♠s ♦❢ ❖❉❊s

✷✼✽

❊①❛♠♣❧❡ ✹✳✺✳✽✿ ❡①♣♦♥❡♥t✐❛❧ ❝❛s❡

❚❤❡ ❢❛♠✐❧② ♦❢ str❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ ❣r❛♣❤s✱ y = Cex ,

❝r❡❛t❡s ❛♥ ❖❉❊s✿

y ′ = Cex x′ .

❚❤✐s ✐s ❛ ♣♦ss✐❜❧❡ ✈❡❝t♦r ✜❡❧❞ ❢♦r t❤✐s ❢❛♠✐❧②✿ 

x′ = 1, y ′ = Cex .

❊①❛♠♣❧❡ ✹✳✺✳✾✿ ❢❛♠✐❧② ♦❢ ❝✐r❝❧❡s

❲❤❛t ❛❜♦✉t t❤❡s❡ ❝♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s❄

✭■♥ t❤❡ ❝❛s❡ ✇❤❡♥ C = 0✱ ✇❡ ❤❛✈❡ t❤❡ ♦r✐❣✐♥✳✮ ❚❤❡② ❛r❡ ❣✐✈❡♥ ❜② x2 + y 2 = C ≥ 0 .

❲❡ ❞✐✛❡r❡♥t✐❛t❡ ✭✐♠♣❧✐❝✐t❧②✮ ✇✐t❤ r❡s♣❡❝t t♦ t✿ 2xx′ + 2yy ′ = 0 .

❲❡ ❝❤♦♦s❡ ✇❤❛t x′ , y ′ ♠✐❣❤t ❜❡ ❡q✉❛❧ t♦ ✐♥ ♦r❞❡r ❢♦r t❤❡ t✇♦ t❡r♠s t♦ ❝❛♥❝❡❧✳ ❚❤✐s ✐s ❛ ♣♦ss✐❜❧❡ ✈❡❝t♦r ✜❡❧❞ ❢♦r t❤✐s ❢❛♠✐❧②✿  x′ = −y, y ′ = x.

❊①❛♠♣❧❡ ✹✳✺✳✶✵✿ ❢❛♠✐❧② ♦❢ ❤②♣❡r❜♦❧❛s

❚❤❡s❡ ❤②♣❡r❜♦❧❛s ❛r❡ ❣✐✈❡♥ ❜② t❤❡s❡ ❡q✉❛t✐♦♥s✿ xy = C .

✭■♥ t❤❡ ❝❛s❡ ✇❤❡♥ C = 0✱ ✇❡ ❤❛✈❡ t❤❡ t✇♦ ❛①❡s✳✮

❚❤❡♥✱ ✇❡ ❤❛✈❡ ❛♥ ❖❉❊✿

x′ y + xy ′ = 0 .

✹✳✻✳

❉✐s❝r❡t❡ s②st❡♠s ♦❢ ❖❉❊s

❚❤✐s ✐s ❛ ♣♦ss✐❜❧❡ ✈❡❝t♦r ✜❡❧❞ ❢♦r t❤✐s ❢❛♠✐❧②✿ 

✷✼✾

x′ = x, y ′ = −y.

◆♦♥❡ ♦❢ t❤❡ ❡①❛♠♣❧❡s ❤❛✈❡ ♣r♦❜❧❡♠s ✇✐t❤ ❡✐t❤❡r ❡①✐st❡♥❝❡ ♦r ✉♥✐q✉❡♥❡ss ✕ ✐♥ ❝♦♥tr❛st t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❖❉❊s✳ ❚❤❡ ♣r♦♦❢s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t t❤❡♦r❡♠s ❧✐❡ ♦✉ts✐❞❡ t❤❡ s❝♦♣❡ ♦❢ t❤✐s ❜♦♦❦✳ ❚❤❡♦r❡♠ ✹✳✺✳✶✶✿ ❊①✐st❡♥❝❡

❙✉♣♣♦s❡ (x0 , y0 ) ✐s ❛ ♣♦✐♥t ♦♥ t❤❡ xy ✲♣❧❛♥❡ ❛♥❞ s✉♣♣♦s❡✿ • H ✐s ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ t❤❛t ❝♦♥t❛✐♥s x0 ✳ • G ✐s ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ t❤❛t ❝♦♥t❛✐♥s y0 ✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t ❢✉♥❝t✐♦♥s z = f (x, y) ❛♥❞ z = g(x, y) ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❛r❡ ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ x ❛♥❞ y ♦♥ H × G✳ ❚❤❡♥ t❤❡ s②st❡♠ ♦❢ ❖❉❊✱ 

x′ = f (x, y), y ′ = g(x, y).

s❛t✐s✜❡s t❤❡ ❡①✐st❡♥❝❡ ♣r♦♣❡rt② ❛t (t0 , x0 , y0 ) ❢♦r ❛♥② t0 ✳

❚❤❡♦r❡♠ ✹✳✺✳✶✷✿ ❯♥✐q✉❡♥❡ss

❙✉♣♣♦s❡ (x0 , y0 ) ✐s ❛ ♣♦✐♥t ♦♥ t❤❡ xy ✲♣❧❛♥❡ ❛♥❞ s✉♣♣♦s❡✿ • H ✐s ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ t❤❛t ❝♦♥t❛✐♥s x0 ✳ • G ✐s ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ t❤❛t ❝♦♥t❛✐♥s y0 ✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t ❢✉♥❝t✐♦♥ z = f (x, y) ❛♥❞ z = g(x, y) ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❛r❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ x ❛♥❞ y ♦♥ H × G✳ ❚❤❡♥ t❤❡ s②st❡♠ ♦❢ ❖❉❊s✱ 

x′ = f (x, y), y ′ = g(x, y).

s❛t✐s✜❡s t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt② ❛t (t0 , x0 , y0 ) ❢♦r ❛♥② t0 ✳

✹✳✻✳ ❉✐s❝r❡t❡ s②st❡♠s ♦❢ ❖❉❊s

❉✐s❝r❡t❡ ❖❉❊s ❛♣♣r♦①✐♠❛t❡ ❛♥❞ ❛r❡ ❛♣♣r♦①✐♠❛t❡❞ ❜② ❝♦♥t✐♥✉♦✉s ❖❉❊s✳ ❚❤❡ s❛♠❡ ✐s tr✉❡ ❢♦r s②st❡♠s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❞✐s❝r❡t❡ s②st❡♠ ❢♦r t❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧ ♣r♦❞✉❝❡s t❤✐s ❛❧♠♦st ❡①❛❝t❧② ❝②❝❧✐❝ ♣❛t❤✿

✹✳✻✳

❉✐s❝r❡t❡ s②st❡♠s ♦❢ ❖❉❊s

✷✽✵

■♥ ♦t❤❡r ✇♦r❞s✱ ❊✉❧❡r✬s ♠❡t❤♦❞ ✐s ❝❛♣❛❜❧❡ ♦❢ tr❛❝✐♥❣ s♦❧✉t✐♦♥s ✈❡r② ❝❧♦s❡ t❤❡ ♦♥❡s ♦❢ t❤❡ ❖❉❊ ✐t ❝❛♠❡ ❢r♦♠✳ ❏✉st ❛s ✐♥ t❤❡

1✲❞✐♠❡♥s✐♦♥❛❧

❝❛s❡✱ t❤❡ ■❱P t❡❧❧s ✉s✿



✇❤❡r❡ ✇❡ ❛r❡ ✭t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✮✱ ❛♥❞



t❤❡ ❞✐r❡❝t✐♦♥ ✇❡ ❛r❡ ❣♦✐♥❣ ✭t❤❡ ❖❉❊✮✳

❏✉st ❛s ❜❡❢♦r❡✱ t❤❡ ✉♥❦♥♦✇♥ s♦❧✉t✐♦♥ ✐s r❡♣❧❛❝❡❞ ✇✐t❤ ✐ts

❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥



❊①❛♠♣❧❡ ✹✳✻✳✶✿ ❢❛♠✐❧② ♦❢ ❝✐r❝❧❡s ▲❡t✬s ❝♦♥s✐❞❡r ❛❣❛✐♥ t❤❡s❡ ❝♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s✿

❚❤❡② ❛r❡ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❖❉❊s✿

❲❡ ❝❤♦♦s❡ t❤❡ ✐♥❝r❡♠❡♥t ♦❢

t✿



x′ = y , y ′ = −x . ∆t = 1 .

❲❡ st❛rt ✇✐t❤ t❤✐s ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✿

t0 = 0,

x0 = 0,

❲❡ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦ ♥✉♠❜❡rs ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥s✿



x′ = 2, y ′ = 0;

y0 = 2 .

✹✳✻✳

❉✐s❝r❡t❡ s②st❡♠s ♦❢ ❖❉❊s

✷✽✶

❚❤✐s ✐s t❤❡ ❞✐r❡❝t✐♦♥ ✇❡ ✇✐❧❧ ❢♦❧❧♦✇✳ ❚❤❡ ✐♥❝r❡♠❡♥ts ❛r❡ 

∆x = 2 · ∆t = 2 · 1 = 2 , ∆y = 0 · ∆t = 0 · 1 = 0 .

❖✉r ♥❡①t ❧♦❝❛t✐♦♥ ♦♥ t❤❡ xy ✲♣❧❛♥❡ ✐s t❤❡♥✿ 

x1 = x0 + ∆x = 0 + 2 = 2 , y1 = y0 + ∆y = 2 + 0 = 2 .

❆ ♥❡✇ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❛♣♣❡❛rs✿ x0 = 2,

y0 = 2 .

❲❡ ❛❣❛✐♥ s✉❜st✐t✉t❡ t❤❡s❡ t✇♦ ♥✉♠❜❡rs ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥s✿ 

x′ = 2 , y ′ = −2 .

♣r♦❞✉❝✐♥❣ t❤❡ ❞✐r❡❝t✐♦♥ ✇❡ ✇✐❧❧ ❢♦❧❧♦✇✳ ❚❤❡ ✐♥❝r❡♠❡♥ts ❛r❡ 

∆x = 2 · ∆t = 2 · 1 = 2 , ∆y = −2 · ∆t = −2 · 1 = −2 .

❖✉r ♥❡①t ❧♦❝❛t✐♦♥ ♦♥ t❤❡ xy ✲♣❧❛♥❡ ✐s t❤❡♥✿ 

x2 = x1 + ∆x = 2 + 2 = 4 , y2 = y1 + ∆y = 2 + (−2) = 0 .

❖♥❡ ♠♦r❡ ■❱P✿ ❚❤❡ ✐♥❝r❡♠❡♥ts ❛r❡

x2 = 0, y2 = −4 . 

∆x = 0 · ∆t = 0 · 1 = 0 , ∆y = −4 · ∆t = −4 · 1 = −4 .



x3 = x2 + ∆x = 4 + 0 = 4 , y3 = y2 + ∆y = 0 − 4 = −4 .

❖✉r ♥❡①t ❧♦❝❛t✐♦♥ ♦♥ t❤❡ xy ✲♣❧❛♥❡ ✐s t❤❡♥✿

❚❤❡s❡ ❢♦✉r ♣♦✐♥ts ❢♦r♠ ❛ ✈❡r② ❝r✉❞❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ♦♥❡ ♦❢ ♦✉r ❝✐r❝✉❧❛r s♦❧✉t✐♦♥s✿

❚❤❡② ❛r❡ ❝❧❡❛r❧② s♣✐r❛❧✐♥❣ ❛✇❛② ❢r♦♠ t❤❡ ♦r✐❣✐♥✳ ■♥ t❡r♠s ♦❢ ♠♦t✐♦♥✱ t❤✐s ✐s ♦✉r ♣❧❛♥✿ ◮ ❆t ♦✉r ❝✉rr❡♥t ❧♦❝❛t✐♦♥ ❛♥❞ ❝✉rr❡♥t t✐♠❡✱ ✇❡ ❡①❛♠✐♥❡ t❤❡ ❖❉❊ t♦ ✜♥❞ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡♥

♠♦✈❡ ❛❝❝♦r❞✐♥❣❧② t♦ t❤❡ ♥❡①t ❧♦❝❛t✐♦♥✳

✹✳✻✳

❉✐s❝r❡t❡ s②st❡♠s ♦❢ ❖❉❊s

✷✽✷

❉❡✜♥✐t✐♦♥ ✹✳✻✳✷✿ ❊✉❧❡r s♦❧✉t✐♦♥ ❚❤❡

❊✉❧❡r s♦❧✉t✐♦♥

✇✐t❤ ✐♥❝r❡♠❡♥t

x′ = f (x, y) y ′ = g(x, y)

 ✐s t❤❡ t✇♦ s❡q✉❡♥❝❡s

{xn } 

✇❤❡r❡

∆t > 0

❛♥❞

{yn }

♦❢ t❤❡ ■❱P



x(t0 ) = x0 y(t0 ) = y0

♦❢ r❡❛❧ ♥✉♠❜❡rs ❣✐✈❡♥ ❜②✿

xn+1 = xn + f (xn , yn ) · ∆t yn+1 = yn + g(xn , yn ) · ∆t

tn+1 = tn + ∆t✳

❖♥❝❡ ❛❣❛✐♥✱ ✐❢ ✇❡ ❞❡r✐✈❡❞ ♦✉r ❖❉❊s ❢r♦♠ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧ ✭✈✐❛ ❜❛❝❦ t♦ ✐t✿

∆t → 0✮✱

❊✉❧❡r✬s ♠❡t❤♦❞ ✇✐❧❧ ❜r✐♥❣ ✉s r✐❣❤t

❖❉❊s

ր ❞✐s❝r❡t❡ ♠♦❞❡❧

same!

←−−−−−−−

ց ❊✉❧❡r✬s ♠❡t❤♦❞

❊①❛♠♣❧❡ ✹✳✻✳✸✿ s♣r❡❛❞s❤❡❡t ▲❡t✬s ♥♦✇ ❝❛rr② ♦✉t t❤✐s ♣r♦❝❡❞✉r❡ ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✳ ❚❤❡ ❢♦r♠✉❧❛s ❢♦r

❂❘❬✲✶❪❈✰❘❬✲✶❪❈❬✶❪✯❘✸❈✶

❛♥❞

xn

❛♥❞

yn

❛r❡ r❡s♣❡❝t✐✈❡❧②✿

❂❘❬✲✶❪❈✲❘❬✲✶❪❈❬✲✶❪✯❘✸❈✶

❚❤❡s❡ ❛r❡ t❤❡ r❡s✉❧ts✿

■♥ ❝♦♥tr❛st t♦ t❤❡ ❝❛s❡ ♦❢ ❛ s✐♥❣❧❡ ❖❉❊✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥s ❞♦ ♥♦t ❜❡❤❛✈❡ ❡rr❛t✐❝❛❧❧② ❝❧♦s❡ t♦ t❤❡

x✲❛①✐s✳

❚❤❡ r❡❛s♦♥ ✐s t❤❛t t❤❡r❡ ✐s ♥♦ ❞✐✈✐s✐♦♥ ❜②

y

❛♥②♠♦r❡✳

❊①❛♠♣❧❡ ✹✳✻✳✹✿ ❢❛♠✐❧② ♦❢ ❤②♣❡r❜♦❧❛s ▲❡t✬s ❝♦♥s✐❞❡r ❛❣❛✐♥ t❤❡s❡ ❤②♣❡r❜♦❧❛s✿

xy = C . ❚❤❡② ❛r❡ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ s②st❡♠✿

 ❆♥ ❊✉❧❡r s♦❧✉t✐♦♥ ✐s s❤♦✇♥ ❜❡❧♦✇✿

x′ = x , y ′ = −y .

✹✳✼✳

◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ s②st❡♠s ♦❢ ❖❉❊s

✷✽✸

❍♦✇❡✈❡r✱ ✐s t❤✐s ❛s②♠♣t♦t✐❝ ❝♦♥✈❡r❣❡♥❝❡ t♦✇❛r❞ t❤❡ x✲❛①✐s ♦r ❞♦ t❤❡② ♠❡r❣❡❄

✹✳✼✳ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ s②st❡♠s ♦❢ ❖❉❊s

❙✐♥❝❡ ❊✉❧❡r✬s ♠❡t❤♦❞ ❞❡♣❡♥❞s ♦♥ t❤❡ ✈❛❧✉❡ ♦❢ h✱ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t✱ t❤❡♥✱ ❡✈❡♥ ✇✐t❤ s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r ✈❛❧✉❡s ♦❢ h✱ t❤❡ r❡s✉❧t r❡♠❛✐♥s ❛ ♠❡r❡ ❛♣♣r♦①✐♠❛t✐♦♥✳ ▼❡❛♥✇❤✐❧❡✱ q✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ❝♦❧❧❡❝ts ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ s♦❧✉t✐♦♥s ✇✐t❤♦✉t s♦❧✈✐♥❣ t❤❡ s②st❡♠ ✕ ❡✐t❤❡r ❛♥❛❧②t✐❝❛❧❧② ♦r ♥✉♠❡r✐❝❛❧❧②✳ ❚❤❡ r❡s✉❧t ✐s ❢✉❧❧② ❛❝❝✉r❛t❡ ❜✉t ✈❡r② ❜r♦❛❞ ❞❡s❝r✐♣t✐♦♥s ♦❢ t❤❡ s♦❧✉t✐♦♥s✳ ❊①❛♠♣❧❡ ✹✳✼✳✶✿ ✶❞ q✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s

▲❡t✬s r❡✈✐❡✇ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ s✐♥❣❧❡ ❖❉❊ ❢r♦♠ ❧❛st s❡❝t✐♦♥✿

y ′ = − tan(y) . ❚❤✐s ✐s ✇❤❛t ✇❡ ❝♦♥❝❧✉❞❡ ❛❜♦✉t t❤❡ str✐♣ [−π/2, π/2]✿ • ❋♦r −π/2 < y < 0✱ ✇❡ ❤❛✈❡ y ′ = − tan y > 0 ❛♥❞✱ t❤❡r❡❢♦r❡✱ y ր✳ • ❋♦r y = 0✱ ✇❡ ❤❛✈❡ y ′ = − tan y = 0 ❛♥❞✱ t❤❡r❡❢♦r❡✱ y ✐s ❛ ❝♦♥st❛♥t s♦❧✉t✐♦♥✳ • ❋♦r 0 < y < π/2✱ ✇❡ ❤❛✈❡ y ′ = − tan y < 0 ❛♥❞✱ t❤❡r❡❢♦r❡✱ y ց✳ ■♥ ❢❛❝t✱ ❡✈❡r② s♦❧✉t✐♦♥ y ✐s ❞❡❝r❡❛s✐♥❣ ✭♦r ✐♥❝r❡❛s✐♥❣✮ t❤r♦✉❣❤♦✉t ✐ts ❞♦♠❛✐♥✳ ❚❤❡ ❝♦♥❝❧✉s✐♦♥s ❛r❡ ❝♦♥✜r♠❡❞ ✇✐t❤ ❊✉❧❡r✬s ♠❡t❤♦❞✿

❲❡ ❝❛♥ ♠❛t❝❤ t❤✐s ❖❉❊ ✇✐t❤ ❛ s②st❡♠✿



x′ = 1 , y ′ = − tan y .

■ts s♦❧✉t✐♦♥s ❤❛✈❡ t❤❡s❡ tr❛❥❡❝t♦r✐❡s ❛s s♦❧✉t✐♦♥s✳

✹✳✼✳

◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ s②st❡♠s ♦❢ ❖❉❊s

✷✽✹

❚❤❡ ❞✐r❡❝t✐♦♥s ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛r❡ ✐ts t❛♥❣❡♥t ✈❡❝t♦rs✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ s②st❡♠ ♦❢ ❖❉❊s✿



x′ = f (x, y) , y ′ = g(x, y) ,

❛r❡ ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ s✐❣♥s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✿

x′ = f (x, y) < 0 x′ = f (x, y) = 0 x′ = f (x, y) > 0 ← • → ′ y = g(x, y) < 0 ↓ ւ ↓ ց ′ y = g(x, y) = 0 • ← • → y ′ = g(x, y) > 0 ↑ տ ↑ ր ❊①❛♠♣❧❡ ✹✳✼✳✷✿ ♠♦r❡ ❝♦♠♣❧❡①

❈♦♥s✐❞❡r ♥❡①t✿

y ′ = cos y ·

p

|y| .

✇❡ ❞❡♠♦♥str❛t❡❞ t❤❛t t❤❡ ♠♦♥♦t♦♥✐❝✐t② ♦❢ t❤❡ s♦❧✉t✐♦♥s ✈❛r✐❡s ✇✐t❤ t❤❡ ❝♦s✐♥❡✿

❲❡ ❝❛♥ s❡❡ t❤❛t ❛❧❧ s♦❧✉t✐♦♥s ♣r♦❣r❡ss ❢♦r✇❛r❞ ❛❧♦♥❣ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢

x❄

❈♦♥s✐❞❡r✿



x✲❛①✐s✳

❲❤❛t ✐❢ ✇❡ ❛❞❞ ✈❛r✐❛❜✐❧✐t② ♦❢ t❤❡

x′ = sin y , y ′ = cos y .

❲❡ ❝♦♥❞✉❝t t❤❡ ✏s✐❣♥ ❛♥❛❧②s✐s✑ ❢♦r ❜♦t❤ ❢✉♥❝t✐♦♥s ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✿ ♣❛t❤ y x′ x = x(t) 2π 0 − x↓ ←←←← π 0 + x↑ →→→→ 0 0 − x↓ ←←←← −π 0

y y ′ y = y(t) ♣❛t❤ 3π/2 0 − y↓ ↓↓↓↓ π/2 0 + y↑ ↑↑↑↑ −π/2 0

◆♦✇ ♣✉t t❤❡♠ t♦❣❡t❤❡r✿

→ ր ↑ տ ←

→ ր ↑ տ ←

❚❤❡ r❡s✉❧ts ❛r❡ ❝♦♥✜r♠❡❞ ✇✐t❤ ❊✉❧❡r✬s ♠❡t❤♦❞✿

→ ր ↑ տ ←

→ ր ↑ տ ←

✹✳✼✳

◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ s②st❡♠s ♦❢ ❖❉❊s

✷✽✺

❊①❛♠♣❧❡ ✹✳✼✳✸✿ tr✐❣

❚❤❡ ♥❡①t ♦♥❡✿



x′ = sin x , y ′ = cos y .

❲❡ ❝♦♥❞✉❝t t❤❡ ✏s✐❣♥ ❛♥❛❧②s✐s✑ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✿

y 3π/2 π/2 −π/2 −3π/2

x −π π 2π 3π ′ y |x 0 + 0 − 0 + 0 0 • → • ← • → • − ↓ ց ↓ ւ ↓ ց ↓ 0 • → • ← • → • + ↑ ր ↑ տ ↑ ր ↑ 0 • → • ← • → • − ↓ ց ↓ ւ ↓ ց ↓ 0 • → • ← • → • ′

❚❤❡ r❡s✉❧ts ❛r❡ ❝♦♥✜r♠❡❞ ✇✐t❤ ❊✉❧❡r✬s ♠❡t❤♦❞✿

❊①❛♠♣❧❡ ✹✳✼✳✹✿ ❞✐s❝♦♥t✐♥✉♦✉s ❘❍❙

❚❤❡ ♥❡①t ♦♥❡ ✐s ❞✐s❝♦♥t✐♥✉♦✉s✿

❋♦r ❊✉❧❡r✬s ♠❡t❤♦❞ ✇❡ ✉s❡ t❤❡

❋▲❖❖❘

x′ = x − y, y ′ = [x + y] . ❢✉♥❝t✐♦♥✿

✹✳✼✳ ◗✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ♦❢ s②st❡♠s ♦❢ ❖❉❊s

✷✽✻

❊①❡r❝✐s❡ ✹✳✼✳✺

❈♦♥✜r♠ t❤❡ ♣❧♦t ❜❡❧♦✇ ❜② ❛♥❛❧②③✐♥❣ t❤✐s s②st❡♠✿ x′ = y, y ′ = sin y · y .

❙✉♣♣♦s❡ t❤❡ s②st❡♠ ✐s t✐♠❡✲✐♥❞❡♣❡♥❞❡♥t✱ x′ = f (x, y), y ′ = g(x, y) .

❚❤❡♥ ✐t ✐s t❤♦✉❣❤t ♦❢ ❛s ❛ ✢♦✇✿ ❧✐q✉✐❞ ✐♥ ❛ ♣♦♥❞ ♦r t❤❡ ❛✐r ♦✈❡r ❛ s✉r❢❛❝❡ ♦❢ t❤❡ ❊❛rt❤✳

❚❤❡♥✱ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s r❡❝♦❣♥✐③❡❞ ❛s ❛ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ✈❡❝t♦r ✜❡❧❞✳ ■♥ ❝♦♥tr❛st t♦ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ✇✐t❤ ♦♥❧② t❤r❡❡ ♠❛✐♥ ♣♦ss✐❜✐❧✐t✐❡s✱ ✇❡ ✇✐❧❧ s❡❡ ❛ ✇✐❞❡ ✈❛r✐❡t② ♦❢ ❜❡❤❛✈✐♦rs ❛r♦✉♥❞ ❛♥ ❡q✉✐❧✐❜r✐✉♠ ✇❤❡♥ t❤❡ s♣❛❝❡ ♦❢ ❧♦❝❛t✐♦♥ ✐s ❛ ♣❧❛♥❡✳ ❲✐t❤ s✉❝❤ ❛ s②st❡♠✱ t❤❡ q✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s ✐s ♠✉❝❤ s✐♠♣❧❡r✳ ■♥ ❢❛❝t✱ t❤❡ ♦♥❡s ❛❜♦✈❡ ❡①❤✐❜✐t ♠♦st ♦❢ t❤❡ ♣♦ss✐❜❧❡ ♣❛tt❡r♥s ♦❢ ❧♦❝❛❧ ❜❡❤❛✈✐♦r✳ ❲❡ ❝♦♥❝❡♥tr❛t❡ ♦♥ ✇❤❛t ✐s ❣♦✐♥❣ ♦♥ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ ❛ ❣✐✈❡♥ ❧♦❝❛t✐♦♥ (x, y) = (a, b)✳

✹✳✽✳ ❚❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥ ❛♥❞ ❧✐♥❡❛r s②st❡♠s

✷✽✼

❚❤❡ ✜rst ♠❛✐♥ ♣♦ss✐❜✐❧✐t② ✐s

f (a, b) 6= 0

♦r

g(a, b) 6= 0 ,

✇❤✐❝❤ ✐s ❡q✉✐✈❛❧❡♥t t♦

F (a, b) =< f (a, b) , g(a, b) >6= 0 . ❚❤❡♥✱ ❢r♦♠ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢

(a, b)✳

f

g ✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t ♥❡✐t❤❡r ❝❤❛♥❣❡s ✐ts s✐❣♥ ✐♥ s♦♠❡ ❞✐s❦ D t❤❛t ❝♦♥t❛✐♥s ♣❛t❤s ❧♦❝❛t❡❞ ✇✐t❤✐♥ D ♣r♦❝❡❡❞ ✐♥ ❛♥ ❛❜♦✉t t❤❡ s❛♠❡ ❞✐r❡❝t✐♦♥✿

❛♥❞

❚❤❡♥✱ t❤❡ s♦❧✉t✐♦♥s ✇✐t❤

❚❤❡ ❜❡❤❛✈✐♦r ✐s ✏❣❡♥❡r✐❝✑✳ ▼♦r❡ ✐♥t❡r❡st✐♥❣ ❜❡❤❛✈✐♦rs ❛r❡ s❡❡♥ ❛r♦✉♥❞ ❛ ③❡r♦ ♦❢

◮ F (a, b) = 0 =⇒ (x, y) = (a, b)

F✿

✐s ❛ st❛t✐♦♥❛r② s♦❧✉t✐♦♥ ✭❛♥ ❡q✉✐❧✐❜r✐✉♠✮✳

❚❤❡♥ t❤❡ ♣❛tt❡r♥ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤❡ ♣♦✐♥t✱ ✐✳❡✳✱ ❛♥ ♦♣❡♥ ❞✐s❦ ♦❢

f

♦r

g✱

D✱

❞❡♣❡♥❞s ♦♥ ✇❤❡t❤❡r t❤✐s ✐s ❛ ♠❛①✐♠✉♠

♦r ❛ ♠✐♥✐♠✉♠✱ ♦r ♥❡✐t❤❡r✳ ❙♦♠❡ ♦❢ t❤❡ ✐❞❡❛s ❝♦♠❡ ❢r♦♠ ❞✐♠❡♥s✐♦♥

1✳

❋♦r ❡①❛♠♣❧❡✱ ❛ st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✱

y→a

❛s

x → +∞ ,

y→a

❛s

x → −∞ ,

✐s ❛ s✐♥❦✿ ✢♦✇ ✐♥ ♦♥❧②✳ ❆♥ ✉♥st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠✱

✐s ❛ s♦✉r❝❡✿ ✢♦✇ ♦✉t ♦♥❧②✳

❆♥ s❡♠✐✲st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠ ❝♦✉❧❞ ♠❡❛♥ t❤❛t s♦♠❡ t❤❡ s♦❧✉t✐♦♥s ❛s②♠♣t♦t✐❝❛❧❧② ❛♣♣r♦❛❝❤ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❛♥❞ ♦t❤❡rs ❞♦ ♥♦t✳ ❆s ②♦✉ ❝❛♥ s❡❡ t❤❡r❡ ❛r❡ ♠❛♥② ♠♦r❡ ♣♦ss✐❜✐❧✐t✐❡s t❤❛♥ ✐♥ ❞✐♠❡♥s✐♦♥

✹✳✽✳ ❚❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥ ❛♥❞ ❧✐♥❡❛r s②st❡♠s

❲❡ ❝❛♥ ❝♦♠❜✐♥❡ t❤❡ t✇♦ ✈❛r✐❛❜❧❡s t♦ ❢♦r♠ ❛ ♣♦✐♥t ♦r ❛ ✈❡❝t♦r ♦♥ t❤❡ ♣❧❛♥❡✿

X = (x, y)

X =< x, y > .

♦r

❲❡ ❝❛♥ ❛❧s♦ ✉s❡ t❤❡ ❝♦❧✉♠♥✲✈❡❝t♦r ♥♦t❛t✐♦♥✿

X=



x y



.

1✳

✹✳✽✳

❚❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥ ❛♥❞ ❧✐♥❡❛r s②st❡♠s

✷✽✽

◆❡①t✱ t❤❡ s❛♠❡ ❤❛♣♣❡♥s t♦ t❤❡✐r ❞❡r✐✈❛t✐✈❡s ❛s ✈❡❝t♦rs✿

 ∆x     ∆X ∆x ∆y ∆t  . = , =  ∆t ∆t ∆t ∆y  ∆t 

❚❤❡♥ t❤❡ s❡t✉♣ ✇❡ ❤❛✈❡ ❜❡❡♥ ✉s✐♥❣ ❢♦r r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s r❡❛♣♣❡❛rs✿

∆X = F (X), X(t0 ) = X0 . ∆t ■♥ t❤✐s s❡❝t✐♦♥✱ ♦✉r ♠❛✐♥ ❝♦♥❝❡r♥ ✇✐❧❧ ❜❡ ❖❉❊s ✇✐t❤ r❡s♣❡❝t t♦







X =< x , y >=



x′ y′

❞❡r✐✈❛t✐✈❡s ✿ 

,

❛♥❞

X ′ = F (X), X(t0 ) = X0 . ❆❧❧ t❤❡ ♣❧♦tt✐♥❣✱ ❤♦✇❡✈❡r✱ ✐s ❞♦♥❡ ✇✐t❤ t❤❡ ❞✐s❝r❡t❡ ❖❉❊s✳

2 ❚❤❡ ✏♣❤❛s❡ s♣❛❝❡✑ R ✐s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ❧♦❝❛t✐♦♥s✳ ❚❤❡♥ t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ ❣✐✈❡♥ ♣❛rt✐❝❧❡ ✐s ❛ ❢✉♥❝t✐♦♥ + 2 X : R → R ♦❢ t✐♠❡ t ≥ 0✳ ▼❡❛♥✇❤✐❧❡✱ t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ ♣❛rt✐❝❧❡ ✐s ❣♦✈❡r♥❡❞ ❜② t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✢♦✇✱ ❛t ❡❛❝❤ ❧♦❝❛t✐♦♥✱ t❤❡ s❛♠❡ ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡✿ t❤❡ ✈❡❧♦❝✐t② ♦❢ ❛ ♣❛rt✐❝❧❡ ✐❢ ✐t ❤❛♣♣❡♥s t♦ ❜❡ ❛t ♣♦✐♥t

X

✐s

F (X)✳

❚❤❡♥ ❡✐t❤❡r t❤❡ ♥❡①t ♣♦s✐t✐♦♥ ✐s ♣r❡❞✐❝t❡❞ t♦ ❜❡

X + F (X)

✕ t❤❛t✬s ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧ ✕ ♦r

F (X)

✐s ❥✉st ❛

t❛♥❣❡♥t ♦❢ t❤❡ tr❛❥❡❝t♦r② ✕ t❤❛t✬s ❛♥ ❖❉❊✳ ❆ ✈❡❝t♦r ✜❡❧❞ s✉♣♣❧✐❡s ❛

❞✐r❡❝t✐♦♥ t♦ ❡✈❡r② ❧♦❝❛t✐♦♥✱ ✐✳❡✳✱ t❤❡r❡ ✐s ❛ ✈❡❝t♦r ❛tt❛❝❤❡❞ t♦ ❡❛❝❤ ♣♦✐♥t ♦❢ t❤❡ ♣❧❛♥❡✿ point 7→ vector .



✈❡❝t♦r ✜❡❧❞

✐♥ ❞✐♠❡♥s✐♦♥

2

✐s ❛♥② ❢✉♥❝t✐♦♥✿

F : R2 → R2 . ❋✉rt❤❡r♠♦r❡✱ ♦♥❡ ❝❛♥ t❤✐♥❦ ♦❢ ❛ ✈❡❝t♦r ✜❡❧❞ ❛s ❛

t✐♠❡✲✐♥❞❡♣❡♥❞❡♥t

X ′ = F (X) . ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ■❱P ❛❞❞s ❛♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✿

X(t0 ) = X0 .

❖❉❊ ♦♥ t❤❡ ♣❧❛♥❡✿

✹✳✽✳

❚❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥ ❛♥❞ ❧✐♥❡❛r s②st❡♠s

✷✽✾

❉❡✜♥✐t✐♦♥ ✹✳✽✳✶✿ s♦❧✉t✐♦♥ ♦❢ s②st❡♠ ♦❢ ❖❉❊s ❆ s♦❧✉t✐♦♥ ♦❢ ❛ s②st❡♠ ♦❢ ❖❉❊s ✐s ❛ ❢✉♥❝t✐♦♥ u ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ I s✉❝❤ t❤❛t ❢♦r ❡✈❡r② t ✐♥ I ✇❡ ❤❛✈❡✿

X ′ (t) = F (X(t))

❉❡✜♥✐t✐♦♥ ✹✳✽✳✷✿ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ ❋♦r ❛ ❣✐✈❡♥ s②st❡♠ ♦❢ ❖❉❊s ❛♥❞ ❛ ❣✐✈❡♥ (t0 , X0 )✱ t❤❡ ❛♥ ■❱P✱ ✐s

X ′ = F (X),

✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠✱ ♦r

X(t0 ) = X0

❛♥❞ ✐ts s♦❧✉t✐♦♥ ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❖❉❊ t❤❛t s❛t✐s✜❡s t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❛❜♦✈❡✳

❉❡✜♥✐t✐♦♥ ✹✳✽✳✸✿ ❊✉❧❡r s♦❧✉t✐♦♥ ❚❤❡

❊✉❧❡r s♦❧✉t✐♦♥

✇✐t❤ ✐♥❝r❡♠❡♥t ∆t > 0 ♦❢ t❤❡ ■❱P✿

X ′ = F (X),

X(t0 ) = X0 ;

✐s ❛ s❡q✉❡♥❝❡ {Xn } ♦❢ ♣♦✐♥ts ♦♥ t❤❡ ♣❧❛♥❡ ❣✐✈❡♥ ❜②✿

Xn+1 = Xn + F (Xn ) · ∆t ✇❤❡r❡ tn+1 = tn + ∆t✳ ❆❧❧ t❤❡ ❞❡✜♥✐t✐♦♥s ❛❜♦✈❡ r❡♠❛✐♥ ✈❛❧✐❞ ✐❢ ✇❡ t❤✐♥❦ ♦❢ X ❛s ❛ ❧♦❝❛t✐♦♥ ✐♥ ❛♥ N ✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ RN ✳ ❲❡ ♥♦✇ ❝♦♥❝❡♥tr❛t❡ ♦♥ ❧✐♥❡❛r s②st❡♠s ✭t❤❛t ♠❛② ❜❡ ❛❝q✉✐r❡❞ ❢r♦♠ ♥♦♥✲❧✐♥❡❛r ♦♥❡s ✈✐❛ ❧✐♥❡❛r✐③❛t✐♦♥✮✳ ❆❧❧ t❤❡ ❢✉♥❝t✐♦♥s ✐♥✈♦❧✈❡❞ ❛r❡ ❧✐♥❡❛r ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❞✐✛❡r❡♥t✐❛❜❧❡❀ ❜♦t❤ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ❛r❡ s❛t✐s✜❡❞✦ ❚❤✐s ✐s t❤❡ ❝❛s❡ ♦❢ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ F ✳ ❆s s✉❝❤✱ ✐t ✐s ❣✐✈❡♥ ❜② ❛ ♠❛tr✐① ❛♥❞ ✐s ❡✈❛❧✉❛t❡❞ ✈✐❛ ♠❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥✿      ax + by x a b . = F (X) = F X = cx + dy y c d ■t ✐s ✇r✐tt❡♥ s✐♠♣❧② ❛s

X′ = F X .

❚❤❡ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ t❤✐s ♠❛tr✐① ✕ t❤❡ ❞❡t❡r♠✐♥❛♥t✱ t❤❡ tr❛❝❡✱ ❛♥❞ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ✕ ✇✐❧❧ ❤❡❧♣ ✉s ❝❧❛ss✐❢② s✉❝❤ ❛ s②st❡♠✳ ❍♦✇❡✈❡r✱ t❤❡ ✜rst ♦❜s❡r✈❛t✐♦♥ ✐s ✈❡r② s✐♠♣❧❡✿ X = 0 ✐s t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♦❢ t❤❡ s②st❡♠✳ ■♥ ❢❛❝t ✇❤❛t ✇❡✬✈❡ ❧❡❛r♥❡❞ ❛❜♦✉t s②st❡♠s ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s t❡❧❧s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣✳

❚❤❡♦r❡♠ ✹✳✽✳✹✿ ❊q✉✐❧✐❜r✐✉♠ ♦❢ ▲✐♥❡❛r ❙②st❡♠ ❲❤❡♥ det F 6= 0✱ X = 0 ✐s t❤❡ ♦♥❧② ❡q✉✐❧✐❜r✐✉♠ ♦❢ t❤❡ s②st❡♠ X ′ = F X ❀ ♦t❤❡r✇✐s❡✱ t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② s✉❝❤ ♣♦✐♥ts✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❤❛t ♠❛tt❡r ✐s ✇❤❡t❤❡r F ✱ ❛s ❛ ❢✉♥❝t✐♦♥✱ ✐s ♦r ✐s ♥♦t ♦♥❡✲t♦✲♦♥❡✳

✹✳✽✳

❚❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥ ❛♥❞ ❧✐♥❡❛r s②st❡♠s

✷✾✵

❊①❛♠♣❧❡ ✹✳✽✳✺✿ ❞❡❣❡♥❡r❛t❡

❚❤❡ ❧❛tt❡r ✐s t❤❡ ✏❞❡❣❡♥❡r❛t❡✑ ❝❛s❡ s✉❝❤ ❛s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ✈❡r② s✐♠♣❧❡ s②st❡♠ ♦❢ ❖❉❊s✿



x′ = 2x y′ = 0

=⇒ =⇒



x = Ce2t y =K

■t ✐s ❡❛s② t♦ s♦❧✈❡✱ ♦♥❡ ❡q✉❛t✐♦♥ ❛t ❛ t✐♠❡✳ ❲❡ ❤❛✈❡ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ ♦♥ t❤❡ t❤❡

x✲❛①✐s

❛♥❞ ❝♦♥st❛♥t ♦♥

y ✲❛①✐s✳

❊①❛♠♣❧❡ ✹✳✽✳✻✿ s❛❞❞❧❡

▲❡t✬s ❝♦♥s✐❞❡r t❤✐s s②st❡♠ ♦❢ ❖❉❊s✿



x′ = −x y ′ = 4y

=⇒ =⇒



x = Ce−t y = Ke4t

❲❡ s♦❧✈❡ ✐t ✐♥st❛♥t❧② ❜❡❝❛✉s❡ t❤❡ t✇♦ ✈❛r✐❛❜❧❡s ❛r❡ ❢✉❧❧② s❡♣❛r❛t❡❞✳ ❲❡ ❝❛♥ t❤✐♥❦ ♦❢ ❡✐t❤❡r ♦❢ t❤❡s❡ t✇♦ s♦❧✉t✐♦♥s ♦❢ t❤❡ t✇♦ ❖❉❊s ❛s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✇❤♦❧❡ s②st❡♠ t❤❛t ❧✐✈❡s ❡♥t✐r❡❧② ✇✐t❤✐♥ ♦♥❡ ♦❢ t❤❡ t✇♦ ❛①❡s✿

❲❡ ❤❛✈❡ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ ♦♥ t❤❡

x✲❛①✐s

❛♥❞ ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛② ♦♥ t❤❡

y ✲❛①✐s✳

❚❤❡ r❡st ♦❢ t❤❡

s♦❧✉t✐♦♥s ❛r❡ s❡❡♥ t♦ t❡♥❞ t♦✇❛r❞ ♦♥❡ ♦❢ t❤❡s❡✳ ❙✐♥❝❡ ♥♦t ❛❧❧ ♦❢ t❤❡ s♦❧✉t✐♦♥s ❣♦ t♦✇❛r❞ t❤❡ ♦r✐❣✐♥✱ ✐t ✐s

✉♥st❛❜❧❡✳

❚❤✐s ♣❛tt❡r♥ ✐s ❝❛❧❧❡❞ ❛ ✏s❛❞❞❧❡✑ ❜❡❝❛✉s❡ t❤❡ ❝✉r✈❡s ❧♦♦❦ ❧✐❦❡ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❛r♦✉♥❞ ❛ s❛❞❞❧❡ ♣♦✐♥t✳ ❍❡r❡✱ t❤❡ ♠❛tr✐① ♦❢

F =



F

−1 0 0 4

✐s ❞✐❛❣♦♥❛❧✿



.

✹✳✽✳

❚❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥ ❛♥❞ ❧✐♥❡❛r s②st❡♠s

✷✾✶

❆❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ ❤❛✈❡✿

X=



x y



=



Ce−t Ke4t



= Ce

−t



1 0



+ Ke

4t



0 1



.

❲❡ ❤❛✈❡ ❡①♣r❡ss❡❞ t❤❡ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ ❛s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ ❜❛s✐s ✈❡❝t♦rs✦ ❊①❛♠♣❧❡ ✹✳✽✳✼✿ ♥♦❞❡

❆ s❧✐❣❤t❧② ❞✐✛❡r❡♥t s②st❡♠ ✐s✿



x′ = 2x y ′ = 4y

❍❡r❡✱

F =



=⇒ =⇒



x = Ce2t y = Ke4t

2 0 0 4



.

❖♥❝❡ ❛❣❛✐♥✱ ❡✐t❤❡r ♦❢ t❤❡s❡ t✇♦ s♦❧✉t✐♦♥s ♦❢ t❤❡ t✇♦ ❖❉❊s ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✇❤♦❧❡ s②st❡♠ t❤❛t ❧✐✈❡s ❡♥t✐r❡❧② ✇✐t❤✐♥ ♦♥❡ ♦❢ t❤❡ t✇♦ ❛①❡s ✭❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ ♦♥ ❜♦t❤ ♦❢ t❤❡ ❛①❡s✮ ❛♥❞ t❤❡ r❡st ♦❢ t❤❡ s♦❧✉t✐♦♥s ❛r❡ s❡❡♥ t♦ t❡♥❞ t♦✇❛r❞ ♦♥❡ ♦❢ t❤❡s❡✳ ❚❤❡ s❧✐❣❤t ❝❤❛♥❣❡ t♦ t❤❡ s②st❡♠ ♣r♦❞✉❝❡s ❛ ✈❡r② ❞✐✛❡r❡♥t ♣❛tt❡r♥✿

❆❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ ❤❛✈❡✿

       2t  Ce x 4t 0 2t 1 . + Ke = = Ce X= 1 Ke4t 0 y

❚❤✐s ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ ❜❛s✐s ✈❡❝t♦rs ✇✐t❤ t✐♠❡✲❞❡♣❡♥❞❡♥t ✇❡✐❣❤ts✳ ❚❤❡ ❡q✉✐❧✐❜r✐✉♠ ✐s ✉♥st❛❜❧❡ ❜✉t ❝❤❛♥❣✐♥❣ 2 ❛♥❞ 4 t♦ −2 ❛♥❞ −4 ✇✐❧❧ r❡✈❡rs❡ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ❝✉r✈❡s ❛♥❞ ♠❛❦❡ ✐t st❛❜❧❡✳ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ ✐s ❢❛st❡r ❛❧♦♥❣ t❤❡ y ✲❛①✐s❀ t❤❛t ✐s ✇❤② t❤❡ s♦❧✉t✐♦♥s ❛♣♣❡❛r t♦ ❜❡ t❛♥❣❡♥t t♦ t❤❡ x✲❛①✐s✳ ■♥ ❢❛❝t✱ ❡❧✐♠✐♥❛t✐♥❣ t ❣✐✈❡s ✉s y = x2 ❛♥❞ s✐♠✐❧❛r ❣r❛♣❤s✳ ❲❤❡♥ t❤❡ t✇♦ ❝♦❡✣❝✐❡♥ts ❛r❡ ❡q✉❛❧✱ t❤❡ ❣r♦✇t❤ ✐s ✐❞❡♥t✐❝❛❧ ❛♥❞ t❤❡ s♦❧✉t✐♦♥s ❛r❡ s✐♠♣❧② str❛✐❣❤t ❧✐♥❡s✿

❲❤❛t ✐❢ t❤❡ t✇♦ ✈❛r✐❛❜❧❡s ❛r❡♥✬t s❡♣❛r❛t❡❞❄ ❚❤❡ ✐♥s✐❣❤t ✐s t❤❛t t❤❡② ❝❛♥ ❜❡ ✕ ❛❧♦♥❣ t❤❡

❡✐❣❡♥✈❡❝t♦rs ♦❢ t❤❡

✹✳✽✳

❚❤❡ ✈❡❝t♦r ♥♦t❛t✐♦♥ ❛♥❞ ❧✐♥❡❛r s②st❡♠s

✷✾✷

♠❛tr✐①✳ ■♥❞❡❡❞✱ t❤❡ ❜❛s✐s ✈❡❝t♦rs ❛r❡ t❤❡ ❡✐❣❡♥✈❡❝t♦rs ♦❢ t❤❡s❡ t✇♦ ❞✐❛❣♦♥❛❧ ♠❛tr✐❝❡s✳ ◆♦✇ ❥✉st ✐♠❛❣✐♥❡ t❤❛t t❤❡ t✇♦ ♣✐❝t✉r❡s ❛r❡ s❦❡✇❡❞✿

▲❡t✬s ❧♦♦❦ ❛t t❤❡♠ ♦♥❡ ❛t ❛ t✐♠❡✳ ❚❤❡ ✐❞❡❛ ✐s ✉♥❝♦♠♣❧✐❝❛t❡❞✿ t❤❡ s②st❡♠ ✇✐t❤✐♥ t❤❡ ❡✐❣❡♥s♣❛❝❡ ✐s 1✲ ❞✐♠❡♥s✐♦♥❛❧✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐t ✐s ❛ s✐♥❣❧❡ ❖❉❊ ❛♥❞ ❝❛♥ ❜❡ s♦❧✈❡❞ t❤❡ ✉s✉❛❧ ✇❛②✳ ❚❤✐s ✐s ❤♦✇ ✇❡ ✜♥❞ s♦❧✉t✐♦♥s✳ ❊✈❡r② s♦❧✉t✐♦♥ X t❤❛t ❧✐❡s ✇✐t❤✐♥ t❤❡ ❡✐❣❡♥s♣❛❝❡✱ ✇❤✐❝❤ ✐s ❛ ❧✐♥❡✱ ✐s ❛ t✲❞❡♣❡♥❞❡♥t ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❡✐❣❡♥✈❡❝t♦r V ✿ X = rV

=⇒ X ′ = (rV )′ = F (rV ) = rF V = rλV =⇒ r′ V = λrV =⇒ r′ = λr =⇒ r = eλt

❚❤❡♦r❡♠ ✹✳✽✳✽✿ ❊✐❣❡♥s♣❛❝❡ ❙♦❧✉t✐♦♥s ■❢

λ

✐s ❛♥ ❡✐❣❡♥✈❛❧✉❡ ❛♥❞

V

❛ ❝♦rr❡s♣♦♥❞✐♥❣ ❡✐❣❡♥✈❡❝t♦r ♦❢ ❛ ♠❛tr✐①

F✱

t❤❡♥

X = eλt V ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❧✐♥❡❛r s②st❡♠

X′ = F X✳

Pr♦♦❢✳

❚♦ ✈❡r✐❢②✱ ✇❡ s✉❜st✐t✉t❡ ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥ ❛♥❞ ✉s❡ ❧✐♥❡❛r✐t② ✭♦❢ ❜♦t❤ ♠❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✛❡r✲ ❡♥t✐❛t✐♦♥✮✿ X = eλt V

=⇒ ❧❡❢t✲❤❛♥❞ s✐❞❡✿ X ′ = (eλt V )′ = (eλt )′ V = λeλt V r✐❣❤t✲❤❛♥❞ s✐❞❡✿ F X = F (eλt V ) = eλt F V = eλt λV

❚❤❡ ❡✐❣❡♥✈❛❧✉❡ ❝❛♥ ❜❡ ❝♦♠♣❧❡① ✦ ❚❤❡ s❡❝♦♥❞ ✐❞❡❛ ✐s t♦ tr② t♦ ❡①♣r❡ss t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❣❡♥❡r❛❧ ❧✐♥❡❛r s②st❡♠ ❛s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t✇♦ s♦❧✉t✐♦♥s ❢♦✉♥❞ t❤✐s ✇❛②✳

✹✳✾✳

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s

✷✾✸

❚❤❡♦r❡♠ ✹✳✽✳✾✿ ❘❡♣r❡s❡♥t❛t✐♦♥ ■♥ ❚❡r♠s ♦❢ ❊✐❣❡♥s♦❧✉t✐♦♥s

❙✉♣♣♦s❡ V1 ❛♥❞ V2 ❛r❡ t✇♦ ❡✐❣❡♥✈❡❝t♦rs ♦❢ ❛ ♠❛tr✐① F t❤❛t ❝♦rr❡s♣♦♥❞ t♦ t✇♦ ❡✐❣❡♥✈❛❧✉❡s λ1 ❛♥❞ λ2✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t t❤❡ ❡✐❣❡♥✈❡❝t♦rs ❛r❡♥✬t ♠✉❧t✐♣❧❡s ♦❢ ′ ❡❛❝❤ ♦t❤❡r✳ ❚❤❡♥ ❛❧❧ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❧✐♥❡❛r s②st❡♠ X = F X ❛r❡ ❣✐✈❡♥ ❛s ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ♥♦♥✲tr✐✈✐❛❧ s♦❧✉t✐♦♥s ✇✐t❤✐♥ t❤❡ ❡✐❣❡♥s♣❛❝❡s✿ X = Ceλ1 t V1 + Keλ2 t V2 ,

✇✐t❤ r❡❛❧ ❝♦❡✣❝✐❡♥ts C ❛♥❞ K ✳ Pr♦♦❢✳ ❙✐♥❝❡ t❤❡s❡ s♦❧✉t✐♦♥s ❝♦✈❡r t❤❡ ✇❤♦❧❡ ♣❧❛♥❡✱ t❤❡ ❝♦♥❝❧✉s✐♦♥ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt②✳

❉❡✜♥✐t✐♦♥ ✹✳✽✳✶✵✿ ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥ ❋♦r ❛ ❣✐✈❡♥ ❡✐❣❡♥✈❡❝t♦r

s♦❧✉t✐♦♥✳

V

✇✐t❤ ❡✐❣❡♥✈❛❧✉❡

λ✱

✇❡ ✇✐❧❧ ❝❛❧❧

eλt V



❝❤❛r❛❝t❡r✐st✐❝

❛❧❧

t❤❡ s♦❧✉t✐♦♥s✳

❊①❡r❝✐s❡ ✹✳✽✳✶✶ ❙❤♦✇ t❤❛t ✇❤❡♥ ❛❧❧ ❡✐❣❡♥✈❡❝t♦rs ❛r❡ ♠✉❧t✐♣❧❡s ♦❢ ❡❛❝❤ ♦t❤❡r✱ t❤❡ ❢♦r♠✉❧❛ ✇♦♥✬t ❣✐✈❡ ✉s

✹✳✾✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s ❲❡ ❝♦♥s✐❞❡r ❛ ❢❡✇ s②st❡♠s ✇✐t❤

♥♦♥✲❞✐❛❣♦♥❛❧ ♠❛tr✐❝❡s✳

❚❤❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐❣❡♥✈❡❝t♦rs

❝♦♠❡ ❢r♦♠ ❈❤❛♣t❡r ✸✳

❊①❛♠♣❧❡ ✹✳✾✳✶✿ ❞❡❣❡♥❡r❛t❡ ▲❡t✬s ❝♦♥s✐❞❡r ❛ ♠♦r❡ ❣❡♥❡r❛❧ s②st❡♠ ♦❢ ❖❉❊s✿



x′ = x +2y, y ′ = 2x +4y,

=⇒ F =



1 2 2 4



.

❊✉❧❡r✬s ♠❡t❤♦❞ s❤♦✇s t❤❡ ❢♦❧❧♦✇✐♥❣ s♦❧✉t✐♦♥s✿

■t ❛♣♣❡❛rs t❤❛t t❤❡ s②st❡♠ ❤❛s ✭❡①♣♦♥❡♥t✐❛❧✮ ❣r♦✇t❤ ✐♥ ♦♥❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ❝♦♥st❛♥t ✐♥ ❛♥♦t❤❡r✳ ❲❤❛t ❛r❡ t❤♦s❡ ❞✐r❡❝t✐♦♥s❄ ▲✐♥❡❛r ❛❧❣❡❜r❛ ❤❡❧♣s✳

✹✳✾✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s

✷✾✹

❋✐rst✱ t❤❡ ❞❡t❡r♠✐♥❛♥t ✐s ③❡r♦ ✿ det F = det





1 2 2 4

= 1 · 4 − 2 · 2 = 0.

❚❤❛t✬s ✇❤② t❤❡r❡ ✐s ❛ ✇❤♦❧❡ ❧✐♥❡ ♦❢ ♣♦✐♥ts X ✇✐t❤ F X = 0✳ ❚❤❡s❡ ❛r❡ st❛t✐♦♥❛r② ♣♦✐♥ts✳ ❚♦ ✜♥❞ t❤❡♠✱ ✇❡ s♦❧✈❡ t❤✐s ❡q✉❛t✐♦♥✿  x +2y = 0, 2x +4y = 0,

=⇒ x = −2y .

❲❡ ❤❛✈❡✱ t❤❡♥✱ ❡✐❣❡♥✈❡❝t♦rs ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ③❡r♦ ❡✐❣❡♥✈❛❧✉❡ λ1 = 0✿ V1 =



2 −1



=⇒ F V1 = 0 .

❙♦✱ s❡❝♦♥❞✱ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ♥♦♥✲③❡r♦ ❡✐❣❡♥✈❛❧✉❡ ✿ det(F − λI) = det





1−λ 2 2 4−λ

= λ2 − 5λ = λ(λ − 5) .

▲❡t✬s ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs ❢♦r λ2 = 5✳ ❲❡ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✿ F V = λ2 V ,

❛s ❢♦❧❧♦✇s✿ FV =



1 2 2 4



x y

❚❤✐s ❣✐✈❡s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ 

x +2y = 5x, 2x +4y = 5y,

=⇒





=5



x y



−4x +2y = 0, 2x −y = 0,

.

=⇒ y = 2x .

❚❤✐s ❧✐♥❡ ✐s t❤❡ ❡✐❣❡♥s♣❛❝❡✳ ❲❡ ❝❤♦♦s❡ t❤❡ ❡✐❣❡♥✈❡❝t♦r t♦ ❜❡✿ V2 =



1 2



.

❊✈❡r② s♦❧✉t✐♦♥ st❛rts ♦✛ t❤❡ ❧✐♥❡ y = −x/2 ❛♥❞ ❝♦♥t✐♥✉❡s ❛❧♦♥❣ t❤✐s ✈❡❝t♦r✳ ■t ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ ❡✐❣❡♥✈❡❝t♦rs✿     X = CV1 + KV2 = C

2 −1

+ Ke5t

❊①❡r❝✐s❡ ✹✳✾✳✷

❋✐♥❞ t❤❡ ❧✐♥❡ ♦❢ st❛t✐♦♥❛r② s♦❧✉t✐♦♥s✳ ❊①❛♠♣❧❡ ✹✳✾✳✸✿ s❛❞❞❧❡

▲❡t✬s ❝♦♥s✐❞❡r t❤✐s s②st❡♠ ♦❢ ❖❉❊s✿ ❍❡r❡✱ t❤❡ ♠❛tr✐① ♦❢ F ✐s ♥♦t ❞✐❛❣♦♥❛❧✿



x′ = x +2y, y ′ = 3x +2y. F =

❊✉❧❡r✬s ♠❡t❤♦❞ s❤♦✇s t❤❡ ❢♦❧❧♦✇✐♥❣✿



1 2 3 2



.

1 2

.

✹✳✾✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s

✷✾✺

❚❤❡ t✇♦ ❧✐♥❡s t❤❡ s♦❧✉t✐♦♥s ❛♣♣❡❛r t♦ ❝♦♥✈❡r❣❡ t♦ ❛r❡ t❤❡ ❡✐❣❡♥s♣❛❝❡s✳ ▲❡t✬s ✜♥❞ t❤❡♠✿ det(F − λI) = det



1−λ 2 3 2−λ



= λ2 − 3λ − 4 .

❚❤❡r❡❢♦r❡✱ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ λ1 = −1, λ2 = 4 .

◆♦✇ ✇❡ ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✳ ❲❡ s♦❧✈❡ t❤❡ t✇♦ ❡q✉❛t✐♦♥s✿

F Vk = λk Vk , k = 1, 2 .

❚❤❡ ✜rst✿ F V1 =





1 2 3 2



x y

= −1

❚❤✐s ❣✐✈❡s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ 

x +2y = −x, 3x +2y = −y,



=⇒

❲❡ ❝❤♦♦s❡ V1 =





x y

2x +2y = 0, 3x +3y = 0,



1 −1



.



1 −1

.

=⇒ x = −y .

❊✈❡r② s♦❧✉t✐♦♥ ✇✐t❤✐♥ t❤✐s ❡✐❣❡♥s♣❛❝❡ ✭t❤❡ ❧✐♥❡ y = −x✮ ✐s ❛ ♠✉❧t✐♣❧❡ ♦❢ t❤✐s ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥✿ X1 = e

❚❤❡ s❡❝♦♥❞ ❡✐❣❡♥✈❛❧✉❡✿ F V2 =

❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠✿ 

x +2y = 4x, 3x +2y = 4y,



λ1 t

V1 = e

1 2 3 2

=⇒



❲❡ ❝❤♦♦s❡ V2 =



−t

x y



=4





.

x y

−3x +2y = 0, 3x −2y = 0, 

1 3/2



.



1 3/2



.

=⇒ x = 2y/3 .

❊✈❡r② s♦❧✉t✐♦♥ ✇✐t❤✐♥ t❤✐s ❡✐❣❡♥s♣❛❝❡ ✭t❤❡ ❧✐♥❡ y = 3x/2✮ ✐s ❛ ♠✉❧t✐♣❧❡ ♦❢ t❤✐s ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥✿ X2 = e

λ2 t

V2 = e

4t



.

❚❤❡ t✇♦ s♦❧✉t✐♦♥s X1 ❛♥❞ X2 ✱ ❛s ✇❡❧❧ ❛s −X1 ❛♥❞ −X2 ✱ ❛r❡ s❤♦✇♥ ❜❡❧♦✇✿

✹✳✾✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s

✷✾✻

❚❤❡ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡s❡ t✇♦ ❜❛s✐❝ s♦❧✉t✐♦♥s✿ X = Ce

λ1 t

V1 + Ke

λ2 t

V2 = Ce

✐✳❡✳✱



❚❤❡ ❡q✉✐❧✐❜r✐✉♠ ✐s ✉♥st❛❜❧❡✳



−t

1 −1



+ Ke

4t



1 3/2



=



Ce−t + Ke4t −Ce−t + 3/2Ke4t

x = Ce−t +Ke4t , −t y = −Ce +3/2Ke4t .

❊①❛♠♣❧❡ ✹✳✾✳✹✿ ♥♦❞❡

▲❡t✬s ❝♦♥s✐❞❡r t❤✐s s②st❡♠ ♦❢ ❖❉❊s✿



❍❡r❡✱ t❤❡ ♠❛tr✐① ♦❢ F ✐s ♥♦t ❞✐❛❣♦♥❛❧✿

x′ = −x −2y, y′ = x −4y.

F =



−1 −2 1 −4



.

❊✉❧❡r✬s ♠❡t❤♦❞ s❤♦✇s t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡ ❛♥❛❧②s✐s st❛rts ✇✐t❤ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✿ det(F − λI) = det



−1 − λ −2 1 −4 − λ



= λ2 − 5λ + 6 .

❚❤❡r❡❢♦r❡✱ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ λ1 = −3, λ2 = −2 .

❚♦ ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✱ ✇❡ s♦❧✈❡ t❤❡ t✇♦ ❡q✉❛t✐♦♥s✿

F Vk = λk Vk , k = 1, 2 .

❚❤❡ ✜rst✿ F V1 =



−1 −2 1 −4



x y



= −1



x y



.



,

✹✳✾✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s

✷✾✼

❚❤✐s ❣✐✈❡s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ 

−x −2y = −3x, x −4y = −3y,

=⇒



2x −2y = 0, x −y = 0,

V1 =



1 1

❲❡ ❝❤♦♦s❡



=⇒ x = y .

.

❊✈❡r② s♦❧✉t✐♦♥ ✇✐t❤✐♥ t❤✐s ❡✐❣❡♥s♣❛❝❡ ✭t❤❡ ❧✐♥❡ y = x✮ ✐s ❛ ♠✉❧t✐♣❧❡ ♦❢ t❤✐s ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥✿ X1 = e

❚❤❡ s❡❝♦♥❞ ❡✐❣❡♥✈❛❧✉❡✿



V1 = e

−1 −2 F V2 = 1 −4

❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠✿ 

λ1 t

−x −2y = −2x, x −4y = −2y,

=⇒

❲❡ ❝❤♦♦s❡

−3t

  1 . 1

    x x . = −2 y y



x −2y = 0, x −2y = 0,

=⇒ x = 2y .

  2 . V2 = 1

❊✈❡r② s♦❧✉t✐♦♥ ✇✐t❤✐♥ t❤✐s ❡✐❣❡♥s♣❛❝❡ ✭t❤❡ ❧✐♥❡ y = x/2✮ ✐s ❛ ♠✉❧t✐♣❧❡ ♦❢ t❤✐s t❤✐s ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥✿ X2 = e

λ2 t

V2 = e

−2t

  2 . 1

❚❤❡ t✇♦ s♦❧✉t✐♦♥s X1 ❛♥❞ X2 ✱ ❛s ✇❡❧❧ ❛s −X1 ❛♥❞ −X2 ✱ ❛r❡ s❤♦✇♥ ❜❡❧♦✇✿

❚❤❡ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡s❡ t✇♦ ❜❛s✐❝ s♦❧✉t✐♦♥s✿ X = Ce

λ1 t

V1 + Ke

λ2 t

V2 = Ce

−3t



1 1



+ Ke

−2t



2 1



.

❚❤❡ ❡q✉✐❧✐❜r✐✉♠ ✐s st❛❜❧❡✳

❉❡✜♥✐t✐♦♥ ✹✳✾✳✺✿ st❛❜❧❡ ♥♦❞❡ ❋♦r ❛ ❧✐♥❡❛r s②st❡♠ X ′ = F X ✱ t❤❡ ❡q✉✐❧✐❜r✐✉♠ s♦❧✉t✐♦♥ X0 = 0 ✐s ❝❛❧❧❡❞ ❛ st❛❜❧❡ ♥♦❞❡ ✐❢ ❡✈❡r② ♦t❤❡r s♦❧✉t✐♦♥ X s❛t✐s✜❡s✿ X(t) → 0 ❛s t → +∞ ❛♥❞ ||X(t)|| → ∞ ❛s t → −∞;

✹✳✶✵✳

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞ ✉♥st❛❜❧❡ ♥♦❞❡

❛♥❞ ❛♥

X(t) → 0 ♣r♦✈✐❞❡❞ ♥♦

X

✷✾✽

✐❢

❛s

t → −∞

❛♥❞

||X(t)|| → ∞

♠❛❦❡s ❛ ❢✉❧❧ r♦t❛t✐♦♥ ❛r♦✉♥❞

❛s

t → +∞;

0✳

❉❡✜♥✐t✐♦♥ ✹✳✾✳✻✿ s❛❞❞❧❡ X ′ = F X ✱ t❤❡ ❡q✉✐❧✐❜r✐✉♠ s♦❧✉t✐♦♥ X0 = 0 ✐s ❝❛❧❧❡❞ ❛ s❛❞❞❧❡ X t❤❛t s❛t✐s❢②✿

❋♦r ❛ ❧✐♥❡❛r s②st❡♠ ✐❢ ✐t ❤❛s s♦❧✉t✐♦♥s

||X(t)|| → ∞

❛s

t → ±∞ .

❚❤❡♦r❡♠ ✹✳✾✳✼✿ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r ❙②st❡♠s ■ F λ2

❙✉♣♣♦s❡ ♠❛tr✐①



■❢



■❢

λ1

❛♥❞

❤❛s t✇♦ r❡❛❧ ❡✐❣❡♥✈❛❧✉❡s

λ1

❛♥❞

❤❛✈❡ t❤❡ s❛♠❡ s✐❣♥✱ t❤❡ s②st❡♠

λ2 ✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡✿ X ′ = F X ❤❛s ❛ ♥♦❞❡✱

st❛❜❧❡

✇❤❡♥ t❤✐s s✐❣♥ ✐s ♥❡❣❛t✐✈❡ ❛♥❞ ✉♥st❛❜❧❡ ✇❤❡♥ t❤✐s s✐❣♥ ✐s ♣♦s✐t✐✈❡✳

λ1

❛♥❞

λ2

❤❛✈❡ t❤❡ ♦♣♣♦s✐t❡ s✐❣♥s✱ t❤❡ s②st❡♠

X′ = F X

❤❛s ❛ s❛❞❞❧❡✳

Pr♦♦❢✳ ❚❤❡ st❛❜✐❧✐t② ✐s s❡❡♥ ✐♥ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥s✱ ❛s

λt

λt

||X|| = ||e V || = e · ||V || →



∞ 0

✐❢ ✐❢

t → +∞✿

λ > 0, λ < 0.

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛st t❤❡♦r❡♠✱ ✇❡ ❤❛✈❡ ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥s✳ ❚❤❡♥✱ ✐♥ t❤❡ ❢♦r♠❡r ❝❛s❡✱ ✇❡ ❤❛✈❡ ♦♥❡ ♦r t❤❡ ♦t❤❡r ♣❛tt❡r♥✱ ❛♥❞ ✐♥ t❤❡ ❧❛tt❡r✱ ❜♦t❤✳ ❚❤❡r❡ ❝❛♥ ❜❡ ♥♦ r♦t❛t✐♦♥ ❜❡❝❛✉s❡ ♥♦ s♦❧✉t✐♦♥ ❝❛♥ ✐♥t❡rs❡❝t ❛♥ ❡✐❣❡♥s♣❛❝❡✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ✉♥✐q✉❡♥❡ss ♣r♦♣❡rt②✳

✹✳✶✵✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞ ❲❤❛t ✐❢ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡

❝♦♠♣❧❡① ❄

❘❡❝❛❧❧ t❤❛t t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ♦❢ ♠❛tr✐①

F

✐s

χ(λ) = det(F − λI) = λ2 − tr F · λ + det F . ❚❤❡ ❞✐s❝r✐♠✐♥❛♥t ♦❢ t❤✐s q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s

D = (tr F )2 − 4 det F . ❲❤❡♥

D > 0✱

✇❡ ❤❛✈❡ t✇♦ ❞✐st✐♥❝t r❡❛❧ ❡✐❣❡♥✈❛❧✉❡s✱ t❤❡ ❝❛s❡ ❛❞❞r❡ss❡❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✳

✇✐t❤ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s ✇❤❡♥❡✈❡r

D < 0✳

❚❤❡ tr❛♥s✐t✐♦♥❛❧ ❝❛s❡ ✐s

❲❡ ❛r❡ ❢❛❝❡❞

D = 0✳

❊①❛♠♣❧❡ ✹✳✶✵✳✶✿ ✐♠♣r♦♣❡r ♥♦❞❡ ■♥ ❝♦♥tr❛st t♦ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ ❛ ♥♦❞❡ ♠❛② ❜❡ ♣r♦❞✉❝❡❞ ❜② ❛ ♠❛tr✐① ✇✐t❤ r❡♣❡❛t❡❞ ✭❛♥❞✱ t❤❡r❡❢♦r❡✱ r❡❛❧✮ ❡✐❣❡♥✈❛❧✉❡s✿

F = ❊✉❧❡r✬s ♠❡t❤♦❞ s❤♦✇s t❤❡ ❢♦❧❧♦✇✐♥❣✿



−1 2 0 −1



.

✹✳✶✵✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞

✷✾✾

❚❤❡ ❛♥❛❧②s✐s st❛rts ✇✐t❤ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✿ det(F − λI) = det



−1 − λ 2 0 −1 − λ



= (−1 − λ)2 .

❚❤❡r❡❢♦r❡✱ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ λ1 = λ2 = −1 .

❚❤❡ ♦♥❧② ❡✐❣❡♥✈❡❝t♦rs ❛r❡ ❤♦r✐③♦♥t❛❧✳ ❚❤❡ s♦❧✉t✐♦♥ ✐s ❣✐✈❡♥ ❜② X = Ce

−t



1 0





+ K te

−t



1 0



+e

−t



? 1



.

❊①❡r❝✐s❡ ✹✳✶✵✳✷

❋✐♥✐s❤ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ✐♥ t❤❡ ❡①❛♠♣❧❡✳ ❲❤❡♥ D < 0✱ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ❝♦♠♣❧❡①✦ ❚❤❡r❡❢♦r❡✱ t❤❡r❡ ❛r❡ ♥♦ ❡✐❣❡♥✈❡❝t♦rs ✭♥♦t r❡❛❧ ♦♥❡s ❛♥②✇❛②✮✳ ❉♦❡s t❤❡ s②st❡♠ X ′ = F X ❡✈❡♥ ❤❛✈❡ s♦❧✉t✐♦♥s❄ ❚❤❡ t❤❡♦r❡♠ ❛❜♦✉t ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥s s❛②s ②❡s✱ t❤❡② ❛r❡ ❝❡rt❛✐♥ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s✳✳✳ ❊①❛♠♣❧❡ ✹✳✶✵✳✸✿ ❝❡♥t❡r

❈♦♥s✐❞❡r



x′ = y, y ′ = −x.

❲❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t t❤❡ s♦❧✉t✐♦♥ ✐s ❢♦✉♥❞ ❜② s✉❜st✐t✉t✐♦♥✿ x′′ = (x′ )′ = y ′ = −x .

❚❤❡r❡❢♦r❡ t❤❡ s♦❧✉t✐♦♥s ❛r❡ t❤❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ sin t ❛♥❞ cos t✳ ❚❤❡ r❡s✉❧t ✐s ❝♦♥✜r♠❡❞ ✇✐t❤ ❊✉❧❡r✬s ♠❡t❤♦❞ ✭✇✐t❤ ❛ ❧✐♠✐t❡❞ ♥✉♠❜❡r ♦❢ st❡♣ t♦ ♣r❡✈❡♥t t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥s t♦ s♣✐r❛❧ ♦✉t✮✿

❆❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r② ❛❜♦✈❡✱ t❤❡ s♦❧✉t✐♦♥s ❛r❡ s✉♣♣♦s❡❞ t♦ ❜❡ ❡①♣♦♥❡♥t✐❛❧ r❛t❤❡r t❤❛♥ tr✐❣♦♥♦♠❡tr✐❝✳ ❇✉t t❤❡ ❧❛tt❡r ❛r❡ ❥✉st ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ✇✐t❤ ✐♠❛❣✐♥❛r② ❡①♣♦♥❡♥ts✳

✹✳✶✵✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞ ▲❡t✬s ♠❛❦❡ t❤✐s s♣❡❝✐✜❝❀ ✇❡ ❤❛✈❡

F = ❛♥❞ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✱

✸✵✵





0 1 −1 0

,

χ(λ) = λ2 + 1 ,

❤❛s t❤❡s❡ ❝♦♠♣❧❡① r♦♦ts✿ λ1,2 = ±i✳ ❚♦ ✜♥❞ t❤❡ ✜rst ❡✐❣❡♥✈❡❝t♦r✱ ✇❡ s♦❧✈❡✿

F V1 =





0 1 −1 0



x y

=i



x y



.

❚❤✐s ❣✐✈❡s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿  y = ix =⇒ y = ix . −x = iy ❲❡ ❝❤♦♦s❡ ❛ ❝♦♠♣❧❡① ❡✐❣❡♥✈❡❝t♦r✿

V1 = ❛♥❞ s✐♠✐❧❛r❧②✿

V2 =





1 i 1 −i

 

,

,

❚❤❡ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ✕ ♦✈❡r t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✕ ♦❢ t❤❡s❡ t✇♦ ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥s✿     1 1 −it λ1 t λ2 t it . + Ke X = Ce V1 + Ke V2 = Ce −i i ❚❤❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞✦ ✳✳✳✐♥ t❤❡ ❝♦♠♣❧❡① ❞♦♠❛✐♥✳ ❲❤❛t ✐s t❤❡ r❡❛❧ ♣❛rt ❄ ▲❡t K = 0✳ ❚❤❡♥ t❤❡ s♦❧✉t✐♦♥ ✐s✿     it     cos t + i sin t e cos t + i sin t 1 it . =C =C =C X = Ce − sin t + i cos t i(cos t + i sin t) i ieit ■ts r❡❛❧ ♣❛rt ✐s✿

Re X = C ❚❤❡s❡ ❛r❡ ❛❧❧ t❤❡ ❝✐r❝❧❡s✳



cos t − sin t



❊①❛♠♣❧❡ ✹✳✶✵✳✹✿ ❢♦❝✉s

▲❡t✬s ❝♦♥s✐❞❡r ❛ ♠♦r❡ ❝♦♠♣❧❡① s②st❡♠ ❖❉❊s✿  ′ x = 3x −13y, y ′ = 5x +y. ❍❡r❡✱ t❤❡ ♠❛tr✐① ♦❢ F ✐s ♥♦t ❞✐❛❣♦♥❛❧✿

F = ❊✉❧❡r✬s ♠❡t❤♦❞ s❤♦✇s t❤❡ ❢♦❧❧♦✇✐♥❣✿



3 −13 5 1



.

.

✹✳✶✵✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞

✸✵✶

❚❤❡ ❛♥❛❧②s✐s st❛rts ✇✐t❤ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧✿ 

χ(λ) = det(F − λI) = det

3 − λ −13 5 1−λ



= λ2 − 4λ + 68 .

❚❤❡r❡❢♦r❡✱ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ λ1,2 = 2 ± 8i .

◆♦✇ ✇❡ ✜♥❞ t❤❡ ❡✐❣❡♥✈❡❝t♦rs✳ ❲❡ s♦❧✈❡ t❤❡ t✇♦ ❡q✉❛t✐♦♥s✿ F Vk = λk Vk , k = 1, 2 .

❚❤❡ ✜rst✿ F V1 =



3 −13 5 1



x y



= (2 + 8i)

❚❤✐s ❣✐✈❡s ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ 

3x −13y = (2 + 8i)x 5x +y = (2 + 8i)y

=⇒



V1 =

F V2 =

❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠✿ 

3x −13y = (2 − 8i)x 5x +y = (2 − 8i)y



3 −13 5 1

=⇒



x y



.

(1 − 8i)x −13y =0 5x +(−1 − 8i)y = 0

❲❡ ❝❤♦♦s❡ ❚❤❡ s❡❝♦♥❞ ❡✐❣❡♥✈❛❧✉❡✿







1 + 8i 5 x y





V2 =



= (2 − 8i)

1 − 8i 5



1 + 8i y. 5

=⇒ x =

1 − 8i y. 5

. 

x y



.

(1 + 8i)x −13y =0 5x +(−1 + 8i)y = 0

❲❡ ❝❤♦♦s❡

=⇒ x =

.

❚❤❡ ❣❡♥❡r❛❧ ❝♦♠♣❧❡① s♦❧✉t✐♦♥ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥s✿ Z = Ce

λ1 t

V1 + Ke

λ2 t

V2 = Ce

(2+8i)t



1 + 8i 5



+ Ke

(2−8i)t



1 − 8i 5



.

✹✳✶✵✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞

✸✵✷

▲❡t✬s ♥♦✇ ❡①❛♠✐♥❡ ❛ s✐♠♣❧❡ r❡❛❧ s♦❧✉t✐♦♥✳ ❲❡ ❧❡t C = 1 ❛♥❞ K = 0✿ 



 1 + 8i  X = Re Z = Re e(2+8i)t   5    1 + 8i  = e2t Re e8it   5





 1 + 8i  = e2t Re(cos 8t + i sin 8t)   5    (cos 8t + i sin 8t)(1 + 8i)  = e2t Re   (cos 8t + i sin 8t)5    cos 8t + i sin 8t + 8i cos 8t − 8 sin 8t  = e2t Re   5 cos 8t + i5 sin 8t    cos 8t − 8 sin 8t  = e2t  . 5 cos 8t

P❧♦tt✐♥❣ t❤✐s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❝♦♥✜r♠s ❊✉❧❡r✬s ♠❡t❤♦❞ r❡s✉❧t✿

❉❡✜♥✐t✐♦♥ ✹✳✶✵✳✺✿ st❛❜❧❡ ❛♥❞ ✉♥st❛❜❧❡ ❢♦❝✉s ❋♦r ❛ ❧✐♥❡❛r s②st❡♠ X ′ = F X ✱ t❤❡ ❡q✉✐❧✐❜r✐✉♠ s♦❧✉t✐♦♥ X0 = 0 ✐s ❝❛❧❧❡❞ ❛ st❛❜❧❡ ❢♦❝✉s ✐❢ ❡✈❡r② ♦t❤❡r s♦❧✉t✐♦♥ X s❛t✐s✜❡s✿ X(t) → 0 ❛s t → +∞ ❛♥❞ ||X(t)|| → ∞ ❛s t → −∞;

❛♥❞ ❛♥ ✉♥st❛❜❧❡ ❢♦❝✉s ✐❢ X(t) → 0 ❛s t → −∞ ❛♥❞ ||X(t)|| → ∞ ❛s t → +∞;

♣r♦✈✐❞❡❞ ❡✈❡r② s✉❝❤ X ♠❛❦❡s ❛ ❢✉❧❧ r♦t❛t✐♦♥ ❛r♦✉♥❞ 0✳

✹✳✶✵✳

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞

✸✵✸

❉❡✜♥✐t✐♦♥ ✹✳✶✵✳✻✿ ❝❡♥t❡r ❋♦r ❛ ❧✐♥❡❛r s②st❡♠

X′ = F X✱

t❤❡ ❡q✉✐❧✐❜r✐✉♠ s♦❧✉t✐♦♥

X0 = 0

✐s ❝❛❧❧❡❞ ❛

❝❡♥t❡r

✐❢ ❛❧❧ s♦❧✉t✐♦♥s ❛r❡ ❝②❝❧❡s✳

❚❤❡♦r❡♠ ✹✳✶✵✳✼✿ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r ❙②st❡♠s ■■ ❙✉♣♣♦s❡ ♠❛tr✐①

F

❤❛s t✇♦ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ❡✐❣❡♥✈❛❧✉❡s

λ1

❛♥❞

λ2 ✳

❚❤❡♥ ✇❡

❤❛✈❡✿



■❢ t❤❡ r❡❛❧ ♣❛rt ♦❢

λ1

❛♥❞

λ2

✐s ♥♦♥✲③❡r♦✱ t❤❡ s②st❡♠

X′ = F X

❤❛s ❛ ❢♦❝✉s✱

st❛❜❧❡ ✇❤❡♥ t❤✐s s✐❣♥ ♦❢ t❤✐s ♥✉♠❜❡r ✐s ♥❡❣❛t✐✈❡ ❛♥❞ ✉♥st❛❜❧❡ ✇❤❡♥ t❤✐s s✐❣♥ ✐s ♣♦s✐t✐✈❡✳



■❢ t❤❡ r❡❛❧ ♣❛rt ♦❢

λ1

❛♥❞

λ2

✐s ③❡r♦✱ t❤❡ s②st❡♠

X′ = F X

❤❛s ❛ ❝❡♥t❡r✳

Pr♦♦❢✳ ❚❤❡ st❛❜✐❧✐t② ✐s s❡❡♥ ✐♥ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ❝❤❛r❛❝t❡r✐st✐❝ s♦❧✉t✐♦♥s✱ ❛s

λt

||X|| = ||e V || = ||e

(a+bi)t

at

t → +∞✿ at

V || = e | cos bt + i sin bt| · ||V || = e ||V || →



∞ 0

✐❢ ✐❢

a > 0, a < 0.

❚❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ ❝❧❛ss✐✜❝❛t✐♦♥ t❤❡♦r❡♠s ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

❚❤❡r❡❜② ✇❡ ❝♦♠♣❧❡t❡ t❤❡ s❡q✉❡♥❝❡✿ ❡❧❡♠❡♥t❛r② ❛❧❣❡❜r❛

−→ ♠❛tr✐① ❛❧❣❡❜r❛ −→ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳

❚♦ s✉♠♠❛r✐③❡✱ ✐♥ ♦r❞❡r t♦ ❝❧❛ss✐❢② ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ❖❉❊s ✈❡❝t♦r ♦♥ t❤❡ ♣❧❛♥❡✱ ✇❡ ❝❧❛ss✐❢②

F

X′ = F X✱

✇❤❡r❡

F

✐s ❛

2×2

♠❛tr✐① ❛♥❞

X

✐s ❛

❛❝❝♦r❞✐♥❣ t♦ ✐ts ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ✈✐s✉❛❧✐③❡ ❤♦✇ t❤❡ ❧♦❝❛t✐♦♥s ♦❢ t❤❡s❡ t✇♦

♥✉♠❜❡rs ✐♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ ✐♥❞✐❝❛t❡ ✈❡r② ❞✐✛❡r❡♥t ❜❡❤❛✈✐♦rs ♦❢ t❤❡ tr❛❥❡❝t♦r✐❡s✳ ✭❚❤❡ ♠✐ss✐♥❣ ♣❛tt❡r♥s ❛r❡ ❜❡tt❡r ✐❧❧✉str❛t❡❞ ❞②♥❛♠✐❝❛❧❧②✱ ❛s t❤❡ ❡①❛❝t ✏♠♦♠❡♥ts✑ ✇❤❡♥ ♦♥❡ ♣❛tt❡r♥ tr❛♥s✐t✐♦♥s ✐♥t♦ ❛♥♦t❤❡r✳✮

❊①❡r❝✐s❡ ✹✳✶✵✳✽ P♦✐♥t ♦✉t ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ t❤❡ ❧♦❝❛t✐♦♥s ♦❢ t❤❡ ❝❡♥t❡r ❛♥❞ t❤❡ ✐♠♣r♦♣❡r ♥♦❞❡✳

✹✳✶✵✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞

✸✵✹

❊①❡r❝✐s❡ ✹✳✶✵✳✾

❍♦✇ ❧✐❦❡❧② ✇♦✉❧❞ ❛ ❣✐✈❡♥ s②st❡♠ ❢❛❧❧ ✐♥t♦ ❡❛❝❤ ♦❢ t❤❡s❡ ✜✈❡ ❝❛t❡❣♦r✐❡s❄ ❲❤❛t ❛❜♦✉t t❤❡ ❝❡♥t❡r ❛♥❞ t❤❡ ✐♠♣r♦♣❡r ♥♦❞❡❄ ❊①❡r❝✐s❡ ✹✳✶✵✳✶✵

❲❤❛t ♣❛r❛♠❡t❡rs ❞❡t❡r♠✐♥❡ t❤❡ ❝❧♦❝❦✇✐s❡ ✈s✳ ❝♦✉♥t❡r✲❝❧♦❝❦✇✐s❡ ❜❡❤❛✈✐♦r❄ ❊①❛♠♣❧❡ ✹✳✶✵✳✶✶✿ ♣r❡❞❛t♦r✲♣r❡②

▲❡t✬s ❝❧❛ss✐❢② t❤❡ ❡q✉✐❧✐❜r✐❛ ♦❢ t❤❡ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧ ✕ ✈✐❛ ❧✐♥❡❛r✐③❛t✐♦♥✳ ❖✉r ♥♦♥✲❧✐♥❡❛r s②st❡♠ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿  ′ x = αx −βxy, y ′ = δxy −γy, ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ ❝♦❡✣❝✐❡♥ts αx, β, δ, γ ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡

X ′ = G(X), ✇✐t❤ G(x, y) = (αx − βxy, δxy − γy) . ❚❤❡ ❏❛❝♦❜✐❛♥ ♦❢ G ✐s



   ∂ ∂ (αx − βxy) (αx − βxy)  ∂x   α − βy −βx  ∂y = G′ (x, y) =  .  ∂  ∂ δy δx − γ (δxy − γy) (δxy − γy) ∂x ∂y

❚❤❡ ♠❛tr✐① ❞❡♣❡♥❞s ♦♥ (x, y) ❜❡❝❛✉s❡ t❤❡ s②st❡♠ ✐s ♥♦♥✲❧✐♥❡❛r✳ ❇② ✜①✐♥❣ ❧♦❝❛t✐♦♥s X = A✱ ✇❡ ❝r❡❛t❡ ❧✐♥❡❛r ✈❡❝t♦r ❖❉❊s✿ X ′ = G′ (A)X . ❋✐rst✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ③❡r♦ ❡q✉✐❧✐❜r✐✉♠✱

x = 0, y = 0 . ❍❡r❡✱ ′

F = G (0, 0) =



α − β · 0 −β · 0 δ·0 δ·0−γ



=



α 0 0 −γ

❚❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ❢♦✉♥❞ ❜② s♦❧✈✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥✿   α−λ 0 = (α − λ)(−γ − λ) = 0 . det 0 −γ − λ



.

❚❤❡r❡❢♦r❡✱

λ1 = α, λ2 = −γ .

❲❡ ❤❛✈❡ t✇♦ r❡❛❧ ❡✐❣❡♥✈❛❧✉❡s ♦❢ ♦♣♣♦s✐t❡ s✐❣♥s✳ ❚❤✐s ✐s ❛ s❛❞❞❧❡ ✦ ■♥❞❡❡❞✱ ❛r♦✉♥❞ t❤✐s ♣♦✐♥t t❤❡ ❢♦①❡s ❞❡❝❧✐♥❡ ✇❤✐❧❡ t❤❡ r❛❜❜✐ts ✐♥❝r❡❛s❡ ✐♥ ♥✉♠❜❡rs✳ ❚❤❡ ♠❛✐♥ ❡q✉✐❧✐❜r✐✉♠ ✐s

x= ❍❡r❡✱

F = G′



γ α , δ β



α γ , y= . δ β

  α γ  βγ α−β· −β · 0 − β δ = δ  = .  α γ δα δ· δ· −γ 0 β δ β 

✹✳✶✵✳

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥t✐♥✉❡❞

✸✵✺

❚❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ❢♦✉♥❞ ❜② s♦❧✈✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥✿

 βγ  −λ − δ  2 det  δα  = λ + αγ = 0 . −λ β 

❚❤❡r❡❢♦r❡✱

λ1 =



❲❡ ❤❛✈❡ t✇♦ ♣✉r❡❧② ✐♠❛❣✐♥❛r② ❡✐❣❡♥✈❛❧✉❡s✳ ❝②❝❧✐❝ ❜❡❤❛✈✐♦r✳

❚❤❡ r❡s✉❧ts ♠❛t❝❤ ♦✉r ♣r❡✈✐♦✉s ❛♥❛❧②s✐s✳

√ αγ i, λ2 = − αγ i . ❚❤✐s ✐s ❛

❝❡♥t❡r ✦

■♥❞❡❡❞✱ ❛r♦✉♥❞ t❤✐s ♣♦✐♥t ✇❡ ❤❛✈❡ ❛

❈❤❛♣t❡r ✺✿ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❖❉❊s

❈♦♥t❡♥ts ✺✳✶ ✺✳✷ ✺✳✸ ✺✳✹ ✺✳✺ ✺✳✻ ✺✳✼ ✺✳✽ ✺✳✾

❱❡❝t♦r✲✈❛❧✉❡❞ ❢♦r♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚❤❡ ♣✉rs✉✐t ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❛s s②st❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❱❡❝t♦r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r✿ ❛ ❞♦✉❜❧❡ s♣r✐♥❣ ❆ ♣❡♥❞✉❧✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ P❧❛♥❡t❛r② ♠♦t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚❤❡ t✇♦✲ ❛♥❞ t❤r❡❡✲❜♦❞② ♣r♦❜❧❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❆ ❝❛♥♥♦♥ ✐s ✜r❡❞✳✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✸✵✻ ✸✵✼ ✸✶✷ ✸✶✺ ✸✷✶ ✸✷✹ ✸✸✵ ✸✸✻ ✸✸✾

✺✳✶✳ ❱❡❝t♦r✲✈❛❧✉❡❞ ❢♦r♠s

t❤❡ ❡①❛♠♣❧❡ ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✕ ❛♥❞ ❡s♣❡❝✐❛❧❧② ♠♦t✐♦♥ ✐♥ s♣❛❝❡ ✕ s✉❣❣❡sts t❤❛t ✇❡ ♠❛② ♥❡❡❞ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s t♦ ❜❡ ♠✉❧t✐✲❞✐♠❡♥s✐♦♥❛❧✳ ❲❡ s❛✇ ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞✱ ❥✉st ❧✐❦❡ ❛ 0✲❢♦r♠✱ ❛t t❤❡ ♥♦❞❡s ♦❢ t❤❡ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❧✐♥❡✱ ❜✉t ✇✐t❤ ✈❛❧✉❡s ✐♥ R2 ✱ ✉♥❧✐❦❡ ❛ 0✲❢♦r♠✳ ❚❤✐s ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❧❛st ❞❡✜♥✐t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✺✳✶✳✶✿ ✈❡❝t♦r✲✈❛❧✉❡❞ ❞✐s❝r❡t❡ ❢♦r♠ ❙✉♣♣♦s❡ n ❛♥❞ m ❛r❡ ❣✐✈❡♥✳ ❚❤❡♥ ❛ ✈❡❝t♦r✲✈❛❧✉❡❞ ❞✐s❝r❡t❡ ❢♦r♠ F ♦❢ ❞❡❣r❡❡ k ✱ ♦r s✐♠♣❧② ❛ k ✲❢♦r♠✱ ✐s ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ k ✲❝❡❧❧s ♦❢ Rn ✇✐t❤ ✈❛❧✉❡s ✐♥ Rm ✳ ❆s ②♦✉ ❝❛♥ s❡❡✱ ✇❡ ✇✐❧❧ ❜❡ ✉s✐♥❣ ❝❛♣✐t❛❧ ❧❡tt❡rs ❢♦r ✈❡❝t♦r✲✈❛❧✉❡❞ ❢♦r♠s ✐♥ ❛❝❝♦r❞❛♥❝❡ ✇✐t❤ ♦✉r ❝♦♥✈❡♥t✐♦♥✳ ◆♦t❡ t❤❛t ❞✐s❝r❡t❡ ❢♦r♠s ❞♦ ♥♦t ❡①❛❝t❧② ♠❛t❝❤ ♦✉r ❧✐st ♦❢ ❢✉♥❝t✐♦♥s✿ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✱ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✱ ✈❡❝t♦r ✜❡❧❞s✱ ❛♥❞ ❢✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s✳ ❋r♦♠ t❤❡ s❛♠❡ ❞♦♠❛✐♥✱ ✇❡ ♣✐❝❦ ❝❡❧❧s ♦❢ ❞✐✛❡r❡♥t ❞✐♠❡♥s✐♦♥s ♣r♦❞✉❝✐♥❣ ❢♦r♠s ♦❢ ❞✐✛❡r❡♥t ❞❡❣r❡❡s✳ ❚❤✐s ✐s ❛♥ ✐❧❧✉str❛t✐♦♥ ♦❢ t✇♦ ✈❡❝t♦r✲✈❛❧✉❡❞ ❢♦r♠s✿ ❛ 0✲❢♦r♠ ❛♥❞ ❛ 1✲❢♦r♠ ✭❢♦r t❤❡ ❧❛tt❡r✱ t❤❡ ✈❡❝t♦rs ❤❛✈❡ t♦ ❜❡ ♠♦✈❡❞ t♦ ♣✉t t❤❡ st❛rt✐♥❣ ♣♦✐♥ts ❛t t❤❡ ♦r✐❣✐♥✮❀ ✐✳❡✳✱ n = 1 ❛♥❞ m = 2✿

✺✳✷✳

❚❤❡ ♣✉rs✉✐t ❝✉r✈❡s

✸✵✼

❚❤❡ ❢♦r♠❡r ♠❛② r❡♣r❡s❡♥t t❤❡ ❧♦❝❛t✐♦♥s ❛♥❞ t❤❡ ❧❛tt❡r t❤❡ ✈❡❧♦❝✐t✐❡s✳ ❇♦t❤ ❝❛♥ ❜❡ s❡❡♥ ❛s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✳ ◆❡①t ✐s ❛♥ ✐❧❧✉str❛t✐♦♥ ♦❢ ❛ r❡❛❧✲✈❛❧✉❡❞ ❛♥❞ ❛ ✈❡❝t♦r✲✈❛❧✉❡❞ 1✲❢♦r♠s❀ ✐✳❡✳✱ n = 2✱ m = 1 ❛♥❞ n = 2✱ m = 2 r❡s♣❡❝t✐✈❡❧②✿

❚❤❡ ❢♦r♠❡r ♠❛② r❡♣r❡s❡♥t ❛ ✢♦✇ ♦❢ ✇❛t❡r ❛❧♦♥❣ ❛ s②st❡♠ ♦❢ ♣✐♣❡s ❛♥❞ t❤❡ ❧❛tt❡r t❤❡ s❛♠❡ ✢♦✇ ✇✐t❤ ♣♦ss✐❜❧❡ ❧❡❛❦s✳ ❇♦t❤ ❝❛♥ ❜❡ s❡❡♥ ❛s ✈❡❝t♦r ✜❡❧❞s✳ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs ❛❧❧♦✇s ✉s t♦ r❡♣r♦❞✉❝❡ t❤❡ ❞❡✜♥✐t✐♦♥s ❢r♦♠ ❈❤❛♣t❡r ✶ ✐♥ t❤❡ ♥❡✇✱ ♠✉❧t✐✲❞✐♠❡♥s✐♦♥❛❧✱ ✐♥ ♦✉t♣✉t✱ ❝♦♥t❡①t✳ ❲❡ ❥✉st ❛ss✉♠❡ t❤❛t ❛ s♣❛❝❡ ♦❢ ✐♥♣✉ts R ✇✐t❤ ❛ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ❛♥❞ ❛ s♣❛❝❡ ♦❢ ♦✉t♣✉ts Rm ❛r❡ ❣✐✈❡♥✳

✺✳✷✳ ❚❤❡ ♣✉rs✉✐t ❝✉r✈❡s

■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✇✐❧❧ ♦❢t❡♥ r❡❢❡r t♦ s②st❡♠s ♦❢ ❖❉❊s ❛s ✈❡❝t♦r

❖❉❊s ♦r s✐♠♣❧② ❖❉❊s✳

■♠❛❣✐♥❡ ❛ ❤♦✉♥❞ r✉♥♥✐♥❣ ❛❢t❡r ❛ r❛❜❜✐t✳ ▲❡t✬s ✐♥✈❡st✐❣❛t❡ t❤❡✐r ♠✉t✉❛❧ ❞②♥❛♠✐❝s✳

❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ❤♦✉♥❞ ✐s ❛❧✇❛②s ❤❡❛❞✐♥❣ ❛t t❤❡ r❛❜❜✐t ✇❤✐❧❡ t❤❡ r❛❜❜✐t ✐s ❤❡❛❞✐♥❣ ❢♦r ✐ts ❤♦❧❡ ❛♥❞ ♥❡✈❡r ❝❤❛♥❣❡s ❞✐r❡❝t✐♦♥s✳

✺✳✷✳

❚❤❡ ♣✉rs✉✐t ❝✉r✈❡s

❲❡ ❜✉✐❧❞ ❛ ❞✐s❝r❡t❡

✸✵✽

♠♦❞❡❧✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ t✐♠❡ ♣r♦❣r❡ss❡s ✐♥ ✐♥❝r❡♠❡♥ts✿ t0 = 0, tn+1 = tn + ∆t .

❚❤❡ ♠❛✐♥ ❛ss✉♠♣t✐♦♥ ✐s✿ ◮ ❉✉r✐♥❣ t❤❡ t✐♠❡ ✐♥t❡r✈❛❧ [tn , tn + ∆t]✱ t❤❡ ❤♦✉♥❞ ✇✐❧❧ ❜❡ r✉♥♥✐♥❣ t♦✇❛r❞ t❤❡ s♣♦t ✇❤❡r❡ t❤❡ r❛❜❜✐t ✇❛s ❛t t✐♠❡ tn ✳

❙✉♣♣♦s❡ t❤❡✐r ❧♦❝❛t✐♦♥s ❛r❡✿ • (x, y) ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ r❛❜❜✐t✱ ❛♥❞ • (p, q) ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ❤♦✉♥❞✳

❚❤❡② ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② ♣❛r❛♠❡tr✐❝

❝✉r✈❡s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ t❤❡ st❛♥❞❛r❞ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ r❡❛❧ ❧✐♥❡✿

• (xn , yn ) = (x(tn ), y(tn )) ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ r❛❜❜✐t ❛❢t❡r n ✐♥❝r❡♠❡♥ts✱ ❛♥❞ • (pn , qn ) = (p(tn ), q(tn )) ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ❤♦✉♥❞ ❛❢t❡r n ✐♥❝r❡♠❡♥ts✳

❖♥ t❤❡ ❧❡❢t ✇❡ ❤❛✈❡✱ ❥✉st ❛s ❜❡❢♦r❡✱ ❛ s✐♠♣❧✐✜❡❞ ♥♦t❛t✐♦♥✳ ❙✉♣♣♦s❡ • s ✐s t❤❡ s♣❡❡❞ ♦❢ t❤❡ r❛❜❜✐t✱ ❛♥❞

• v ✐s t❤❡ s♣❡❡❞ ♦❢ t❤❡ ❤♦✉♥❞✱ v > s✳

❊①❛♠♣❧❡ ✺✳✷✳✶✿ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ♣❛t❤

❲❡ ❝r❡❛t❡ ❛ ♠♦❞❡❧ ❢♦r t❤❡ ❤♦✉♥❞ t❤❛t ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❛ s♣❡❝✐✜❝ ❝❤♦✐❝❡ ♦❢ t❤❡ ♣❛t❤ ♦❢ t❤❡ r❛❜❜✐t✿ ♦♥❡ st❡♣ ❛t ❛ t✐♠❡✳

❲❡ ❝♦♠♣✉t❡ ❝♦♥s❡❝✉t✐✈❡❧②✿ • t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦✿ xn − pn ❛♥❞ yn − qn ✱ ✇❤✐❝❤ ✐s t❤❡ ❞✐r❡❝t✐♦♥ ❢r♦♠ t❤❡ ❤♦✉♥❞ t♦✇❛r❞s t❤❡ r❛❜❜✐t❀ • t❤❡ ✉♥✐t ✈❡❝t♦r ✐♥ t❤✐s ❞✐r❡❝t✐♦♥ ✭❞✐✈✐❞✐♥❣ ❜② t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡♠✮❀ • t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ❤♦✉♥❞ ✭♠✉❧t✐♣❧②✐♥❣ t❤✐s ✈❡❝t♦r ❜② t❤❡ ❤♦✉♥❞s✬ s♣❡❡❞ v ✮❀ ❛♥❞ • t❤❡ ♥❡✇ ❧♦❝❛t✐♦♥ (pn+1 , qn+1 ) ♦❢ t❤❡ ❤♦✉♥❞ ✭❛❞❞✐♥❣ t❤❡ ✈❡❧♦❝✐t② t✐♠❡s t✐♠❡ ✐♥t❡r✈❛❧ h = ∆t t♦ t❤❡ ❧❛st ❧♦❝❛t✐♦♥✮✳

❆❜♦✈❡ ❛♥❞ ❜❡❧♦✇ ❧❡❢t s❤♦✇ t❤❡ ♣✉rs✉✐t ♦❢ ❛ r❛❜❜✐t ❢♦❧❧♦✇✐♥❣ ❛ ❝✐r❝✉❧❛r ♣❛t❤✳ ❇❡❧♦✇ r✐❣❤t✱ ✐t ❢♦❧❧♦✇s ❛ s✐♥✉s♦✐❞✿

✺✳✷✳

❚❤❡ ♣✉rs✉✐t ❝✉r✈❡s

✸✵✾

❊①❡r❝✐s❡ ✺✳✷✳✷

■♠♣❧❡♠❡♥t t❤✐s ♠♦❞❡❧ ❢♦r t❤❡ ❝❛s❡ ♦❢ t❤❡ r❛❜❜✐t r✉♥♥✐♥❣ ♥♦rt❤✲✇❡st✿

❊①❡r❝✐s❡ ✺✳✷✳✸

■♠♣❧❡♠❡♥t t❤✐s ♠♦❞❡❧ ❢♦r t❤❡ ❝❛s❡ ♦❢ t❤❡ r❛❜❜✐t ❢♦❧❧♦✇✐♥❣ ❛ ❝✐r❝❧❡✱ s✐♥✉s♦✐❞✱ s♣✐r❛❧✱ ❡t❝✳ ❲❡ ♥♦✇ ❛♣♣r♦❛❝❤ t❤✐s ❛♥❛❧②t✐❝❛❧❧②✳ ❚♦ ♠❛❦❡ t❤✐s ♣♦ss✐❜❧❡✱ ✇❡ ❝❤♦♦s❡ ❛ s✐♠♣❧❡ ❝❛s❡✿ t❤❡ r❛❜❜✐t ✐s r✉♥♥✐♥❣ ❛❧♦♥❣ t❤❡ y ✲❛①✐s ❢r♦♠ t❤❡ ♦r✐❣✐♥✳ ❚❤❡♥✱ 

xn+1 = 0 , yn+1 = yn + s∆t .

❋✉rt❤❡r♠♦r❡✱ • ∆xn = ∆x (cn ) • ∆yn = ∆y (cn ) • ∆pn = ∆p (cn ) • ∆qn = ∆q (cn )

❍❡r❡ cn ❛r❡✱ s❛②✱ t❤❡ ♠✐❞✲♣♦✐♥ts ♦❢ t❤❡ ✐♥t❡r✈❛❧s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❋♦r t❤❡ ❤♦✉♥❞✱ ✇❡ ❤❛✈❡✿



pn+1 = pn + ∆pn , qn+1 = qn + ∆qn .

❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ❤♦✉♥❞ ✐s ❧♦❝❛t❡❞ t♦ t❤❡ r✐❣❤t ♦❢ t❤❡ y ✲❛①✐s ❛♥❞✱ t❤❡r❡❢♦r❡✱ ♠♦✈✐♥❣ t♦ ✐ts ❧❡❢t✱ ∆xn < 0✳ ❆s t❤❡ ❤♦✉♥❞ ♠♦✈❡s ❢r♦♠ (pn , qn ) ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ (xn , yn )✱ t❤❡ t✇♦ r✐❣❤t tr✐❛♥❣❧❡s ❛r❡ s✐♠✐❧❛r✿ • ✜rst✱ t❤❡ ♦♥❡ ✇✐t❤ t❤❡ s✐❞❡s pn − xn ❛♥❞ yn − qn ✱ ❛♥❞ • s❡❝♦♥❞✱ t❤❡ ♦♥❡ ✇✐t❤ t❤❡ s✐❞❡s −∆pn ❛♥❞ ∆qn ✳

✺✳✷✳

❚❤❡ ♣✉rs✉✐t ❝✉r✈❡s

✸✶✵

❚❤❡s❡ ♦❜s❡r✈❛t✐♦♥s ❣✐✈❡ ✉s t❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧ ❢♦r t❤❡ ❤♦✉♥❞✳ ❲❡ ❤❛✈❡ ❢♦r t❤❡ t❛♥❣❡♥t ♦❢ t❤❡ ❜❛s❡ ❛♥❣❧❡ ♦❢ t❤❡s❡ tr✐❛♥❣❧❡s✿ τn =

∆qn y n − qn = . pn − xn −∆pn

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ❛ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❢♦r ∆qn ✱

∆qn = −τn · ∆pn ,

✐❢ ✇❡ ❝❛♥ ❥✉st ✜♥❞ ∆pn ✳ ❚❤❡ ❤②♣♦t❡♥✉s❡ ♦❢ t❤❡ ❧❛tt❡r tr✐❛♥❣❧❡ ✐s (∆pn )2 + (∆qn )2 = (v · ∆t)2 .

❲❡ ❞❡r✐✈❡ ❢r♦♠ t❤❡ ❧❛st ❡q✉❛t✐♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣✿ 1+

τn2

=1+



∆qn ∆pn

2

=



v · ∆t ∆pn

2

.

❙♦❧✈✐♥❣ t❤✐s ❡q✉❛t✐♦♥ ❢♦r ∆pn ✱ ✇❡ ❝❤♦♦s❡ t❤❡ ♥❡❣❛t✐✈❡ s✐❣♥ ❢♦r t❤❡ sq✉❛r❡ r♦♦t✿ v ∆pn = − p ∆t . 1 + τn2 ❲❛r♥✐♥❣✦ ■❢ t❤❡ r❛❜❜✐t ❣❡ts t♦ t❤❡ r✐❣❤t ♦❢ t❤❡ ❤♦✉♥❞✱ ♦♥❡ ❤❛s t♦ ❝❤❛♥❣❡ t❤❡ s✐❣♥✳

❊①❛♠♣❧❡ ✺✳✷✳✹✿ s♣r❡❛❞s❤❡❡t

▲❡t✬s ❝♦♥✜r♠ ♦✉r ❛♥❛❧②s✐s ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✳ ❚❤❡ s❡tt✐♥❣ ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ∆t = 0.1 s❡❝♦♥❞s • s = 30 ❢❡❡t ♣❡r s❡❝♦♥❞ • v = 31 ❢❡❡t ♣❡r s❡❝♦♥❞ • (x0 , y0 ) = (0, 0) ❢❡❡t • (p0 , q0 ) = (100, 50) ❢❡❡t ❚❤❡ ♠❛✐♥ ❢♦r♠✉❧❛s ❛r❡ t❤❡ ♦♥❡s ❢♦r ∆pn ❛♥❞ ∆qn ✿ ❂✲✭❘✸❈✻✮✴✭❙◗❘❚✭✶✰❘❈❬✲✶❪✂ ✷✮✮✯❘✶❈✷

❚❤❡s❡ ✐s t❤❡ s✐♠✉❧❛t✐♦♥✿

❂✲❘❈❬✲✷❪✯❘❈❬✲✶❪

✺✳✷✳

❚❤❡ ♣✉rs✉✐t ❝✉r✈❡s

✸✶✶

❚❤❡ r❡s✉❧ts ♠❛t❝❤ t❤♦s❡ ❛❝q✉✐r❡❞ ✈✐❛ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧✳

❋r♦♠ t❤❡ t✇♦ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s✱ ✇❡ ❞❡r✐✈❡ ✈✐❛ s✉❜st✐t✉t✐♦♥✿

∆t → 0✱

❚❤❡♥✱ ✈✐❛

v v ∆pn ∆t =⇒ = −p ∆pn = − p 2 ∆t 1 + τn 1 + τn2 ∆qn vτn ∆qn = −τn · ∆pn =⇒ =p ∆t 1 + τn2 ✇❡ ❛rr✐✈❡ ❛t t✇♦ t✐♠❡✲❞❡♣❡♥❞❡♥t ❖❉❊s✿

 dp v  , = −√  dt 1 + τ2 dq vτ   , =√ dt 1 + τ2

✇❤❡r❡

τ= ❛♥❞

(x, y)

✐s t❤❡

✜①❡❞

♣❛r❛♠❡tr✐❝ ❝✉r✈❡ r❡♣r❡s❡♥t✐♥❣ t❤❡ ♠♦t✐♦♥ ♦❢ t❤❡ r❛❜❜✐t✳

◆♦✇✱ ❧❡t✬s s❡❡ ✇❤❛t t❤✐s ❛♥❛❧②s✐s ❧♦♦❦s ❧✐❦❡ ✐♥ t❤❡ t❤❡ ❤♦✉♥❞ ❛♥❞

∆H

♦❢

H

R

y−q , p−x

✈❡❝t♦r ♥♦t❛t✐♦♥✳

❙✉♣♣♦s❡✱ ❛s ❜❡❢♦r❡✱

H

✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢

✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ r❛❜❜✐t✳ ❚❤❡ ❧❛tt❡r ✐s ❦♥♦✇♥✳ ❋♦r t❤❡ ❢♦r♠❡r✱ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ✈❡❝t♦r

❤❛s t♦ ❜❡ ❝♦❧❧✐♥❡❛r t♦ t❤❡ ✈❡❝t♦r ❢r♦♠

H

t♦

R✳

❚❤❡r❡❢♦r❡✱

∆H · HR = ||∆H|| · ||HR|| = S||HR|| . ❚❤❡ ❢♦r♠✉❧❛ ❛♣♣❧✐❡s t♦ ❛♥② ♠✉t✉❛❧ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ r❛❜❜✐t ❛♥❞ t❤❡ ❤♦✉♥❞ ❛s ✇❡❧❧ t♦ ♣✉rs✉✐ts ✐♥ ❛ s♣❛❝❡ ♦❢ ❛♥② ❞✐♠❡♥s✐♦♥✳

✺✳✸✳

❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❛s s②st❡♠s

✸✶✷

❊①❡r❝✐s❡ ✺✳✷✳✺

❉❡r✐✈❡ ❛♥ ❡①♣❧✐❝✐t ❖❉❊s ❢♦r t❤❡ ❝❛s❡ ✇❤❡♥ t❤❡ r❛❜❜✐t ❢♦❧❧♦✇s ❛ str❛✐❣❤t ♣❛t❤✳

✺✳✸✳ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❛s s②st❡♠s

❲❡ ♣r❡✈✐♦✉s❧② ❞✐s❝✉ss❡❞ ❤♦✇ ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ♠♦❞❡❧ ♠♦t✐♦♥ ♦❢ ♦❜❥❡❝ts ❛✛❡❝t❡❞ ❜② ❢♦r❝❡s✳ ❆ s♣❡❝✐❛❧ ❝❛s❡ ✐s ❛ ❧✐♥❡❛r ❖❉❊✿ ∆x ∆2 x = a + bx ❛♥❞ x′′ = ax′ + bx . 2 ❍❡r❡✱ ✇❡ ❤❛✈❡✿

∆t

∆t

• x ✐s t❤❡ ♣♦s✐t✐♦♥✳

∆x ❛♥❞ x′ ✐s t❤❡ ✈❡❧♦❝✐t②✳ ∆t ∆2 x • ❛♥❞ x′′ ✐s t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✳ ∆t2 ▼❡❛♥✇❤✐❧❡ a ❛♥❞ b ❛r❡ ❝♦♥st❛♥t ♥✉♠❜❡rs t❤❛t ❡①♣r❡ss t❤❡ ♣r♦♣♦rt✐♦♥❛❧ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ❢♦r❝❡ ✭✐✳❡✳✱ t❤❡ •

❛❝❝❡❧❡r❛t✐♦♥✮ ♦♥ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ♣♦s✐t✐♦♥ r❡s♣❡❝t✐✈❡❧②✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ♦❜❥❡❝t ♠❛② ❜❡ ❛tt❛❝❤❡❞ t♦ t❤❡ ✇❛❧❧ ❜② ❛ s♣r✐♥❣ ❛♥❞✱ ❛t t❤❡ s❛♠❡ t✐♠❡✱ ❜❡ ❛✛❡❝t❡❞ ❜② t❤❡ ❢r✐❝t✐♦♥ ♦❢ t❤❡ s✉r❢❛❝❡✳

■♥ ❢❛❝t✱ ✐t ♠❛❦❡s s❡♥s❡ t♦ ❛ss✉♠❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ❚❤❡ s♣r✐♥❣ ❤❛s ✐ts ❍♦♦❦❡✬s ❢♦r❝❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❢r♦♠ t❤❡ ❡q✉✐❧✐❜r✐✉♠✱ ✇✐t❤ ❝♦❡✣❝✐❡♥t b✱ ❛♥❞ s✐♥❝❡ ✐t ♣✉❧❧s ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦♣♣♦s✐t❡ t♦ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t b ≤ 0❀ ❛♥❞

• ❚❤❡ ❢r✐❝t✐♦♥ ❢♦r❝❡ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✈❡❧♦❝✐t②✱ ✇✐t❤ ❝♦❡✣❝✐❡♥t a✱ ❛♥❞ s✐♥❝❡ ✐t ♣✉❧❧s ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦♣♣♦s✐t❡ t♦ t❤❡ ✈❡❧♦❝✐t②✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t a ≤ 0✳

❚❤❡s❡ ✇✐❧❧ ❜❡ ♦✉r ❛ss✉♠♣t✐♦♥s✿

a ≤ 0, b < 0 .

Pr❡✈✐♦✉s❧②✱ ✇❡ ✉s❡❞ t❤❡ ❞✐s❝r❡t❡ ❡q✉❛t✐♦♥ ❛❜♦✈❡ t♦ t❤❡ ♠♦t✐♦♥ ♦❢ s♣r✐♥❣ ❛♥❞ ♦t❤❡r ❞②♥❛♠✐❝s✳ ❲❤❛t ❛❜♦✉t t❤❡ ♦t❤❡r ❡q✉❛t✐♦♥❄ ■♥ t❤❡ s✐♠♣❧❡st ❝❛s❡ a = 0✱ ✐t ✇❛s ❛❧s♦ s♦❧✈❡❞✳ ■♥ ♦✉r ❛tt❡♠♣t t♦ s♦❧✈❡ t❤✐s ❡q✉❛t✐♦♥ ✐♥ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ✇❡ r❡❛❧✐③❡ t❤❛t ✇❡ ❤❛✈❡ ♥♦ ♠❡t❤♦❞s ❞❡✈❡❧♦♣❡❞ ❢♦r ❡q✉❛t✐♦♥s ♦❢ s❡❝♦♥❞ ♦r❞❡r✦ ❲❤❡♥ ✇❡ ❡♥❝♦✉♥t❡r ❛ ❝♦♠♣❧❡t❡❧② ♥❡✇ s✐t✉❛t✐♦♥✱ ✇❡ s❤♦✉❧❞ ❛❧✇❛②s tr② t♦ r❡❞✉❝❡ ✐t t♦ s♦♠❡t❤✐♥❣ ❢❛♠✐❧✐❛r✳ ❲❡ ✇✐❧❧ ❛♣♣❧② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❡✈❡r

tr✐❝❦✳ ❲❡ ✐♥tr♦❞✉❝❡ ❛♥ ❡①tr❛ ✭❞❡♣❡♥❞❡♥t✮ ✈❛r✐❛❜❧❡✿ y = x′ .

✺✳✸✳

❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❛s s②st❡♠s

✸✶✸

■♥ t❡r♠s ♦❢ ♠♦t✐♦♥✱ t❤❡ tr✐❝❦ ✐s ♥♦t❤✐♥❣ ❜✉t r❡✲✐♥tr♦❞✉❝✐♥❣ t❤❡ ✈❡❧♦❝✐t② ❜❛❝❦ ✐♥t♦ t❤❡ ♣✐❝t✉r❡✳ ❚❤❡ r❡s✉❧t ✐s ❛ s②st❡♠ ♦❢ t✇♦ ❖❉❊s✿  ′ x = y, ′ y = bx + ay. ❚❤❡ r❡s✉❧t ✐s ❛ tr❛❞❡✲♦✛✿ ✐♥❝r❡❛s✐♥❣ t❤❡ ❞✐♠❡♥s✐♦♥ ✕ ❢r♦♠ 1 t♦ 2 ✕ ❜✉t ❞❡❝r❡❛s✐♥❣ t❤❡ ♦r❞❡r ✕ ❢r♦♠ 2 ❢♦r 1 ✕ ♦❢ t❤❡ s②st❡♠✳ ❚❤✐s 2 × 2 ❧✐♥❡❛r ✈❡❝t♦r ❖❉❊✱   0 1 ′ , X = F X, ✇✐t❤ F = b a ❝❛♥ ♥♦✇ ❜❡ s✉❜❥❡❝t❡❞ t♦ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ❛♥❛❧②s✐s ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❧❛st ❝❤❛♣t❡r✳ ❋✐rst✱ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s✿

χ(λ) = λ2 − aλ − b .

❙✉♣♣♦s❡ λ1 ❛♥❞ λ2 ❛r❡ ✐ts t✇♦ r♦♦ts✿

√ a2 + 4b a . λ1,2 = ± 2 2 ❚❤❡♥ t❤❡ ♦✉t❝♦♠❡s ❞❡♣❡♥❞ ♦♥ t❤❡ s✐❣♥ ♦❢ t❤❡ ❞✐s❝r✐♠✐♥❛♥t✱ D = a2 + 4b . ❈♦♥s✐❞❡r t❤❡ ❝❛s❡ ♦❢ D ≥ 0✳ ❙✐♥❝❡ a ≤ 0✱ ✇❡ ❤❛✈❡

a λ1 = − 2



a2 + 4b < 0. 2

❙✐♥❝❡ b < 0✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t

a λ1 = + 2



√ a a −a a2 + 4b a2 < + = + =0 2 2 2 2 2

❚❤❡r❡❢♦r❡✱ ❛♥ ✐♠♣♦rt❛♥t ❝♦♥❝❧✉s✐♦♥ ✐s t❤❛t

t❤❡ r❡❛❧ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ❛❧✇❛②s ♥❡❣❛t✐✈❡✳

❆❝❝♦r❞✐♥❣ t♦ ♦✉r ❝❧❛ss✐✜❝❛t✐♦♥ t❤❡♦r❡♠✱ t❤❡r❡ ❛r❡ t❤r❡❡ ♠❛✐♥ ❝❛s❡s✿

• ❈❛s❡ ✶✿ t✇♦ ❞✐st✐♥❝t r❡❛❧ r♦♦ts❀

• ❈❛s❡ ✷✿ ♦♥❡ r❡❛❧ r❡♣❡❛t❡❞ r♦♦t❀

• ❈❛s❡ ✸✿ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ r♦♦ts✳

❈❛s❡ ✶ ✐s t❤❡ ❝❛s❡ ♦❢

❚❤❡ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ ✐s ❣✐✈❡♥ ❜②

D = a2 + 4b > 0 . x = Ceλ1 t + Keλ2 t .

❆s λ1 , λ2 < 0✱ ✇❡ ❤❛✈❡ ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛② ✭❛ st❛❜❧❡ ♥♦❞❡ ♦❢ t❤❡ ✈❡❝t♦r ❖❉❊✮✿ t❤❡ ❢r✐❝t✐♦♥ ❜r✐♥❣s t❤❡ ❖❉❊ ❜❛❝❦ t♦ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ✇✐t❤♦✉t ♦s❝✐❧❧❛t✐♥❣✳ ❍❡r❡ ❊✉❧❡r✬s ♠❡t❤♦❞ ✐s r✉♥ ✇✐t❤ a = −8, b = −1✿

✺✳✸✳

❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❛s s②st❡♠s

✸✶✹

❈❛s❡ ✷ ✐s t❤❡ ❝❛s❡ ♦❢

D = a2 + 4b = 0 . ❆❣❛✐♥✱ ❛s

λ = λ1 = λ2 < 0 ✱

✇❡ s❡❡ ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛② ✭❛ st❛❜❧❡ ✐♠♣r♦♣❡r ♥♦❞❡ ♦❢ t❤❡ ❖❉❊✮✿ t❤❡ ❖❉❊✬s

♠♦t✐♦♥ ❞✐❡s ♦✉t✱ ❡✈❡♥ t❤♦✉❣❤ ✐t ♠✐❣❤t ♦✈❡rs❤♦♦t✳ ❚❤❡ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ ✐s ❣✐✈❡♥ ❜②

x = (C + Kt)eλt . ❍❡r❡ ❊✉❧❡r✬s ♠❡t❤♦❞ ✐s r✉♥ ✇✐t❤

a = −2, b = −1✿

❈❛s❡ ✸ ✐s t❤❡ ❝❛s❡ ♦❢

D = a2 + 4b < 0 . ❚❤❡ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ ✐s ❣✐✈❡♥ ❜②

x = Ceαt cos(βt) + Keαt sin(βt) , ✇❤❡r❡

α = Re λ1,2 = ✐s ♥♦♥✲♥❡❣❛t✐✈❡ ❜② ❛ss✉♠♣t✐♦♥ ❛♥❞

β = Im λ1,2 = ± ✐s ♥♦♥✲③❡r♦ ❜❡❝❛✉s❡

D 6= 0✳

❲❤❡♥

a 2 √

D 2

a < 0✱ ✇❡ s❡❡ ❛♥ ♦s❝✐❧❧❛t✐♥❣ ❞❡❝❛② ✭❛ st❛❜❧❡ ❢♦❝✉s ♦❢ t❤❡ ❖❉❊✮✿ t❤❡ ❢r✐❝t✐♦♥

✐s str♦♥❣ ❡♥♦✉❣❤ t♦ ❜r✐♥❣ t❤❡ ❖❉❊ t♦ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❡✈❡♥t✉❛❧❧② ❜✉t ✐s♥✬t str♦♥❣ ❡♥♦✉❣❤ t♦ st♦♣ t❤❡ ❖❉❊ ❢r♦♠ ♦s❝✐❧❧❛t✐♥❣✳ ❍❡r❡ ❊✉❧❡r✬s ♠❡t❤♦❞ ✐s r✉♥ ✇✐t❤

a = 0✱ ✇❡ a = 0, b = −1✿

❲❤❡♥

a = −.5, b = −1✿

s❡❡ ♣✉r❡ ♦s❝✐❧❧❛t✐♦♥ ✭❛ ❝❡♥t❡r ♦❢ t❤❡ ❖❉❊✮✿ ♥♦ ❢r✐❝t✐♦♥✳

❍❡r❡ ❊✉❧❡r✬s ♠❡t❤♦❞ ✐s r✉♥ ✇✐t❤

✺✳✹✳

✸✶✺

❱❡❝t♦r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r✿ ❛ ❞♦✉❜❧❡ s♣r✐♥❣

▲♦♦❦✐♥❣ ❛t t❤❡s❡ ✐❧❧✉str❛t✐♦♥ ❛s ❛ s❡q✉❡♥❝❡ r❡✈❡❛❧s t❤❡ ❡✛❡❝t ♦❢ ❣r♦✇✐♥❣ ♦❢ t❤❡ ❢r✐❝t✐♦♥ ❝♦❡✣❝✐❡♥t b✳ ❊①❡r❝✐s❡ ✺✳✸✳✶

P♦✐♥t ♦✉t ✐♥ ✇❤❛t ✇❛② t❤❡ ❧❛st ✐❧❧✉str❛t✐♦♥ ❞♦❡s♥✬t ♠❛t❝❤ t❤❡ ❞❡s❝r✐♣t✐♦♥ ❛♥❞ ❡①♣❧❛✐♥ ✇❤②✳

✺✳✹✳ ❱❡❝t♦r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r✿ ❛ ❞♦✉❜❧❡ s♣r✐♥❣

❲❤❛t ♦❢ t❤❡ ♥❛t✉r❡ ♦❢ t❤❡ ❢♦r❝❡s r❡♠❛✐♥s t❤❡ s❛♠❡ ❜✉t t❤❡ ♠♦t✐♦♥ ✐s

♦♥ t❤❡ ♣❧❛♥❡



■♥ t❤❡ s✐♠♣❧❡st s✐t✉❛t✐♦♥✱ ✇❡ ❤❛✈❡ t❤✐s✿

• ❚✇♦ ❞✐✛❡r❡♥t s♣r✐♥❣s ❛r❡ ❛tt❛❝❤❡❞ t♦ t❤❡ ♦❜❥❡❝t ❛❧♦♥❣ t❤❡ t✇♦ ❛①❡s✳ • ❚✇♦ ❞✐✛❡r❡♥t ❦✐♥❞s ♦❢ ❢r✐❝t✐♦♥ ❛r❡ ♣r♦❞✉❝❡❞ ❛❧♦♥❣ t❤❡ t✇♦ ❛①❡s✳

❆s t❤❡ ♦❜❥❡❝t ♠♦✈❡s✱ ✇❡ ❤❛✈❡ t✇♦ ❡✛❡❝ts✿

• ❚✇♦ s♣r✐♥❣s ❡①❡rt t✇♦ ❢♦r❝❡s ❛❧♦♥❣ t❤❡ ❛①❡s ✕ ♥❡❣❛t✐✈❡❧② ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❢r♦♠ t❤❡ ❡q✉✐❧✐❜r✐✉♠✳ • ❚✇♦ ❦✐♥❞s ♦❢ ❢r✐❝t✐♦♥ ❡①❡rt t✇♦ ❢♦r❝❡s ❛❧♦♥❣ t❤❡ ❛①❡s ✕ ♥❡❣❛t✐✈❡❧② ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✈❡❧♦❝✐t②✳

✺✳✹✳

✸✶✻

❱❡❝t♦r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r✿ ❛ ❞♦✉❜❧❡ s♣r✐♥❣

❚❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ❛♥❞ t❤❡ ✈❡❧♦❝✐t✐❡s ♦❢ t❤❡ ♦❜❥❡❝t ✐♥ t❤❡ t✇♦ ❞✐r❡❝t✐♦♥s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❡❛❝❤ ♦t❤❡r✳ ❚❤❡r❡❢♦r❡✱ t❤❡r❡ ✇✐❧❧ ❜❡ t✇♦ ❧✐♥❡❛r ❖❉❊ ♦❢ ♦r❞❡r 2 ♦❢ t❤❡ s❛♠❡ ❦✐♥❞s ❛s ❜❡❢♦r❡✿ x′′ = ax′ + bx ❛♥❞ y ′′ = cy ′ + dy ,

✇❤❡r❡

a ≤ 0 ❛♥❞ b ≤ 0 ❛♥❞ c ≤ 0 ❛♥❞ d ≤ 0 .

■♥ t❤✐s ❝❛s❡✱ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ❛♣♣❧✐❡s✱ s❡♣❛r❛t❡❧②✳ ❲❡ ❝♦♠❜✐♥❡ t❤❡ t✇♦ s❝❛❧❛r ❡q✉❛t✐♦♥s ✐♥t♦ ♦♥❡ ✈❡❝t♦r ❡q✉❛t✐♦♥✿ 

x′′ = ax′ +bx y ′′ = cy ′ +dy



❧❡❛❞✐♥❣ t♦

●❡♥❡r❛❧❧②✱ ✇❡ ❤❛✈❡ ❛ ✈❡❝t♦r ❖❉❊✿

x′′ y ′′



=



a 0 0 c



x′ y′



+



b 0 0 d



x y



X ′′ = AX ′ + BX .

❍❡r❡✱ X=



x y



,



X =



x′ y′



′′

,

X =

x′′ y ′′





❛r❡ t❤❡ ♣♦s✐t✐♦♥ ✈❡❝t♦r✱ t❤❡ ✈❡❧♦❝✐t②✱ ❛♥❞ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✱ r❡s♣❡❝t✐✈❡❧②✳ ❆❧s♦✱ A ❛♥❞ B ❛r❡ t✇♦ 2 × 2 ♠❛tr✐❝❡s t❤❛t ❡①♣r❡ss t❤❡ ✭❧✐♥❡❛r✮ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ❢♦r❝❡ ✭✐✳❡✳✱ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✮ ♦♥ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ ♣♦s✐t✐♦♥ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡② ❞♦♥✬t ❤❛✈❡ t♦ ❜❡ ❞✐❛❣♦♥❛❧ ❛♥②♠♦r❡✳ ■♥ ❝♦♥tr❛st t♦ t❤❡ ❧❛st s❡❝t✐♦♥✱ ✇❡ ✇✐❧❧ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ❞✐s❝r❡t❡ ❖❉❊s✿ ∆X ∆2 X = A + BX . ∆t2 ∆t

❍❡r❡✱ ❛ s♦❧✉t✐♦♥ X ✐s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❞❡✜♥❡❞ ♦♥ t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧✱ t0 , t1 , ... ✭✐✳❡✳✱ ❛ ❞✐s❝r❡t❡ 0✲❢♦r♠✮✱ s♦ t❤❛t ✐ts ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛♥❞ t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥✿ 

 ∆2 x ∆2 X  ∆t2  = 2 . ∆y ∆t2 ∆t2

 ∆x ∆X  ∆t  = , ∆y ∆t ∆t







 x  X =  , y

■♥ ♦r❞❡r t♦ s✐♠♣❧✐❢② t❤❡ ♥♦t❛t✐♦♥✱ ✇❡ ❞❡✜♥❡ t❤r❡❡ s❡q✉❡♥❝❡s ♦❢ ✈❡❝t♦rs ✐♥ R2 ✿ ∆2 X (tn ) ∆t2 ∆X (tn ) Xn′ = ∆t Xn′′ =

Xn = X( tn )

❚❤❡ ❖❉❊ t❤❡♥ ♣r♦❞✉❝❡s t❤❡s❡ t❤r❡❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❢♦r t❤♦s❡ t❤r❡❡ s❡q✉❡♥❝❡s ♦❢ ✈❡❝t♦rs✿ ′′ Xn+1 = AXn′ + BXn ′ ′′ Xn+1 = Xn′ + Xn+1 ∆t ′ Xn+1 = Xn + Xn+1 ∆t

❲❡ ❛♣♣❧② t❤❡♠ ❜❡❧♦✇✳ ❚❤❡ st❛rt✐♥❣ ♣♦✐♥t ✐s t❤❡ s✐♠♣❧❡st ❝❛s❡ ♦❢ t❤❡ t✇♦ s♣r✐♥❣s✳ ❍❡r❡✱ t❤❡ ♠❛tr✐❝❡s ❛r❡ ❞✐❛❣♦♥❛❧ ✿ A=

✇✐t❤



ax 0 0 ay



,

B=



bx 0 0 by

ax , ay ≤ 0 ❛♥❞ bx , by < 0 .



,

✺✳✹✳ ❱❡❝t♦r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r✿ ❛ ❞♦✉❜❧❡ s♣r✐♥❣

✸✶✼

❇❡❧♦✇✱ ✇❡ ✇✐❧❧ ❜❡ ❣r❛❞✉❛❧❧② ❛❞❞✐♥❣ ♥❡✇ ❢♦r❝❡s t♦ t❤❡ s❡t✉♣ ❜② ❛❞❞✐♥❣ ❛ ♥❡✇ ♥♦♥✲③❡r♦ ❡♥tr② t♦ t❤❡ ♠❛tr✐❝❡s✳ ❚❤❡ r❡s✉❧ts ❛r❡ ✐❧❧✉str❛t❡❞ ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐s❝r❡t❡ ♠♦❞❡❧s ♣r♦❞✉❝❡❞ ❜② ❊✉❧❡r✬s ♠❡t❤♦❞✳ ❚❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✇✐❧❧ ❜❡ t❤❡ s❛♠❡✿

X0 =



1 0



,

X0′

=



1 1



.

x✲❛①✐s✿     −0.5 0 0 0 . , B= A= 0 0 0 0

❲❡ st❛rt ✇✐t❤ ♥♦t❤✐♥❣ ❜✉t ❛ s♣r✐♥❣ ❛❧♦♥❣ t❤❡

❚❤❡ r❡s✉❧t ✐s ❛♥ ♦s❝✐❧❧❛t✐♦♥ ❛❧♦♥❣ t❤❡

x✲❛①✐s

❛♥❞ ✉♥✐❢♦r♠ ♠♦t✐♦♥ ❛❧♦♥❣ t❤❡

y ✲❛①✐s✿   0 0 , A= 0 0

y ✲❛①✐s✳

❲❡ ♥♦✇ ❛❞❞ ❛♥ ✐❞❡♥t✐❝❛❧ s♣r✐♥❣ ❢♦r t❤❡

❆s ❛ r❡s✉❧t✱ ❜♦t❤

x

❛♥❞

y

B=



−0.5 0 0 −0.5



.

♦s❝✐❧❧❛t❡ ❛t t❤❡ s❛♠❡ ♣❡r✐♦❞ ❛♥❞ ❝♦♠❡ ❜❛❝❦ s✐♠✉❧t❛♥❡♦✉s❧②❀ ✐t✬s ❛ ❝②❝❧❡✳ ❚❤❡ ♣❛t❤

❛♣♣❡❛rs t♦ ❜❡ ❛♥ ❡❧❧✐♣s❡ ❝❡♥t❡r❡❞ ❛t t❤❡ ♦r✐❣✐♥✳ ❲❡ ❛❧s♦ ♥♦t✐❝❡ t❤❛t t❤❡ ♠♦t✐♦♥ ✐s s❧♦✇s ❞♦✇♥ ❛t t❤❡ t✐♣s ♦❢ t❤✐s ❡❧❧✐♣s❡✳

❊①❡r❝✐s❡ ✺✳✹✳✶ ❲❤② ✐s♥✬t t❤❡ tr❛❥❡❝t♦r② ❛ ❝✐r❝❧❡❄

▲❡t✬s ♠❛❦❡ t❤❡ ❧❛tt❡r s♣r✐♥❣ ♠♦r❡ r✐❣✐❞ ✿

A=



0 0 0 0



,

B=



−0.5 0 0 −1



.

✺✳✹✳

✸✶✽

❱❡❝t♦r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r✿ ❛ ❞♦✉❜❧❡ s♣r✐♥❣

❆s ❛ r❡s✉❧t✱ x ❛♥❞ y st✐❧❧ ♦s❝✐❧❧❛t❡ ❜✉t ❛t ❞✐✛❡r❡♥t ❢r❡q✉❡♥❝✐❡s✳ ❚❤❡② ♠❛② ❝♦♠❡ ❜❛❝❦ s✐♠✉❧t❛♥❡♦✉s❧② ❛❢t❡r s❡✈❡r❛❧ ♣❡r✐♦❞s✳ ❚❤❛t ✇✐❧❧ ❝r❡❛t❡ ❛ ❝②❝❧❡✳ ❲❡ ♥♦✇ ❛❞❞ ❢r✐❝t✐♦♥ ✐♥ t❤❡ x✲❞✐r❡❝t✐♦♥✿ A=



−0.3 0 0 0



,

B=



−0.5 0 0 −1.



.

❲❤✐❧❡ y st✐❧❧ ♦s❝✐❧❧❛t❡s✱ t❤❡ x✲♣♦s✐t✐♦♥✬s ♦s❝✐❧❧❛t✐♦♥ ❞✐♠✐♥✐s❤❡s ❛s ✐t ❛♣♣r♦❛❝❤❡s 0✳ ❲❡ ❛❧s♦ ❛❞❞ ❢r✐❝t✐♦♥ ✐♥ t❤❡ y ✲❞✐r❡❝t✐♦♥ ♥♦✇✿ A=



−0.3 0 0 −0.2



,

B=



−0.5 0 0 −1.0



.

❇♦t❤ x ♦r y ♦s❝✐❧❧❛t✐♦♥s ❞✐♠✐♥✐s❤ ❛♥❞ t❤❡ ♣♦✐♥t ❛♣♣r♦❛❝❤❡s 0✳ ❯♣ ✉♥t✐❧ ♥♦✇✱ x ❛♥❞ y ❤❛s ❜❡❡♥ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ t❤❡ t✇♦ ❖❉❊s ❛r❡ s♦❧✈❛❜❧❡ s❡♣❛r❛t❡❧②✳ ❚❤✐s ✐s ✇❤② ❡✈❡r②t❤✐♥❣ ✇❡ s❡❡ ✐♥ t❤❡ s✐♠✉❧❛t✐♦♥s ❛r❡ ❝♦♥✜r♠❡❞ ❜② t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✳ ❋r♦♠ ♥♦✇ ♦♥✱ t❤✐s ❝❧❛ss✐✜❝❛t✐♦♥ ❞♦❡s♥✬t ❛♣♣❧② ❛♥②♠♦r❡✳ ◆♦t❡ ❛❧s♦ t❤❛t t❤❡ ❝❧❡✈❡r tr✐❝❦ ♦❢ ✐♥tr♦❞✉❝✐♥❣ t❤❡ ✈❡❧♦❝✐t✐❡s ❛s ♥❡✇ ✈❛r✐❛❜❧❡s ✇✐❧❧ ♣r♦❞✉❝❡ 4 × 4 ♠❛tr✐❝❡s✱ t❤❡ ❛♥❛❧②s✐s ♦❢ ✇❤✐❝❤ ❧✐❡s ♦✉ts✐❞❡ t❤❡ s❝♦♣❡ ♦❢ t❤✐s t❡①t✳ ❚❤✐s ✐s ✇❤② ✇❡ ♣r♦❝❡❡❞ ✇✐t❤ ❞✐s❝r❡t❡ ♠♦❞❡❧s ♦♥❧②✳

✺✳✹✳

✸✶✾

❱❡❝t♦r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r✿ ❛ ❞♦✉❜❧❡ s♣r✐♥❣

❲❡ ✐♥t❡r♠✐① t❤❡ t✇♦ ✈❛r✐❛❜❧❡s ♥♦✇ ❜② ♠❛❦✐♥❣ t❤❡ ❢r✐❝t✐♦♥ ✈❛r② ✐♥ t❤❡ ❞✐r❡❝t✐♦♥s ♦t❤❡r t❤❛♥ t❤❡ t✇♦ ❛①❡s ✭❢♦r ❡①❛♠♣❧❡✱ t❤❡ s♣r✐♥❣s ♠❛② ❜❡ ♣❧❛❝❡❞ ✉♥❞❡r ✈❛r✐♦✉s ❛♥❣❧❡s✮✿ A=



−0.3 −0.3 0.6 −0.2



,

B=



−0.5 0 0 −1



.

❚❤❡ ♣❛tt❡r♥ ♦❢ ❜❛❝❦✲❛♥❞✲❢♦rt❤ ❝♦♥✈❡r❣❡♥❝❡ t♦✇❛r❞ 0 r❡♠❛✐♥s✳ ❆s ❛ ✜♥❛❧ ❡①♣❡r✐♠❡♥t✱ ✇❡ r❡♠♦✈❡ t❤❡ y ✲❛①✐s s♣r✐♥❣✿ A=



−0.3 −0.3 0.6 −0.2



,

B=



−0.5 0 0 0



.

❚❤❡ ♦s❝✐❧❧❛t✐♦♥ ♦❢ t❤❡ y ✲❝♦♦r❞✐♥❛t❡ ♠♦✈❡s ❛✇❛② ❢r♦♠ 0✳ ▲❡t✬s ❧♦♦❦ ❛t ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❡♥❡r❣② ✐♥ t❤✐s ❞✐s❝r❡t❡ ♠♦❞❡❧✳ ❇✉t ✜rst ❧❡t✬s s❡❡ ✇❤❛t t❤❡ ❞❛t❛ t❡❧❧s ❛❜♦✉t t❤❡ ❝♦♥t✐♥✉♦✉s ♠♦t✐♦♥✳ ❋♦r s✐♠♣❧✐❝✐t② ✇❡ ❝♦♥t✐♥✉❡ t♦ ❛ss✉♠❡ t❤❛t t❤❡ ♠❛ss ✐s ❡q✉❛❧ t♦ 1✳ ❚❤❡♥ t❤❡ t❤✐s ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✿

❦✐♥❡t✐❝ ❡♥❡r❣②

✐s ❦♥♦✇♥ t♦ ❜❡

1 K(t) = ||X ′ (t)||2 . 2

❊✉❧❡r✬s ♠❡t❤♦❞ s❛♠♣❧❡s t❤❡ ✈❡❧♦❝✐t② ❛♥❞ ❣✐✈❡s ✉s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✱ t❤❡ ❡♥❡r❣② ✿ 1 Kn = ||Xn′ ||2 , 2

✐♥ ❡❛❝❤ r♦✇ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t✳ ◆❡①t✱ s✉♣♣♦s❡ t❤❛t t❤❡ ❜❡ ❢♦✉♥❞ ❛s t❤❡ ✇♦r❦ ♦❢ t❤✐s ❢♦r❝❡✿ Z t W (t) = −

0

s❛♠♣❧❡❞ ❦✐♥❡t✐❝

✳ ❚❤❡♥ t❤❡ ♣♦t❡♥t✐❛❧

❢♦r❝❡ ✐s ❝♦♥s❡r✈❛t✐✈❡

❡♥❡r❣②

❝❛♥

X ′′ · X ′ dt .

❲❤❛t ❞♦❡s t❤❡ ❞❛t❛ t❡❧❧ ✉s ❛❜♦✉t t❤✐s ❢✉♥❝t✐♦♥❄ ❖♥❝❡ ❛❣❛✐♥✱ ❊✉❧❡r✬s ♠❡t❤♦❞ s❛♠♣❧❡s t❤❡ ✈❡❧♦❝✐t② ❛♥❞ ❣✐✈❡s ✉s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✱ t❤❡ s❛♠♣❧❡❞ ♣♦t❡♥t✐❛❧ ❡♥❡r❣② ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✳ ❖✈❡r ❛ ♣❡r✐♦❞ ♦❢ t✐♠❡ ❢r♦♠ tk−1 t♦ tk ✭❧❡♥❣t❤ ∆t✮✱ t❤❡ ♣♦t❡♥t✐❛❧ ❡♥❡r❣② ❣r♦✇s ❜②✿ Wk = −Xk′′ · Xk′ ∆t .

✺✳✹✳

✸✷✵

❱❡❝t♦r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r✿ ❛ ❞♦✉❜❧❡ s♣r✐♥❣

❲❡ ❛❞❞ t❤❡s❡ ✐♥ ❛❧❧ r♦✇s ✉♣ t♦ t❤❡ nt❤ t♦ ✜♥❞ t❤❡ ❝✉rr❡♥t s❛♠♣❧❡❞ Pn =

n X

♣♦t❡♥t✐❛❧ ❡♥❡r❣②



Wk .

k=0

❚❤❡ ♥❡①t ❝♦❧✉♠♥ ❝♦♥t❛✐♥s t❤❡ s✉♠ ♦❢ t❤❡ t✇♦✱ t❤❡ t♦t❛❧

s❛♠♣❧❡❞ ❡♥❡r❣②



n X 1 ′ 2 Xk′′ · Xk′ ∆t . En = Kn + Pn = ||Xn || − 2 k=0

❆s ②♦✉ ❝❛♥ s❡❡✱ ✐t✬s ♥♦t ❝♦♥s❡r✈❡❞✦

❚❤❡ r❡s✉❧t ✐s t♦ ❜❡ ❡①♣❡❝t❡❞ ❢r♦♠ s❛♠♣❧✐♥❣ ✭❛ ❞✐s❝r❡t❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢✮ ❛ ❝♦♥t✐♥✉♦✉s ♠♦t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✺✳✹✳✷✿ ❞✐s❝r❡t❡ ❖❉❊ ♦❢ s❡❝♦♥❞ ♦r❞❡r ❆ ❞✐s❝r❡t❡

❖❉❊ ♦❢ s❡❝♦♥❞ ♦r❞❡r

✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s✿

′′ Xn+1 = f (Xn , Xn′ ) ′ ′′ Xn+1 = Xn′ + Xn+1 ∆t ′ Xn+1 = Xn + Xn+1 ∆t

❢♦r s♦♠❡ ❢✉♥❝t✐♦♥ f ❛♥❞ ❛ ♥♦♥✲③❡r♦ ♥✉♠❜❡r ∆t✳ ■ts ♦❢ ✈❡❝t♦rs t❤❛t s❛t✐s✜❡s t❤❡s❡ ❡q✉❛t✐♦♥s✳

s♦❧✉t✐♦♥

✐s t❤r❡❡ s❡q✉❡♥❝❡s

❚❤❡ ❞✐s❝r❡t❡ ❖❉❊ ❝♦♥s❡r✈❡s ❡♥❡r❣② ✐❢ ✇❡ ❥✉st ❧♦♦❦ ❛t t❤❡ ✇♦r❦ t❤❡ r✐❣❤t ✇❛②✳ ❲❡ r❡❝♦♠♣✉t❡ t❤❡ ♣♦t❡♥t✐❛❧ ❡♥❡r❣② ✐♥ t❤❡ ♥❡①t ❝♦❧✉♠♥ ✇✐t❤ t❤❡ ❝✉rr❡♥t ✈❡❧♦❝✐t② Xk′ r❡♣❧❛❝❡❞ ✇✐t❤ t❤❡ ❛✈❡r❛❣❡ ♦❢ t❤❡ ❝✉rr❡♥t ❛♥❞ t❤❡ ♣r❡✈✐♦✉s ✈❡❧♦❝✐t✐❡s✿ 1 ′ + Xk′ ) ∆t . Wk = Xk′′ · (Xk−1 2

❚❤❡ ❧❛st ❝♦❧✉♠♥ ❝♦♥✜r♠s t❤❡ ❝♦♥s❡r✈❛t✐♦♥ ♦❢ ❡♥❡r❣②✳ ❚❤✐s ❢❛❝t ✐s ❛❧s♦ ❡❛s② t♦ ♣r♦✈❡ ❛❧❣❡❜r❛✐❝❛❧❧② ❢♦r ❛❧❧ ❧✐♥❡❛r ♦r ♥♦♥✲❧✐♥❡❛r ❞✐s❝r❡t❡ s②st❡♠s✳

❚❤❡♦r❡♠ ✺✳✹✳✸✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❊♥❡r❣② ❚❤❡

❡♥❡r❣②

✐s ❝♦♥st❛♥t✳

♦❢ ❛ s♦❧✉t✐♦♥ X ♦❢ ❛ ❞✐s❝r❡t❡ ❖❉❊ ♦❢ s❡❝♦♥❞ ♦r❞❡r ❞❡✜♥❡❞ ❛s n X 1 1 ′ ′ 2 En = ||Xn || − + Xk′ ) ∆t , Xk′′ · (Xk−1 2 2 k=0

✺✳✺✳

❆ ♣❡♥❞✉❧✉♠

✸✷✶

Pr♦♦❢✳

1 1 1 ′ ′′ ′ En+1 − En = ||Xn+1 ||2 − ||Xn′ ||2 − Xn+1 · (Xn′ + Xn+1 ) ∆t 2 2 2   1 ′ 1 1 ′ ′ ||Xn+1 ||2 − ||Xn′ ||2 − Xn+1 − Xn′ · (Xn+1 + Xn′ ) ∆t = 2 ∆t 2   1 1 ′ ′ ′ = ||Xn+1 ||2 − ||Xn′ ||2 − Xn+1 · Xn+1 − Xn′ · Xn′ 2 2 = 0.

✺✳✺✳ ❆ ♣❡♥❞✉❧✉♠

❉❡s❝r✐♣t✐♦♥✿ ✏❛♥ ♦❜❥❡❝t ✐s ❛tt❛❝❤❡❞ t♦ ❛ ✜①❡❞ ♣♦✐♥t ✇✐t❤ ❛ str✐♥❣ ❛♥❞ s✉❜❥❡❝t❡❞ t♦ t❤❡ ❣r❛✈✐t②✑✳ ▲❡t

x

❜❡ t❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s ❛♥❞

z

t❤❡ ✈❡rt✐❝❛❧✳

❲❡ ❝♦♠♣✉t❡ ✇❤❛t✬s ♥❡❝❡ss❛r②✱ ✐✳❡✳✱ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✱ ❢♦r t❤❡ s♣r❡❛❞s❤❡❡t ❞✐s❝✉ss❡❞ ❛❜♦✈❡✳ ❙✉♣♣♦s❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ str✐♥❣ ✐s

L✳

❋✐rst✱ t❤❡

❣r❛✈✐t② ✐s t❤❡ ❝♦♥st❛♥t ✈❡❝t♦r

G =< 0, −gm > .

❚❤❡♥ t❤❡

0

r❡s✐st❛♥❝❡ R ♦❢ t❤❡ str✐♥❣ ✐s ❢♦✉♥❞ ❛s t❤❡ ♥❡❣❛t✐✈❡ ♦❢ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ ❣r❛✈✐t② ♦♥ t❤❡ ❧✐♥❡ ❢r♦♠

t♦ t❤❡ ❝✉rr❡♥t ❧♦❝❛t✐♦♥

X =< x, z >✿

R=− ❚❤❡♥ ✇❡ ✜♥❞ t❤❡

G·X G·X X = − X. ||X||2 L2

t❛♥❣❡♥t✐❛❧ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❣r❛✈✐t② ✿

F = R + G. ❖r✱ ✐t ✐s s✐♠♣❧②✿

F = G sin θ , ✇❤❡r❡

θ

✐s t❤❡ ❛♥❣❧❡ ♦❢ t❤❡ str✐♥❣ ✇✐t❤ t❤❡ ✈❡rt✐❝❛❧✳ ◆♦t❡ t❤❛t t❤❡ ❤♦r✐③♦♥t❛❧ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❢♦r❝❡ ✐s ③❡r♦

✇❤❡♥ t❤❡ str✐♥❣ ✐s ✈❡rt✐❝❛❧ ❛♥❞ ✇❤❡♥ ✐t ✐s ❤♦r✐③♦♥t❛❧ ❛♥❞ ✐t r❡✈❡rs❡s ✐ts ❞✐r❡❝t✐♦♥✳ ▼❡❛♥✇❤✐❧❡✱ t❤❡ ✈❡rt✐❝❛❧ ❝♦♠♣♦♥❡♥t ✐s ❛❧✇❛②s ♥❡❣❛t✐✈❡✳ ❆♥❞ t❤❡ t❛♥❣❡♥t✐❛❧ ❛❝❝❡❧❡r❛t✐♦♥ ✐s✿

A = F/m .

✺✳✺✳

❆ ♣❡♥❞✉❧✉♠

✸✷✷

❚❤❡s❡ q✉❛♥t✐t✐❡s ❛r❡ ❝♦♠♣✉t❡❞ ✐♥ t❤❛t ♦r❞❡r ❥✉st ❛s ❜❡❢♦r❡✳ ❚❤❡ ✉♣❞❛t❡❞ ✈❛❧✉❡s ♦❢ t❤❡ ✈❡❧♦❝✐t✐❡s ❛♥❞ t❤❡ ❧♦❝❛t✐♦♥s ❛r❡ ❛❧s♦ ❢♦✉♥❞ ❡①❝❡♣t z ✐s ❢♦✉♥❞ ❢r♦♠ x ❜②✿

x 2 + y 2 = L2 , ✇❤❡r❡ L ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ str✐♥❣✳ ❊q✉✐✈❛❧❡♥t❧②✱

x = L cos θ, y = L sin θ . ❲❡ ✉s❡ ❛ s♣r❡❛❞s❤❡❡t t♦ ❡✈❛❧✉❛t❡ t❤❡s❡ ❝♦❧✉♠♥ ❜② ❝♦❧✉♠♥ ❛♥❞ t❤❡♥ ❢♦r ❡❛❝❤ ✐t❡r❛t✐♦♥ ♦❢ t✿

❚❤❡ ✈✐s✉❛❧✐③❛t✐♦♥ s✉❣❣❡sts t❤❛t ✇❡ ✐♥❞❡❡❞ ❤❛✈❡ ❛ ♣❡♥❞✉❧✉♠✳ ❋♦r t❤❡ 3✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✱ ✇❡ ❣✐✈❡ t❤❡ ♣❡♥❞✉❧✉♠ ❛♥♦t❤❡r ✕ ❤♦r✐③♦♥t❛❧ ✕ ❞❡❣r❡❡ ♦❢ ❢r❡❡❞♦♠✱ y ✳ ❚❤❡ ✈❡❝t♦r ❛♥❛❧②s✐s r❡♠❛✐♥s t❤❡ s❛♠❡ ✇✐t❤ G =< 0, 0, −gm > ❛♥❞ X =< x, y, z >✳ ❚❤❡ s♣r❡❛❞s❤❡❡t ♦♥❧② r❡q✉✐r❡s ❛❞❞✐♥❣ ❝♦❧✉♠♥s ❢♦r y ✳ ❚❤❡ ✈❛❧✉❡s ♦❢ z ✐s ❢♦✉♥❞ ❢r♦♠✿

x 2 + y 2 + z 2 = L2 . ❇❡❧♦✇ ✇❡ ♣❧♦t t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ < x, y > ✇✐t❤ ❛ ♥♦♥✲③❡r♦ ❤♦r✐③♦♥t❛❧ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②✿

✺✳✺✳

✸✷✸

❆ ♣❡♥❞✉❧✉♠

❲❡ ♦❜s❡r✈❡ t❤❛t t❤❡ ♣❡♥❞✉❧✉♠ s✇✐♥❣s ❛s ❜❡❢♦r❡ ❜✉t t❤❡ ♣❧❛♥❡ ♦❢ t❤❡ s✇✐♥❣s ✐s ❝♦♥t✐♥✉♦✉s❧② r♦t❛t✐♥❣✳ ❲❡ ♥♦✇ ❣♦ ❜❛❝❦ t♦ t❤❡ 2✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ❛♥❞ ❛❞❞r❡ss ✐t ❛♥❛❧②t✐❝❛❧❧②✳ ❲❡ ❛♣♣❧② ◆❡✇t♦♥✬s ❧❛✇ t♦ t❤❡ t❛♥❣❡♥t✐❛❧ ❛①✐s ♦♥❧②✿ F = −mg sin θ = ma =⇒ a = −g sin θ .

❍♦✇ ✐s t❤✐s ❧✐♥❡❛r ❛❝❝❡❧❡r❛t✐♦♥ a ❛❧♦♥❣ t❤❡ t❛♥❣❡♥t r❡❧❛t❡❞ t♦ t❤❡ ❝❤❛♥❣❡ ✐♥ ❛♥❣❧❡ θ❄ ▲❡t s ❜❡ t❤❡ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡t❡r✳ ❙✐♥❝❡ t❤❡ ❝✉r✈❡ ✐s ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s L✱ ✇❡ ❤❛✈❡✿ s = Lθ .

◆♦✇✱ t❤❡ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡t❡r ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✱ ✐✳❡✳✱ s = s(t)✳ ❲❡ ❞✐✛❡r❡♥t✐❛t❡ t✇✐❝❡ ✇✐t❤ r❡s♣❡❝t t♦ t✿ s = Lθ =⇒ v = s′ = Lθ′ =⇒ a = x′′ = Lθ′′ .

❚❤❡r❡❢♦r❡✱ ♦r

Lθ′′ = −g sin θ , θ′′ +

❚❤✐s ✐s ❛ ♥♦♥✲❧✐♥❡❛r ❖❉❊ ✇✐t❤ r❡s♣❡❝t t♦ θ✳

g sin θ = 0. L

❚♦ ❝♦♥✜r♠ ♦✉r s✐♠✉❧❛t✐♦♥s✱ ✇❡ ♣❧♦t ❛ ❢❡✇ s♦❧✉t✐♦♥s ✉s✐♥❣ ❊✉❧❡r✬s ♠❡t❤♦❞ ❥✉st ❛s ✐♥ t❤❡ ❧❛st ❝❤❛♣t❡r✿

❍❡r❡ θ ✐s ♣❧♦tt❡❞ ❛❣❛✐♥st α = θ′ ❀ ✐✳❡✳✱ ✇❡ ❛r❡ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠✿ (

θ′ = α, g α′ = − sin θ. L

❚❤❡ ❊✉❧❡r✬s s♦❧✉t✐♦♥s ❛r❡♥✬t ♣❡r✐♦❞✐❝ ❜✉t r❛t❤❡r ❛♣♣❡❛r t♦ ❜❡ s♣✐r❛❧s✦ ▲❡t✬s ❧✐♥❡❛r✐③❡ ✦ ❚❤❡r❡ ✐s ♦♥❧② ♦♥❡ ❡q✉✐❧✐❜r✐✉♠ θ = α = 0✳ ◆♦✇✱  d sin θ (0) = 1 . dθ

✺✳✻✳

P❧❛♥❡t❛r② ♠♦t✐♦♥

✸✷✹

❚❤❡r❡❢♦r❡✱ t❤❡ ❧✐♥❡❛r✐③❡❞ s②st❡♠s ✐s✿

(

θ′ = α, g α′ = − θ. L

❲❡ ❝❛♥ ❧♦♦❦ ❛t t❤❡ ❡✐❣❡♥✈❛❧✉❡s✿

A=

"

0 1 g − 0 L

❆❝❝♦r❞✐♥❣ t♦ t❤❡

#

=⇒ χA (λ) = det(A − λI) = det

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r ❙②st❡♠s ■■

"

−λ 1 g − −λ L

#

g = λ + = 0 =⇒ λ1,2 = ± L 2

✐♥ ❈❤❛♣t❡r ✷✱ t❤❡ s②st❡♠ ❤❛s ❛

❝❡♥t❡r

❛t

0✦

r

g i. L

❖❢ ❝♦✉rs❡✱

✇❡ ❝❛♥ ❥✉st s♦❧✈❡ t❤❡ ❧✐♥❡❛r✐③❡❞ s②st❡♠✿

g θ = α = − θ =⇒ θ = θ0 cos L ′′



r

g t L



,

✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✿

θ(0) = θ0

❛♥❞

θ′ (0) = 0 .

❙♦✱ ♦✉r ❝♦♥❝❧✉s✐♦♥ ✐s t❤❛t ✇❤❡♥ s♠❛❧❧ t❤❡ s✇✐♥❣s ❛r❡ ❝❧♦s❡ t♦ ♣❡r✐♦❞✐❝ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❡r✐♦❞✿

2π ❚❤❡ ♣❡r✐♦❞ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❛♠♣❧✐t✉❞❡

r

g . L

θ0 ✦

✺✳✻✳ P❧❛♥❡t❛r② ♠♦t✐♦♥

❆ ❢❛♠✐❧✐❛r ♣r♦❜❧❡♠ ❛❜♦✉t ❛ ❜❛❧❧ t❤r♦✇♥ ✐♥ t❤❡ ❛✐r ❤❛s ❛ s♦❧✉t✐♦♥✿ ✐ts tr❛ ❥❡❝t♦r② ✐s ❛

♣❛r❛❜♦❧❛✳

❍♦✇❡✈❡r✱ ✇❡

❛❧s♦ ❦♥♦✇ t❤❛t ✐❢ ✇❡ t❤r♦✇ r❡❛❧❧②✲r❡❛❧❧② ❤❛r❞ ✭❧✐❦❡ ❛ r♦❝❦❡t✮ t❤❡ ❜❛❧❧ ✇✐❧❧ st❛rt t♦ ♦r❜✐t t❤❡ ❊❛rt❤ ❢♦❧❧♦✇✐♥❣ ❛♥

❡❧❧✐♣s❡✳

❚❤❡ ♠♦t✐♦♥ ♦❢ t✇♦ ♣❧❛♥❡ts ✭♦r ❛ st❛r ❛♥❞ ❛ ♣❧❛♥❡t✱ ♦r ❛ ♣❧❛♥❡t ❛♥❞ ❛ s❛t❡❧❧✐t❡✱ ❡t❝✳✮ ✐s ❣♦✈❡r♥❡❞ ❜② ❛ s✐♥❣❧❡ ❢♦r❝❡✿ t❤❡

❣r❛✈✐t②✳

❘❡❝❛❧❧ ❤♦✇ t❤✐s ❢♦r❝❡ ♦♣❡r❛t❡s✳

◆❡✇t♦♥✬s ▲❛✇ ♦❢ ●r❛✈✐t②✿ ❚❤❡ ❢♦r❝❡ ♦❢ ❣r❛✈✐t② ❜❡t✇❡❡♥ t✇♦ ♦❜❥❡❝ts ✐s ✶✳ ♣r♦♣♦rt✐♦♥❛❧ t♦ ❡✐t❤❡r ♦❢ t❤❡✐r ♠❛ss❡s✱ ✷✳ ✐♥✈❡rs❡❧② ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ sq✉❛r❡ ♦❢ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡✐r ❝❡♥t❡rs✱ ❛♥❞ ✸✳ ❞✐r❡❝t❡❞ ❛❧♦♥❣ t❤❡ s❡❣♠❡♥t ❜❡t✇❡❡♥ t❤❡♠✳

✺✳✻✳

✸✷✺

P❧❛♥❡t❛r② ♠♦t✐♦♥

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❢♦r❝❡ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦r♠✉❧❛✿ F =G

mM , r2

✇❤❡r❡✿ • F ✐s t❤❡ ❢♦r❝❡ ❜❡t✇❡❡♥ t❤❡ ♦❜❥❡❝ts✳ • G ✐s t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❝♦♥st❛♥t✳

• m ✐s t❤❡ ♠❛ss ♦❢ t❤❡ ✜rst ♦❜❥❡❝t✳

• M ✐s t❤❡ ♠❛ss ♦❢ t❤❡ s❡❝♦♥❞ ♦❜❥❡❝t✳

• r ✐s t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❝❡♥t❡rs ♦❢ t❤❡ ♠❛ss❡s✳

▲❡t✬s ❝♦♥♥❡❝t t❤✐s ❢❛♠✐❧✐❛r ♣❤②s✐❝s ❧❛✇ t♦ ❛♥♦t❤❡r✳

❑❡♣❧❡r✬s ▲❛✇s ♦❢ P❧❛♥❡t❛r② ▼♦t✐♦♥✿ ❚❤❡ ♠♦t✐♦♥ ♦❢ ♣❧❛♥❡ts ❛r♦✉♥❞ t❤❡ ❙✉♥ ❢♦❧❧♦✇s t❤❡s❡ ❧❛✇s✿ ✶✳ ❚❤❡ ♦r❜✐t ♦❢ ❛ ♣❧❛♥❡t ✐s ❛♥ ❡❧❧✐♣s❡ ✇✐t❤ t❤❡ ❙✉♥ ❛t ♦♥❡ ♦❢ t❤❡ t✇♦ ❢♦❝✐✳ ✷✳ ❆ ❧✐♥❡ s❡❣♠❡♥t ❥♦✐♥✐♥❣ ❛ ♣❧❛♥❡t ❛♥❞ t❤❡ ❙✉♥ s✇❡❡♣s ♦✉t ❡q✉❛❧ ❛r❡❛s ❞✉r✐♥❣ ❡q✉❛❧ ✐♥t❡r✈❛❧s ♦❢ t✐♠❡✳ ✸✳ ❚❤❡ sq✉❛r❡ ♦❢ t❤❡ ♦r❜✐t❛❧ ♣❡r✐♦❞ ♦❢ ❛ ♣❧❛♥❡t ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❝✉❜❡ ♦❢ t❤❡ s❡♠✐✲♠❛❥♦r ❛①✐s ♦❢ ✐ts ♦r❜✐t✳ ❲❡ ❦♥♦✇ t❤❡ ✈❡❝t♦r ❢♦r♠ ♦❢ ◆❡✇t♦♥✬s ❧❛✇ ✭✇✐t❤ t❤❡ ✜rst ♦❜❥❡❝t ❧♦❝❛t❡❞ ❛t t❤❡ ♦r✐❣✐♥✮✿ X . ||X||3

F = −GmM

❚❤❡♥✱ ✇❡ ❤❛✈❡ ❛ ✈❡❝t♦r ❖❉❊s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❢♦r t❤❡ ❧♦❝❛t✐♦♥ X ♦❢ t❤❡ s❡❝♦♥❞ ♦❜❥❡❝t✿ X ′′ = −GM

❲❡ ✇✐❧❧ ❝❛❧❧ ✐t

X . ||X||3

✳ ■t ✐s ♥♦♥✲❧✐♥❡❛r✳

◆❡✇t♦♥✬s ❖❉❊ ♦❢ ♣❧❛♥❡t❛r② ♠♦t✐♦♥

❆s ❜❡❢♦r❡✱ ❢r♦♠ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ Xn′′ t❤❡ ✈❡❧♦❝✐t② Xn′ ❛♥❞ t❤❡♥ ❢r♦♠ t❤❡ ✈❡❧♦❝✐t② t❤❡ ♣♦s✐t✐♦♥ Xn ❛r❡ ❝♦♠♣✉t❡❞ ❜② ✇❤❛t ❛♠♦✉♥ts t♦ ❊✉❧❡r✬s ♠❡t❤♦❞✳ ❚❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧ ♦❢ ♣❧❛♥❡t❛r② ♠♦t✐♦♥ ✐s ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧ ♦❢ s❡❝♦♥❞ ♦r❞❡r ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s✿ Xn ||Xn ||3 ′ ′′ = Xn + Xn+1 ∆t ′ = Xn + Xn+1 ∆t

′′ Xn+1 = −GM ′ Xn+1 Xn+1

❚❤❡ r❡s✉❧t ✐s ❛s ♣r❡❞✐❝t❡❞✿ t❤❡ ♣♦✐♥ts ❢♦r♠ ✇❤❛t ❧♦♦❦s ❧✐❦❡ ❛♥ ❡❧❧✐♣s❡ ✇✐t❤ ♦♥❡ ♦❢ t❤❡ ❢♦❝✐ ❛t t❤❡ ♦r✐❣✐♥ ✭❋✐rst ❑❡♣❧❡r✬s ▲❛✇✮✳

✺✳✻✳ P❧❛♥❡t❛r② ♠♦t✐♦♥

✸✷✻

■♥ ♦r❞❡r t♦ ❝♦♥✜r♠ t❤✐s ✐❞❡❛✱ ✇❡ ✜rst ❞r❛✇ ❛ ❧✐♥❡ ❢r♦♠ t❤❡ ♦r✐❣✐♥ ✭t❤❡ ✜rst ❢♦❝✉s✮ t♦ t❤❡ ♣♦✐♥t ♦♥ t❤❡ ❝✉r✈❡ ✇❤❡r❡ ✐t ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ t❛♥❣❡♥t✳ ❲❡ t❤❡♥ ❝♦♥t✐♥✉❡ t❤✐s ❧✐♥❡ t♦ t❤❡ ♦t❤❡r ❡♥❞ ♦❢ t❤❡ ❝✉r✈❡ ❛♥❞ ❞✐s❝♦✈❡r t❤❛t ✐♥❞❡❡❞ t❤❡ t❛♥❣❡♥t ✐s ❛❣❛✐♥ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤✐s ❝❤♦r❞✳ ❲❡ t❤❡♥ ♠❡❛s✉r❡ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❢♦❝✉s t♦ t❤❡ ✜rst ♣♦✐♥t ❛♥❞ ♣❧♦t t❤❡ s❛♠❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ s❡❝♦♥❞ ♣♦✐♥t ❛❧♦♥❣ t❤❡ ❝❤♦r❞✳ ❚❤✐s ❣✐✈❡s ✉s t❤❡ s❡❝♦♥❞ ❢♦❝✉s ✭❧❡❢t✮✳

❲❡ ❢✉rt❤❡r t❡st ♦✉r ❝♦♥❥❡❝t✉r❡ ❜② ♣❧♦tt✐♥❣ t❤❡ ❧✐♥❡s r❡♣r❡s❡♥t✐♥❣ r❛②s ♦❢ ❧✐❣❤t t❤❛t st❛rt ❛t ♦♥❡ ❢♦❝✉s ❛♥❞ t❤❡♥ ❜♦✉♥❝❡ ♦✛ t❤❡ ❝✉r✈❡ t♦ t❤❡ ♦t❤❡r ❢♦❝✉s ✭r✐❣❤t✮✳ ❊①❡r❝✐s❡ ✺✳✻✳✶

■s t❤✐s r❡❛❧❧② ❛♥ ❡❧❧✐♣s❡❄ ❇② ❝❤❛♥❣✐♥❣ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✇❡ ❝❛♥ ♣r♦❞✉❝❡ ♦t❤❡r ♣❛tt❡r♥s ♦❢ ❜❡❤❛✈✐♦r s✉❝❤ ❛s ✇❤❛t ❧♦♦❦s ❧✐❦❡ ❛ ❤②♣❡r❜♦❧❛✿

◆♦✇ ✇❤❛t ❛❜♦✉t t❤❡ ❙❡❝♦♥❞ ❑❡♣❧❡r✬s ▲❛✇❄

✺✳✻✳ P❧❛♥❡t❛r② ♠♦t✐♦♥

✸✷✼

❲❡ ❝❛♥ s❡❡ ✐♥ ♦✉r s✐♠✉❧❛t✐♦♥ t❤❛t t❤❡ ♣♦✐♥ts ❛✇❛② ❢r♦♠ t❤❡ ♦r✐❣✐♥ ✭t❤❡ ✜rst ♦❜❥❡❝t✮ ❛r❡ ❝❧♦s❡r s♣❛❝❡❞ t❤❛♥ t❤❡ ♦♥❡s ❝❧♦s❡r t♦ t❤❡ ♦r✐❣✐♥✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ♠♦t✐♦♥ ✐s ❢❛st❡r ✇❤❡♥ t❤❡② ❛r❡ ❝❧♦s❡r✳ ❚❤❛t ✐s ✇❤② t❤❡ ❧♦♥❣❡r tr✐❛♥❣❧❡s ❛r❡ t❤✐♥♥❡r✳

❲❡ ❝♦♥✜r♠ t❤❛t t❤❡ ❛r❡❛s ♦❢ t❤❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ s❛♠❡ ❜② ❝♦♠♣✉t✐♥❣ t❤❡♠ ✐♥ t❤❡ ❧❛st ❝♦❧✉♠♥ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t ✈✐❛ t❤❡ P❛r❛❧❧❡❧♦❣r❛♠ ❋♦r♠✉❧❛ ✿ An+1

1 = det [Xn+1 Xn ] , 2

✇❤❡r❡ Xn+1 ❛♥❞ Xn ❛r❡ ❣✐✈❡♥ ❜② t❤❡✐r ❝♦❧✉♠♥ ✈❡❝t♦rs✳

❲❡ ✇✐❧❧ ❞❡r✐✈❡ t❤❡ ❧❛✇ ✐♥ ❛ ♠♦r❡ ❣❡♥❡r❛❧ s❡tt✐♥❣ t❤❛♥ t❤❡ ♦r✐❣✐♥❛❧✳ ❋✐rst ✇❡ ♥♦t❡ t❤❛t ✐t ♠✐❣❤t ❛♣♣❧② t♦ ❛❧❧ ❞✐s❝r❡t❡ tr❛❥❡❝t♦r✐❡s ♥♦t ❥✉st ❡❧❧✐♣s❡s✳ ❊①❡r❝✐s❡ ✺✳✻✳✷

Pr♦✈❡ t❤❛t ❛♥ ♦❜❥❡❝t ♠♦✈✐♥❣ ❛❧♦♥❣ ❛ str❛✐❣❤t ❧✐♥❡ ❛t ❛ ❝♦♥st❛♥t s♣❡❡❞ s❛t✐s✜❡s t❤❡ ❛r❡❛ ♣r♦♣❡rt② ✇✐t❤ r❡s♣❡❝t t♦ ❛♥② ✜①❡❞ ♣♦✐♥t✳ ❍✐♥t✿ ✉s❡ ❡❧❡♠❡♥t❛r② ❣❡♦♠❡tr②✳

✺✳✻✳ P❧❛♥❡t❛r② ♠♦t✐♦♥

✸✷✽

❙❡❝♦♥❞✱ ✇❡ ✇✐❧❧ s❡❡ t❤❛t t❤❡ ❧❛✇ ❛♣♣❧✐❡s t♦ ♠♦r❡ ❣❡♥❡r❛❧ ❢♦r❝❡s t❤❛♥ ❥✉st t❤❡ ❣r❛✈✐t②✳ ❚❤❡ ❣r❛✈✐t② ✐s ❛ ❝❡♥tr❛❧ ❢♦r❝❡✱ ✐✳❡✳✱ ♦♥❡ t❤❛t ✐s ❞✐r❡❝t❡❞ ❛❧♦♥❣ t❤❡ ❧✐♥❡ ❜❡t✇❡❡♥ t❤❡ ❧♦❝❛t✐♦♥ ❛♥❞ t❤❡ ♦r✐❣✐♥✳ ❚❤❡♥ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❢♦r s✉❝❤ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧s ✐s✿ ′′ Xn+1 = L(||Xn ||)Xn ,

❢♦r s♦♠❡ ❢✉♥❝t✐♦♥ L✳

❚❤❡♦r❡♠ ✺✳✻✳✸✿ ❉✐s❝r❡t❡ ❙❡❝♦♥❞ ❑❡♣❧❡r✬s ▲❛✇

■❢ X ✐s ❛ tr❛❥❡❝t♦r② ♦❢ t❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧ ✇✐t❤ ❝❡♥tr❛❧ ❢♦r❝❡✱ t❤❡ s❡❣♠❡♥t ❢r♦♠ t❤❡ ❢♦❝✉s t♦ t❤❡ ❧♦❝❛t✐♦♥ s✇❡❡♣s ❛♥ ❡q✉❛❧ ❛r❡❛ ✐♥ ❛♥ ❡q✉❛❧ t✐♠❡✳ ❈♦♥✈❡rs❡❧②✱ ✐❢ t❤✐s ❛r❡❛ ♣r♦♣❡rt② ✐s s❛t✐s✜❡❞ ❜② ❡❛❝❤ tr❛❥❡❝t♦r② ♦❢ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧ ♦❢ ♦r❞❡r t✇♦✱ t❤❡ ♠♦❞❡❧ ✐s ♣r♦❞✉❝❡❞ ❜② ❛ ❝❡♥tr❛❧ ❢♦r❝❡✳ Pr♦♦❢✳

❲❡ ❝♦♥s✐❞❡r t❤❡ ♠♦❞❡❧✿

′′ Xn+1 = f (Xn ) ,

✇❤❡r❡ f ✐s s♦♠❡ ❢✉♥❝t✐♦♥✳ ❚❤❡♥✱ ′ Xn+1 = Xn′ + Xn′′ ∆t = Xn′ + f (Xn )∆t .

❲❡ ✇✐❧❧ ✉s❡ t✇✐❝❡ t❤❡ P❛r❛❧❧❡❧♦❣r❛♠ ❋♦r♠✉❧❛ ❢♦r t❤❡ ♦r✐❡♥t❡❞ ❛r❡❛ A ♦❢ t❤❡ tr✐❛♥❣❧❡ s♣❛♥♥❡❞ ❜② ✈❡❝t♦rs M ❛♥❞ N ✿ A=

1 det [M N ] . 2

❍♦✇❡✈❡r✱ t❤❡ ✈❡❝t♦rs ❝❤♦s❡♥ ✇✐❧❧ ❜❡ ❞✐✛❡r❡♥t ❢r♦♠ t❤♦s❡ ✉s❡❞ ✐♥ t❤❡ s♣r❡❛❞s❤❡❡t✳ ❋✐rst✱ s✉♣♣♦s❡ t❤❡ tr✐❛♥❣❧❡ ✇✐t❤ ❛r❡❛ An ✐s ❢♦r♠❡❞ ❜② Xn ❛♥❞ Xn′ ∆t❀ t❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ❜② t❤❡ ❧✐♥❡❛r✐t② ♦❢ t❤❡ ❞❡t❡r♠✐♥❛♥t✿ ∆t det [Xn Xn′ ] . 2 ′ ❙❡❝♦♥❞✱ s✉♣♣♦s❡ t❤❡ tr✐❛♥❣❧❡ ✇✐t❤ ❛r❡❛ An+1 ✐s ❢♦r♠❡❞ ❜② Xn ❛♥❞ Xn+1 ∆t❀ t❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ t❤❡ An =

❢♦❧❧♦✇✐♥❣ ❜② t❤❡ ❧✐♥❡❛r✐t② ❛♥❞ t❤❡ ❛❞❞✐t✐✈✐t② ♦❢ t❤❡ ❞❡t❡r♠✐♥❛♥t✿

  ∆t ′ det Xn Xn+1 2 ∆t det [Xn (Xn′ + f (Xn )∆t)] = 2  ∆t = det [Xn Xn′ ] + ∆t det [Xn f (Xn )] . 2

An+1 =

❚❤❡r❡❢♦r❡✱ ❋✐♥❛❧❧②✱ An+1

∆t2 An+1 − An = det [Xn f (Xn )] . 2 = An ✐❢ ❛♥❞ ♦♥❧② ✐❢ Xn ❛♥❞ f (Xn ) ❛r❡ ♣❛r❛❧❧❡❧✳

❊①❡r❝✐s❡ ✺✳✻✳✹

❉❡r✐✈❡ ❑❡♣❧❡r✬s ❙❡❝♦♥❞ ▲❛✇ ❢r♦♠ t❤✐s t❤❡♦r❡♠✳ ❍✐♥t✿ ✉s❡ ✇❤❛t ②♦✉ ❦♥♦✇ ❛❜♦✉t t❤❡ ❛❝❝✉r❛❝② ♦❢ ❊✉❧❡r✬s ♠❡t❤♦❞✳ ❲❡ t❤✉s ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ♠♦t✐♦♥ ❛❧♦♥❣ t❤❡ ❡❧❧✐♣s❡ ✭♦r t❤❡ ♣❛r❛❜♦❧❛✱ ♦r t❤❡ ❤②♣❡r❜♦❧❛ s❡❡♥ ❜❡❧♦✇✮ ❞♦❡s ♥♦t ♠❛t❝❤ ✐ts st❛♥❞❛r❞ ♣❛r❛♠❡tr✐③❛t✐♦♥✱ x = a sin ωt, y = b sin ωt✳ ❚❤❡ ❢♦r♠✉❧❛s ❢✉❧❧② ❛♣♣❧② t♦ t❤❡ 3✲❞✐♠❡♥s✐♦♥❛❧ s✐t✉❛t✐♦♥✳ P❧♦tt✐♥❣ t❤❡ ♦r❜✐t ❛❧♦♥❣ t❤❡ t❤r❡❡ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡s ♣r♦❞✉❝❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿

✺✳✻✳

P❧❛♥❡t❛r② ♠♦t✐♦♥

✸✷✾

❊①❡r❝✐s❡ ✺✳✻✳✺

■♠♣❧❡♠❡♥t ❛ s✐♠✉❧❛t✐♦♥ ♦❢ ♣❧❛♥❡t❛r② ♠♦t✐♦♥ ✐♥ t❤❡

3✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✳

❉❡♠♦♥str❛t❡ t❤❛t t❤❡ ♠♦t✐♦♥

✐s ♣❧❛♥❛r✳

❲❡ st❛t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐t❤♦✉t ♣r♦♦❢✿ ❚❤❡♦r❡♠ ✺✳✻✳✻✿ P❧❛♥❡t❛r② ▼♦t✐♦♥ ✐♥ P♦❧❛r ❈♦♦r❞✐♥❛t❡s ❆♥② tr❛ ❥❡❝t♦r② ♦❢ t❤❡ ◆❡✇t♦♥✬s ❖❉❊ ♦❢ ♣❧❛♥❡t❛r② ♠♦t✐♦♥ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✲ ✐♥❣ ✐♥ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s✿

r= ✇❤✐❝❤ ✐s

• • • •

e = 1✱ 0 < e < 1✱ ♣❛r❛❜♦❧❛ ❢♦r e = 1✱ ❤②♣❡r❜♦❧❛ ❢♦r e > 1✱

❛ ❝✐r❝❧❡ ❢♦r

❛♥ ❡❧❧✐♣s❡ ❢♦r ❛ ❛

✇✐t❤ t❤❡ ❢♦❝✉s ❛t t❤❡ ♦r✐❣✐♥✳

❚❤❡ ❣r❛♣❤s ❛r❡ ♣❧♦tt❡❞ ❢♦r

e = 0, .5, 1, 2✿

p , 1 + e · cos θ

✺✳✼✳ ❚❤❡ t✇♦✲ ❛♥❞ t❤r❡❡✲❜♦❞② ♣r♦❜❧❡♠s

✸✸✵

❚❤❡ ♣r♦♦❢ ♦❢ t❤✐s t❤❡♦r❡♠ ❧✐❡s ❜❡②♦♥❞ t❤❡ s❝♦♣❡ ♦❢ t❤✐s ❜♦♦❦✳ ❋✐♥❛❧❧②✱ ❧❡t✬s ❣♦ ❜❛❝❦ t♦ t❤❡ q✉❡st✐♦♥ ♣♦s❡❞ ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ s❡❝t✐♦♥✿ ✇❤❛t ✐s t❤❡ ❝♦rr❡❝t tr❛❥❡❝t♦r②❄

▲❡t✬s r❡✈✐❡✇ ✇❤❛t ✇❡ ❝♦♥❝❧✉❞❡❞ ❛❜♦✉t t❤❡s❡ t✇♦ s❡tt✐♥❣s ✐♥ ❈❤❛♣t❡r ✹❍❉✲✷✳



❲❤❡♥ t❤❡ ❊❛rt❤ ✐s s❡❡♥ ❛s ✏❧❛r❣❡✑ ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ s✐③❡ ♦❢ t❤❡ tr❛❥❡❝t♦r②✱ t❤❡ ❣r❛✈✐t② ❢♦r❝❡s ❛r❡ ❛ss✉♠❡❞ t♦ ❜❡ ♣❛r❛❧❧❡❧ ✐♥ ❛❧❧ ❧♦❝❛t✐♦♥s✳ ❚❤❡♥ t❤❡ tr❛❥❡❝t♦r② ✐s ❛ ♣❛r❛❜♦❧❛ ❛s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ t❤❡



x✲❛①✐s

❛❧✐❣♥❡❞ ✇✐t❤ t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ ❊❛rt❤ ❛♥❞

y ✲❛①✐s

✇✐t❤ t❤❡ ❢♦r❝❡✳

❲❤❡♥ t❤❡ ❊❛rt❤ ✐s s❡❡♥ ❛s ❡✐t❤❡r ✏s♠❛❧❧✑ ♦r ❛t ❧❡❛st ♣❡r❢❡❝t❧② s♣❤❡r✐❝❛❧✱ t❤❡ ❣r❛✈✐t② ❢♦r❝❡s ❛r❡ ❛ss✉♠❡❞ t♦ ❣♦ r❛❞✐❛❧❧② t♦✇❛r❞ ✐ts ❝❡♥t❡r✳ ❚❤❡♥ t❤❡ tr❛❥❡❝t♦r② ✐s ❛♥ ❡❧❧✐♣s❡ ✭♦r ❛ ❤②♣❡r❜♦❧❛ ♦r ❛ ♣❛r❛❜♦❧❛✮ ✇✐t❤ ✐ts ❢♦❝✉s ❧♦❝❛t❡❞ ❛t t❤❡ ❝❡♥t❡r✳

❲❤❡♥ t❤❡ s✐③❡ ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ s❤❛♣❡ ♦❢ t❤❡ ❊❛rt❤ ♠❛tt❡r✱ t❤✐♥❣s ❣❡t ❝♦♠♣❧✐❝❛t❡❞✳✳✳

✺✳✼✳ ❚❤❡ t✇♦✲ ❛♥❞ t❤r❡❡✲❜♦❞② ♣r♦❜❧❡♠s

❲❡ ❤❛✈❡ ✐❣♥♦r❡❞ t❤❡ ❡✛❡❝t ♦❢ t❤❡ ❊❛rt❤✬s ❣r❛✈✐t② ♦♥ t❤❡ ❙✉♥✳ ❚❤❡ r❡❛s♦♥ ✐s t❤❛t t❤❡ ♠❛ss s✐❣♥✐✜❝❛♥t❧② ❧❛r❣❡r t❤❛♥ t❤❡ ♠❛ss

q

p

♦❢ t❤❡ ❙✉♥ ✐s

♦❢ t❤❡ ❊❛rt❤✿

p >> q . ❚❤✐s ✐s t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t ❛❧❧♦✇s ✉s t♦ ♣❧❛❝❡ t❤❡ ❙✉♥ ❛t t❤❡ ♦r✐❣✐♥ ❛s ❛ st❛t✐♦♥❛r② ♦❜❥❡❝t✳ ❲❤❛t ✐❢ ✇❡ ❤❛✈❡ t❤❡ t✇♦ ♣❧❛♥❡ts ♦❢ ❝♦♠♣❛r❛❜❧❡ s✐③❡s ❄

✺✳✼✳

✸✸✶

❚❤❡ t✇♦✲ ❛♥❞ t❤r❡❡✲❜♦❞② ♣r♦❜❧❡♠s

❙✉♣♣♦s❡ ♥♦✇ ✇❡ ❤❛✈❡ t✇♦ ♣❧❛♥❡ts ✇✐t❤ ♠❛ss❡s p, q ✇✐t❤ ❧♦❝❛t❡❞ ❛t U, V r❡s♣❡❝t✐✈❡❧②✳ ❊✐t❤❡r ♦❢ t❤❡ t✇♦ ♦❜❥❡❝ts ✐s ❛✛❡❝t❡❞ ❜② t❤❡ ❣r❛✈✐t② ♦❢ t❤❡ ♦t❤❡r✳ ❚❤❡♥ t❤❡ t✇♦ ✐♥t❡r❛❝t✐♦♥s ❛♣♣❡❛r ✐♥ t❤❡ t✇♦ ❞❡♣❡♥❞❡♥t ❖❉❊s✿ U −V ||U − V ||3 V −U V ′′ = −Gp ||V − U ||3 U ′′ = −Gq

❚❤✐s ✐s ❝❛❧❧❡❞ t❤❡ t✇♦✲❜♦❞②



♣r♦❜❧❡♠

▲❡t✬s ❛♣♣❧② ❊✉❧❡r✬s ♠❡t❤♦❞ t♦ s❡❡ ✇❤❛t ❝❛♥ ❤❛♣♣❡♥✳ ❲❡ st❛rt ✇✐t❤ t✇♦ ✐❞❡♥t✐❝❛❧ ♣❧❛♥❡ts✳ ❚❤❡② s❡❡♠ t♦ ❝✐r❝❧❡ ❡❛❝❤ ♦t❤❡r✿

✳✳✳✉♥t✐❧ ✇❡ r❡❛❧✐③❡ t❤❛t ❡✐t❤❡r ♦♥❡ ❝✐r❝❧❡s ❛ ❝❡rt❛✐♥ ♣♦✐♥t✱ t❤❡ s❛♠❡ ♣♦✐♥t ❢♦r ❜♦t❤✳ ❚❤✐s ♣♦✐♥t ❧✐❡s ❤❛❧❢✲✇❛② ❜❡t✇❡❡♥ t❤❡♥✱ 1 C = (U + V ) . 2

❚❤✐s ❢❛❝t ✐s ❝♦♥✜r♠❡❞ ✐♥ t❤❡ ❧❛st ❝♦❧✉♠♥ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t✳

▲❡t✬s ❞♦✉❜❧❡ t❤❡ ♠❛ss ♦❢ t❤❡ ✜rst ♣❧❛♥❡t✳ ❲✐t❤ ❡✈❡r②t❤✐♥❣ ❡❧s❡ r❡♠❛✐♥✐♥❣ t❤❡ s❛♠❡✱ t❤❡ t✇♦ ♣❧❛♥❡ts s❡❡♠ t♦ ❞❛♥❝❡ ❛✇❛② ✇❤✐❧❡ st✐❧❧ ❝✐r❝❧✐♥❣ ❡❛❝❤ ♦t❤❡r✳ ❚❤✐s t✐♠❡✱ t❤❡ ♠✐❞✲♣♦✐♥t ✐s ❛❧s♦ ❝✐r❝❧✐♥❣✳

■♥ ❢❛❝t✱ ✐t ❞♦❡s♥✬t s❡❡♠ t♦ r❡✈❡❛❧ ❛♥②t❤✐♥❣ ❛❜♦✉t ✇❤❛t ✐s ❣♦✐♥❣ ♦♥✳ ❲❤❛t ✐❢ ✇❡ ❧♦♦❦ ❛t t❤❡ ❝❡♥t❡r ♦❢ t❤❡ t✇♦✱ M=

p q U+ V, p+q p+q

✐♥st❡❛❞❄ ■t s❡❡♠s t♦ ❜❡ ♠♦✈✐♥❣ ❛❧♦♥❣ ❛ str❛✐❣❤t ❧✐♥❡✦

♦❢ ♠❛ss

✺✳✼✳

❚❤❡ t✇♦✲ ❛♥❞ t❤r❡❡✲❜♦❞② ♣r♦❜❧❡♠s

✸✸✷

❋✉rt❤❡r♠♦r❡✱ ❧❡t✬s ♣❧♦t t❤❡ t✇♦ tr❛❥❡❝t♦r✐❡s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss✱ ✐✳❡✳✱

P =U −M

❛♥❞

Q=V −M.

❚❤❡② s❡❡♠ t♦ tr❛❝❡ ❡❧❧✐♣s❡s✳ ▲❡t✬s st❛t❡ ❛♥❞ ♣r♦✈❡ t❤❡s❡ ❝♦♥❥❡❝t✉r❡s✳

❚❤❡♦r❡♠ ✺✳✼✳✶✿ ❈❡♥t❡r ♦❢ ▼❛ss ❚✇♦✲❜♦❞② ❙②st❡♠

❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss M ♦❢ t❤❡ t✇♦✲❜♦❞② s②st❡♠ s❛t✐s✜❡s t❤❡ ✈❡❝t♦r ❖❉❊✿ M ′′ = 0 ,

❛♥❞✱ t❤❡r❡❢♦r❡✱ ♠♦✈❡s ❛❧♦♥❣ ❛ str❛✐❣❤t ❧✐♥❡ ❛t ❛ ❝♦♥st❛♥t s♣❡❡❞✳ Pr♦♦❢✳ ❲❡ s✐♠♣❧② s✉❜st✐t✉t❡✿

M

′′

′′ q p U+ V = p+q p+q p q = U ′′ + V ′′ p+q  p+q    q U −V V −U p + −Gq −Gp = p+q ||U − V ||3 p+q ||V − U ||3   V −U U −V pq + = −G 3 p + q ||U − V || ||U − V ||3 

= 0. ❚❤❡♦r❡♠ ✺✳✼✳✷✿ ▲♦❝❛t✐♦♥s ✐♥ ❚✇♦✲❜♦❞② ❙②st❡♠

■❢ U ❛♥❞ V ❛r❡ t❤❡ ❧♦❝❛t✐♦♥s ♦❢ t❤❡ t✇♦ ♣❧❛♥❡ts ✐♥ t❤❡ t✇♦✲❜♦❞② s②st❡♠ ❛♥❞ M ✐s ✐ts ❝❡♥t❡r ♦❢ ♠❛ss M ✱ t❤❡♥ P = U − M ❛♥❞ Q = V − M s❛t✐s❢② t❤❡ ◆❡✇t♦♥✬s ❖❉❊ ♦❢ ♣❧❛♥❡t❛r② ♠♦t✐♦♥ ❛♥❞✱ t❤❡r❡❢♦r❡✱ tr❛❝❡ ❡❧❧✐♣s❡s✱ ♣❛r❛❜♦❧❛s✱ ♦r ❤②♣❡r❜♦❧❛s✳

✺✳✼✳

✸✸✸

❚❤❡ t✇♦✲ ❛♥❞ t❤r❡❡✲❜♦❞② ♣r♦❜❧❡♠s

Pr♦♦❢✳

❲❡ ✜rst ♦❜s❡r✈❡ t❤❛t M ❧♦❝❛t❡❞ ♦♥ t❤❡ ❧✐♥❡ ❜❡t✇❡❡♥ U ❛♥❞ V ✐♥ ♣r♦♣♦rt✐♦♥ t♦ t❤❡✐r ♠❛ss❡s✳ ❚❤❡r❡❢♦r❡✱ U − V ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ U − M ✿ M=

p q p+q U+ V =⇒ U − V = (U − M ) . p+q p+q q

◆♦✇✱ ✇❡ s✐♠♣❧② s✉❜st✐t✉t❡✿ P ′′ = (U − M )′′ = U ′′ − M ′′ = −Gq = −Gq

U −V −0 ||U − V ||3 p+q (U − M ) q

|| p+q (U − M )||3 q

U −M 2 p+q ||(U − M )||3 q 2  U −M q = −Gq p+q ||(U − M )||3 q3 P = −G . 2 (p + q) ||P ||3 = −Gq 

■♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ ✇❡ ❣♦ ❜❛❝❦ t♦ t❤❡ ❝❛s❡ ♦❢ t✇♦ ✐❞❡♥t✐❝❛❧ ♣❧❛♥❡ts ❛♥❞ s✐♠♣❧② ✐♥❝r❡❛s❡ t❤❡ y ✲❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ s❡❝♦♥❞ ♣❧❛♥❡t ❢r♦♠ 1 t♦ 2✳ ❲✐t❤ ❛ ❝❡rt❛✐♥ ❛♠♦✉♥t ♦❢ ❝✐r❝❧✐♥❣✱ t❤❡② st❛rt t♦ ♠♦✈❡ ❛✇❛② ❢r♦♠ ❡❛❝❤ ♦t❤❡r✳

❊✈❡♥t✉❛❧❧②✱ t❤❡② ❛r❡ ✢②✐♥❣ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥s✦ ❏✉st ❛s ✐♥ t❤❡ ❧❛st ❝❛s❡✱ ♣❧♦tt✐♥❣ t❤❡ tr❛❥❡❝t♦r✐❡s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ❤❡❧♣ t♦ r❡✈❡❛❧ t❤❡ ♣❛tt❡r♥✳ ❚❤✉s✱ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✱ t❤❡s❡ t❤r❡❡ s❡❡♠ t♦ ❜❡ t❤❡ ♠♦st ✐♥t❡r❡st✐♥❣ ♦✉t❝♦♠❡s✳ ■♥ t❤❡ ✜rst✱ t✇♦ ✐❞❡♥t✐❝❛❧ ♣❧❛♥❡ts ❞❛♥❝❡ ❛r♦✉♥❞ ❡❛❝❤ ♦t❤❡r ✭✐♥ r❡❛❧✐t②✱ ❛r♦✉♥❞ t❤❡ ❝♦♠❜✐♥❡❞ ❝❡♥t❡r ♦❢ ♠❛ss✮✳ ■♥ t❤❡ s❡❝♦♥❞✱ t❤❡② ❞❛♥❝❡ ❛✇❛② t♦❣❡t❤❡r ✭st✐❧❧ ❛r♦✉♥❞ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss✱ ✇❤✐❝❤ ✐s ♠♦✈✐♥❣✮✳ ❋✐♥❛❧❧②✱ t❤❡ t✇♦ r✉♥ ❛✇❛② ❢r♦♠ ❡❛❝❤ ♦t❤❡r ❜❡❝❛✉s❡ ❜❡②♦♥❞ s♦♠❡ ❞✐st❛♥❝❡ t❤❡② ❢❡❡❧ ❛❧♠♦st ♥♦ ♣✉❧❧✳✳✳

✺✳✼✳

✸✸✹

❚❤❡ t✇♦✲ ❛♥❞ t❤r❡❡✲❜♦❞② ♣r♦❜❧❡♠s

❊①❡r❝✐s❡ ✺✳✼✳✸

❲❤② ❞♦♥✬t ❛♥② ♦❢ t❤❡ ♣❧❛♥❡ts ✐♥ t❤❡ s♦❧❛r s②st❡♠ ❜❡❤❛✈❡ t❤✐s ✇❛②❄ ❲❤❛t ♦t❤❡r ♣♦ss✐❜✐❧✐t✐❡s ❝❛♥ ②♦✉ t❤✐♥❦ ♦❢❄ ❲❤❛t ❛❜♦✉t t❤r❡❡

♣❧❛♥❡ts



❋✐rst ❧❡t✬s r❡✈✐❡✇ ❛ r❡❧❛t❡❞ ♣r♦❜❧❡♠ ❞✐s❝✉ss❡❞ ✐♥ ❈❤❛♣t❡r ✹❍❉✲✷✳ ❚❤❡ ❢❛❝t t❤❛t t❤❡ ❊❛rt❤ ♦r❜✐ts t❤❡ ❙✉♥ ❛♥❞ t❤❡ ▼♦♦♥ ♦r❜✐ts ❛r♦✉♥❞ t❤❡ ❊❛rt❤ ♠❛② ❜❡ ✐❧❧✉str❛t❡❞ ✇✐t❤ t❤❡ ♣✐❝t✉r❡ ♦♥ ❧❡❢t ✇❤✐❧❡ t❤❡ ❛❝t✉❛❧ ❞❛t❛ ❛❜♦✉t t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡ ♦r❜✐ts ❛♥❞ t❤❡ ♣❡r✐♦❞ ♦❢ t❤❡ ▼♦♦♥ ♣r♦❞✉❝❡ t❤❡ ♣✐❝t✉r❡ ♦♥ r✐❣❤t✿

❙✉♣♣♦s❡ ♥♦✇ t❤❛t t❤❡ t❤r❡❡ ♣❧❛♥❡ts ❤❛✈❡ ♠❛ss❡s p, q, m✳ ■❢ ✇❡ ❛ss✉♠❡ t❤❛t p >> q >> m ,

s✉❝❤ ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ ❙✉♥✲❊❛rt❤✲▼♦♦♥✱ t❤❡ ❡✛❡❝t ♦❢ t❤❡ ❣r❛✈✐t② ♦❢ t❤❡ s❡❝♦♥❞ ♦♥ t❤❡ ✜rst ❛♥❞ t❤❡ t❤✐r❞ ♦♥ t❤❡ s❡❝♦♥❞ ✐s ♥❡❣❧✐❣✐❜❧❡✱ ❥✉st ❛s ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✳ ❙✉♣♣♦s❡ • U = 0 ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ❙✉♥ ✜①❡❞ ❛t t❤❡ ♦r✐❣✐♥✱

• V ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ❊❛rt❤ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❙✉♥✱ ❛♥❞ • X ✐s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ▼♦♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❊❛rt❤✳

❚❤❡♥ t❤❡ t✇♦ ✐♥t❡r❛❝t✐♦♥s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❡❛❝❤ ♦t❤❡r ❛♥❞ ✇❡ ❤❛✈❡ t✇♦ ✐♥❞❡♣❡♥❞❡♥t ◆❡✇t♦♥✬s ❖❉❊s ♦❢ ♣❧❛♥❡t❛r② ♠♦t✐♦♥✿ V ||V ||3 X X ′′ = −Gq ||X||3 V ′′ = −Gp

❇♦t❤ s♦❧✉t✐♦♥s ❢♦❧❧♦✇ t❤❡✐r r❡s♣❡❝t✐✈❡ ❡❧❧✐♣s❡s✳ ❲❡ t❤❡♥ ❛❞❞ t❤❡♠ ❛s ✈❡❝t♦rs✱ W =V +X,

t♦ ♣r♦❞✉❝❡ t❤❡ ♣❛t❤ ♦❢ t❤❡ t❤✐r❞ ♣❧❛♥❡t ❛r♦✉♥❞ t❤❡ ✜rst✳ ●❡♥❡r✐❝❛❧❧②✱ ✐t ❧♦♦❦s ❧✐❦❡ t❤✐s✿

✺✳✼✳

❚❤❡ t✇♦✲ ❛♥❞ t❤r❡❡✲❜♦❞② ♣r♦❜❧❡♠s

✸✸✺

❲❡ ♥❡①t ♠❛❦❡ ♦♥❡ ❣❡♥❡r❛❧✐③❛t✐♦♥✿ t❤❡ ♠❛ss ♦❢ t❤❡ s❡❝♦♥❞ ♣❧❛♥❡t ✐s ♥♦t ♥❡❣❧✐❣✐❜❧② s♠❛❧❧ r❡❧❛t✐✈❡ t♦ t❤❛t ♦❢ t❤❡ ✜rst ❛♥②♠♦r❡✳ ❲❡✱ t❤❡♥✱ ❝❛♥✬t ✜① t❤❡ ✜rst ❛t t❤❡ ♦r✐❣✐♥ ❛♥②♠♦r❡✳ ❲❡ ❤❛✈❡ ❛ t✇♦✲❜♦❞② s②st❡♠ ❥✉st ❛s ❛❜♦✈❡✿ U −V ||U − V ||3 V −U V ′′ = −Gp ||V − U ||3 W −U W −V W ′′ = −Gp −q 3 ||W − U || ||W − V ||3

U ′′ = −Gq

■♥ t❤❡ ♠❡❛♥t✐♠❡✱ t❤❡ t❤✐r❞ ♣❧❛♥❡t ✐s ❛✛❡❝t❡❞ ❜② t❤❡ ❣r❛✈✐t② ♦❢ t❤❡ ✜rst t✇♦✳ ■t ✐s ❝❛❧❧❡❞ t❤❡ r❡str✐❝t❡❞

t❤r❡❡

❜♦❞② ♣r♦❜❧❡♠

❲❤✐❧❡ t❤❡ t✇♦ ✜rst ♣❧❛♥❡ts ❞♦ t❤❡ s❛♠❡✱ t❤❡ ✈❛r✐❡t② ♦❢ ❜❡❤❛✈✐♦rs ♦❢ t❤❡ t❤✐r❞ ✐s ❡♥♦r♠♦✉s ❡✈❡♥ ✐♥ ❞✐♠❡♥s✐♦♥ 2✿

❚❤❡r❡ ❛r❡ ♠❛♥② ♣❡r✐♦❞✐❝ tr❛❥❡❝t♦r✐❡s ❜✉t ✇❡ ✇✐❧❧ ♣♦✐♥t ♦✉t ♦♥❧② ♦♥❡✳ ❇❡❧♦✇ ✇❡ ❝♦♥✜r♠ t❤❛t ✐t ✐s ♣♦ss✐❜❧❡ t♦ ♣❧❛❝❡ ❛ s❛t❡❧❧✐t❡ ♦♥ t❤❡ ▼♦♦♥✬s ♦r❜✐t✱ 60 ❞❡❣r❡❡s ❛❤❡❛❞✱ t❤❛t ✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ r❡✈♦❧✈❡ ✐♥ t❤✐s ❢❛s❤✐♦♥ ✐♥❞❡✜♥✐t❡❧②✳

✺✳✽✳

❆ ❝❛♥♥♦♥ ✐s ✜r❡❞✳✳✳

✸✸✻

◆♦✇✱ ✐♥ ❣❡♥❡r❛❧✳ ❙✉♣♣♦s❡ ♥♦✇ ✇❡ ❤❛✈❡ t❤r❡❡ ♣❧❛♥❡ts ✇✐t❤ ♠❛ss❡s

m1 , m2 , m3 ❧♦❝❛t❡❞ ❛t V1 , V2 , V3 r❡s♣❡❝t✐✈❡❧②✳

❊❛❝❤ ♦❢ t❤❡ t❤r❡❡ ♦❜❥❡❝ts ✐s ❛✛❡❝t❡❞ ❜② t❤❡ ❣r❛✈✐t② ♦❢ ❡✐t❤❡r ♦❢ t❤❡ ♦t❤❡r t✇♦✳ ❚❤❡♥ t❤❡s❡ t❤r❡❡ ✐♥t❡r❛❝t✐♦♥ ❛♣♣❡❛r ✐♥ t❤❡ t❤r❡❡ ❞❡♣❡♥❞❡♥t ✈❡❝t♦r ❖❉❊s✿

V1 − V3 V1 − V2 − Gm3 3 ||V1 − V2 || ||V1 − V3 ||3 V2 − V1 V2 − V3 V2′′ = −Gm1 − Gm3 3 ||V2 − V1 || ||V2 − V3 ||3 V3 − V1 V3 − V2 V3′′ = −Gm1 − Gm2 3 ||V3 − V1 || ||V3 − V2 ||3

V1′′ = −Gm2

❚❤❡ ❣❡♥❡r❛❧ t❤r❡❡ ❜♦❞② ♣r♦❜❧❡♠ ❤❛s ♥♦ ❛♥❛❧②t✐❝ s♦❧✉t✐♦♥✳

✺✳✽✳ ❆ ❝❛♥♥♦♥ ✐s ✜r❡❞✳✳✳ ❲❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t ❤✐st♦r② ♦❢ ❜❛❧❧✐st✐❝s ✐s t❤❛t t❤❡ ♣❛r❛❜♦❧✐❝ tr❛❥❡❝t♦r✐❡s ✇❡r❡ ❦♥♦✇♥ s✐♥❝❡ ●❛❧✐❧❡♦ ❛♥❞✱ ❢✉rt❤❡r♠♦r❡✱ ◆❡✇t♦♥ ✇♦r❦❡❞ ♦✉t t❤❡ ♣❤②s✐❝s ❜❡❤✐♥❞ ❛♥❞ t❤❡ ♠❛t❤❡♠❛t✐❝s ✭❝❛❧❝✉❧✉s✮ ❜❡❤✐♥❞ t❤✐s ✐❞❡❛✳ ◆♦♥❡t❤❡❧❡ss✱ ❛s r❡❝❡♥t❧② ❛s ♠✐❞✲✶✾t❤ ❝❡♥t✉r② t❤❡② ❛✐♠❡❞ ❝❛♥♥♦♥s ❜❛s❡❞ ❡♥t✐r❡❧② ♦♥ t❤❡ ✐♥❢♦r♠❛t✐♦♥ t❤❛t ❝♦♠❡s ❢r♦♠ ❡①♣❡r✐♠❡♥t❛t✐♦♥✱ ✇✐t❤♦✉t ♠❛t❤❡♠❛t✐❝s✳ ❍❡r❡ ✐s ❛♥ ❡①❝❡r♣t ❢r♦♠

❚❤❡ ❊♠♣❡r♦r ◆❛♣♦❧❡♦♥✬s ◆❡✇ ❙②st❡♠ ♦❢ ❋✐❡❧❞ ❆rt✐❧❧❡r②

❇❡❧♦✇ ✐s t❤❡ ✏r❛♥❣❡ t❛❜❧❡✑ ❢r♦♠ t❤❡

❚❤❡ ❈♦♥❢❡❞❡r❛t❡ ❖r❞♥❛♥❝❡ ▼❛♥✉❛❧

✭✶✽✺✹✮✿

❞✉r✐♥❣ t❤❡ ❆♠❡r✐❝❛♥ ❈✐✈✐❧ ❲❛r

✭✶✽✻✵s✮✳ ❈❧❡❛r❧②✱ t❤❡ ♥✉♠❜❡rs ❝♦♠❡ ❢r♦♠ s❤♦♦t✐♥❣ ❛♥❞ t❤❡♥ ♠❡❛s✉r✐♥❣ t❤❡ ❞✐st❛♥❝❡✿

✺✳✽✳ ❆ ❝❛♥♥♦♥ ✐s ✜r❡❞✳✳✳

✸✸✼

❲✐t❤ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t t❤❡ ❞②♥❛♠✐❝s ♦❢ ♣r♦❥❡❝t✐❧❡s✱ ✇❡ ❝❛♥ tr② t♦ r❡♣r♦❞✉❝❡ t❤❡s❡ r❡s✉❧ts✳ ❲❡ st❛rt ✇✐t❤ t❤❡ s❛♠❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠✿ 

x′′ = 0 , y ′′ = −32



x′ (0) = s0 cos α , y ′ (0) = s0 sin α



x(0) = 0 , y(0) = y0

✇❤❡r❡ s0 ✐s t❤❡ ✐♥✐t✐❛❧ s♣❡❡❞ ❛♥❞ α ✐s t❤❡ ❛♥❣❧❡ ♦❢ t❤❡ ❜❛rr❡❧✳ ❊①❛♠♣❧❡ ✺✳✽✳✶✿ ❞❛t❛

❚❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ ❤❛s ❛❧r❡❛❞② ❜❡❡♥ s♦❧✈❡❞✿ 

x = s0 cos α · t, y = y0 +s0 sin α · t −16t2 .

❲❡ ♥❡❡❞ t♦ ✜♥❞ s✉❝❤ ❛♥ x t❤❛t y = 0✳ ❲❡ ✜♥❞ t❤❡ t✐♠❡ t♦ r❡❛❝❤ t❤❡ ❣r♦✉♥❞ ✜rst ✭❝❤♦♦s✐♥❣ t❤❡ ♣♦s✐t✐✈❡ ✈❛❧✉❡✮✿ t=

−s0 sin α ±

p  p (s0 sin α)2 − 4(−16)y0 1  = s0 sin α + (s0 sin α)2 + 64y0 . 2(−16) 32

❲❡ ❝♦♥s✐❞❡r t❤❡ ✶✷✲♣♦✉♥❞ ✜❡❧❞ ❤♦✇✐t③❡r✳ ❙✉♣♣♦s❡ t❤❡ ♠✉③③❧❡ ✈❡❧♦❝✐t② ✐s ❦♥♦✇♥✱ s0 = 1054 ❢❡❡t ♣❡r s❡❝♦♥❞✳ ❲❡ ❛❧s♦ ❡st✐♠❛t❡ t❤❡ ✐♥✐t✐❛❧ ❡❧❡✈❛t✐♦♥ ✭t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ ❝❛♥♥♦♥✮ ❛t y0 = 5 ❢❡❡t✳ ❚❤❡♥ t❤❡ ❞✐st❛♥❝❡s s❤♦✉❧❞ ❜❡✿ ❡❧❡✈❛t✐♦♥ ✐♥ ❞❡❣r❡❡s t✐♠❡ ✐♥ s❡❝♦♥❞s✱ t ❞✐st❛♥❝❡ ✐♥ ❢❡❡t✱ x ❞✐st❛♥❝❡ ✐♥ ②❛r❞s t❡st ❞✐st❛♥❝❡ 0 1 2 3 4 5

0.56 1.38 2.43 3.54 4.66 5.80

589 1451 2557 3722 4902 6085

❲❡ s❡❡ ❛ ♠✐s♠❛t❝❤ ✇✐t❤ t❤❡ ❞❛t❛ t❤❛t ❣r♦✇s ❢❛st ✇✐t❤ t❤❡ ❛♥❣❧❡✳

196 484 852 1241 1634 2028

195 539 640 847 975 1072

✺✳✽✳ ❆ ❝❛♥♥♦♥ ✐s ✜r❡❞✳✳✳

✸✸✽

❊①❡r❝✐s❡ ✺✳✽✳✷

❊①♣❧❛✐♥ ✇❤② t❤❡ r❡❞ ❝✉r✈❡ ❧♦♦❦s str❛✐❣❤t✳ ❚❤❡ r❡❛s♦♥ ✇❤② t❤❡ ◆❡✇t♦♥✐❛♥ ♠❡❝❤❛♥✐❝s ♦✈❡r❡st✐♠❛t❡s t❤❡ ❞✐st❛♥❝❡ ✐s ❦♥♦✇♥❀ ✐t ✐s t❤❡ ❛✐r✲r❡s✐st❛♥❝❡✳ ■♥ ❢❛❝t✱ ◆❡✇t♦♥ ❤✐♠s❡❧❢ ✇♦r❦❡❞ ♦✉t ❛❧❧ ♥❡❝❡ss❛r② ❞❡t❛✐❧s ♦❢ ❜❛❧❧✐st✐❝s ✐♥❝❧✉❞✐♥❣ t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ❛✐r r❡s✐st❛♥❝❡✳ ❍❡ ❛ss✉♠❡❞ ❤♦✇❡✈❡r t❤❛t t❤❡ r❡s✐st❛♥❝❡ ❢♦r❝❡ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ s♣❡❡❞✳ ❍❛❧❢ ❛ ❝❡♥t✉r② ❧❛t❡r ❊✉❧❡r t❤♦✉❣❤t✱ ❢♦r ❜❛❧❧✐st✐❝s✱ ✐t ✐s ❜❡tt❡r t♦ t❛❦❡ t❤❡ r❡s✐st❛♥❝❡ ❢♦r❝❡ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ sq✉❛r❡ ♦❢ t❤❡ s♣❡❡❞✳ ❚❤❛t ✇❛s ✶✵✵ ②❡❛rs ❜❡❢♦r❡ t❤♦s❡ t❡sts✦ ❚❤❡ ❞r❛❣ ❡q✉❛t✐♦♥ ❣✐✈❡s t❤❡ ❢♦r❝❡ FD ♦❢ ❞r❛❣ ❡①♣❡r✐❡♥❝❡❞ ❜② ❛♥ ♦❜❥❡❝t ❞✉❡ t♦ ✐ts ♠♦✈❡♠❡♥t t❤r♦✉❣❤ t❤❡ ❛✐r✿

||FD || = 21 ρCA · s2 , ✇❤❡r❡

• ρ ✐s t❤❡ ❞❡♥s✐t② ♦❢ t❤❡ ❛✐r✱

• A ✐s t❤❡ ❝r♦ss s❡❝t✐♦♥❛❧ ❛r❡❛ ♦❢ t❤❡ ♣r♦❥❡❝t✐❧❡✱

• C ✐s t❤❡ ❞r❛❣ ❝♦❡✣❝✐❡♥t ✕ ❛ ❞✐♠❡♥s✐♦♥❧❡ss ❝♦❡✣❝✐❡♥t t❤❛t ❞❡♣❡♥❞s ♦♥ t❤❡ ♣r♦❥❡❝t✐❧❡✬s ❣❡♦♠❡tr②✱ ❛♥❞ • s ✐s t❤❡ s♣❡❡❞ ♦❢ t❤❡ ♣r♦❥❡❝t✐❧❡✳

❚❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤✐s ❢♦r❝❡ FD ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❛t ♦❢ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ♦♣♣♦s✐t❡ t♦ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ✈❡❧♦❝✐t② X ′ ✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❣❡♥❡r❛t❡❞ ❜② t❤✐s ❢♦r❝❡ ✐s

− 21 ρCA

1 sX ′ , M

✇❤❡r❡ M ✐s t❤❡ ♠❛ss ♦❢ t❤❡ ♣r♦❥❡❝t✐❧❡✳ ❚❤❡♥ ♦✉r ■❱P ❜❡❝♦♠❡s✿     x′′ =

   y ′′

1 − 12 ρCA sx′ M , 1 ′ 1 = −32 − 2 ρCA sy M

    x′ (0) = s0 cos α    y ′ (0) = s0 sin α

,

    x(0) = 0

.

   y(0) = y0

❲✐t❤ t❤✐s ✉♣❞❛t❡❞ ❞②♥❛♠✐❝s ♦❢ ♣r♦❥❡❝t✐❧❡s✱ ❧❡t✬s tr② ❛❣❛✐♥ t♦ r❡♣r♦❞✉❝❡ t❤❡ t❡st r❡s✉❧ts✳ ❊①❛♠♣❧❡ ✺✳✽✳✸✿ ❞❛t❛

❲❡ ♥❡❡❞ s♦♠❡ ✐♥❢♦r♠❛t✐♦♥ ❢♦r t❤❡ ❞r❛❣ ❡q✉❛t✐♦♥✿ • t❤❡ ✇❡✐❣❤t M = 8.9 ♣♦✉♥❞s • t❤❡ ❛r❡❛ A = πr2 ♦❢ t❤❡ ❝r♦ss s❡❝t✐♦♥ ♦❢ t❤❡ ❝❛♥♥♦♥❜❛❧❧✱ ✇❤❡r❡ t❤❡ r❛❞✐✉s ✐s r = 4.62/2 = 2.31 ✐♥❝❤❡s • t❤❡ ❞❡♥s✐t② ♦❢ t❤❡ ❛✐r ρ = 0.074887 ♣♦✉♥❞s ♣❡r ❝✉❜✐❝ ❢♦♦t • t❤❡ ❞r❛❣ ❝♦❡✣❝✐❡♥t ♦❢ ❛ s♣❤❡r❡ C = 0.47 ❲❡ ✉s❡ ❊✉❧❡r✬s ♠❡t❤♦❞✿

✺✳✾✳

❇♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s

✸✸✾

❲❡ r❡♣❡❛t ✐t ❢♦r ❡❛❝❤ ❡❧❡✈❛t✐♦♥✱ s✐① t✐♠❡s✿ ❡❧❡✈❛t✐♦♥ ✐♥ ❞❡❣r❡❡s

❞✐st❛♥❝❡ ✐♥ ②❛r❞s

t❡st ❞✐st❛♥❝❡

0 1 2 3 4 5

187 405 629 822 984 1124

195 539 640 847 975 1072

▼✉❝❤ ❜❡tt❡r✦

✺✳✾✳ ❇♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s

❊①❛♠♣❧❡ ✺✳✾✳✶✿ s❛♠♣❧❡ ❇❱P

❙❡tt✐♥❣ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❛s✐❞❡ ❢♦r ❛ ♠♦♠❡♥t✱ ❧❡t✬s ❝♦♥s✐❞❡r t❤✐s s✐♠♣❧❡ ♣r♦❜❧❡♠✿ ◮ ❋r♦♠ ❛❧❧ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s f (x) = ax2 +bx+c✱ ✜♥❞ t❤❡ ♦♥❡s ✇✐t❤ f (0) = 20,

f ′ (0) =

−3✳

❍❡r❡✱ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ ❛r❡ ♣r♦✈✐❞❡❞ ❢♦r ❛ ♣❛rt✐❝✉❧❛r ✈❛❧✉❡ ♦❢ t❤❡ ✈❛r✐❛❜❧❡✱

x = 0✳ [0, ∞]✳

❚❤❡s❡ ❛r❡ ❝❛❧❧❡❞

✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳

❚❤❡ ✇♦r❞ ✏✐♥✐t✐❛❧✑ r❡❢❡rs t♦ t❤❡ st❛rt✐♥❣ ♣♦✐♥t✱

0✱

♦❢ t❤❡ r❛②

▲❡t✬s ❝❤❛♥❣❡ t❤❡ ♣r♦❜❧❡♠✿

◮ ❋r♦♠ ❛❧❧ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s f (x) = ax2 +bx+c✱ ✜♥❞ ♦♥❡s ✇✐t❤ f (0) = 20, f (5) = 30✳ ■♥ t❤✐s ❝❛s❡✱ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ♣r♦✈✐❞❡❞ ❢♦r t✇♦ ✈❛❧✉❡s ♦❢ x ❛♥❞ ♥♦ ✈❛❧✉❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛r❡ ♣r♦✈✐❞❡❞✳ ❚❤❡s❡ ❛r❡ ❝❛❧❧❡❞ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❚❤❡ ✇♦r❞ ✏❜♦✉♥❞❛r②✑ r❡❢❡rs t♦ t❤❡ ❡♥❞✲♣♦✐♥ts✱ 0 ❛♥❞ 5✱ ❛❧s♦ ❦♥♦✇♥ ❛s t❤❡ ❜♦✉♥❞❛r② ♣♦✐♥ts✱ ♦❢ t❤❡ ✐♥t❡r✈❛❧ [0, 5]✳

✺✳✾✳

❇♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s

✸✹✵

❊①❡r❝✐s❡ ✺✳✾✳✷

❋✐♥❞ ❛❧❧ t❤❡s❡ ♣♦❧②♥♦♠✐❛❧s✳ ❊①❛♠♣❧❡ ✺✳✾✳✸✿ ✉♥✐t❛r② q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s

▲❡t✬s ❧✐♠✐t ♦✉rs❡❧✈❡s t♦ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s ✇✐t❤ a = 1✿ • ❋r♦♠ ❛❧❧ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s f (x) = x2 + bx + c✱ ✜♥❞ t❤❡ ♦♥❡ ✇✐t❤ f (0) = 20, f ′ (0) = −3✳ • ❋r♦♠ ❛❧❧ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s f (x) = x2 + bx + c✱ ✜♥❞ t❤❡ ♦♥❡ ✇✐t❤ f (0) = 20, f (5) = 30✳ ❚❤❡♥ ✇❡ ❤❛✈❡ ❛ ❝❡rt❛✐♥ ✉♥✐q✉❡♥❡ss✱ ✐✳❡✳✱ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ❛♥s✇❡r t♦ t❤❡ q✉❡st✐♦♥❀ t❤❡ s♦❧✉t✐♦♥ ✐♥ t❤❡ ❢♦r♠❡r ❝❛s❡ ✐s✱ ♦❢ ❝♦✉rs❡✱ c = 20, b = −3✳ ❋♦r t❤❡ ❧❛tt❡r ❝❛s❡ ✇❡ ♥❡❡❞ ♠♦r❡ ❛❧❣❡❜r❛✿ c = 20 =⇒ 52 + b · 5 + 20 = 30 =⇒ b = (30 − 52 − 20)/5 = 3 .

❊①❛♠♣❧❡ ✺✳✾✳✹✿ ❝❛♥♥♦♥

▲❡t✬s r❡✈✐❡✇ ❛ ❢❛♠✐❧✐❛r ♣r♦❜❧❡♠ ❢r♦♠ ❱♦❧✉♠❡s ✶ ❛♥❞ ✷✳ ❋r♦♠ ❛ 200 ❢❡❡t ❡❧❡✈❛t✐♦♥✱ ❛ ❝❛♥♥♦♥ ✐s ✜r❡❞ ❤♦r✐③♦♥t❛❧❧② ❛t 200 ❢❡❡t ♣❡r s❡❝♦♥❞✳ ❍♦✇ ❢❛r ✇✐❧❧ t❤❡ ❝❛♥♥♦♥❜❛❧❧ ❣♦❄

❲❡ ❦♥♦✇ t❤✐s ❛s ❛♥ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠✳ ■♥❞❡❡❞✱ t❤❡ ❖❉❊ ❤❛s ❜❡❡♥ s♦❧✈❡❞✿ 

x = 200t, y = 200 −16t2 .

✺✳✾✳ ❇♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s

✸✹✶

◆♦✇ ✇❡ ❥✉st ♥❡❡❞ t♦ ✜♥❞ t❤❡ s♦❧✉t✐♦♥ t❤❛t s❛t✐s✜❡s t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✿ • t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥✿ X(0) = (x(0), y(0)) = (0, 200)❀ • t❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②✿ X ′ (0) = (x′ (0), y ′ (0)) = (200, 0)✳

❲❡ s❝r♦❧❧ ❞♦✇♥ t❤❡ s♣r❡❛❞s❤❡❡t t♦ ✜♥❞ t❤❡ r♦✇ ✇✐t❤ y ❝❧♦s❡ t♦ 0✿ ❛r♦✉♥❞ t1 = 3.55 s❡❝♦♥❞s✱ ✇✐t❤ t❤❡ ✈❛❧✉❡ ♦❢ x ❛t t❤❡ t✐♠❡ ❝❧♦s❡ t♦ 710 ❢❡❡t✳ ❆❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ s♦❧✈❡ ❛♥ ❡q✉❛t✐♦♥ ❛♥❞ t❤❡♥ s✉❜st✐t✉t❡✿ r √ √ 200 5 2 5 2 2 y(t1 ) = 200 − 16t1 = 0 =⇒ t1 = = =⇒ x1 = x(t1 ) = 200t1 = 200 ≈ 707 . 16 2 2 ◆♦✇ ✇❤❛t ❝♦✉❧❞ ❜❡ t❤❡ ♠❡❛♥✐♥❣ ♦❢ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✐♥ t❤✐s s❡tt✐♥❣❄ ❲❡ ❛❧r❡❛❞② ❤❛✈❡ ♦♥❡✱ t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥✳ ❚❤❡ s❡❝♦♥❞ ♠❛② ❜❡ t❤❡ ✜♥❛❧ ❧♦❝❛t✐♦♥ ✇❤❡♥✱ ❢♦r ❡①❛♠♣❧❡✱ ✇❡ ❛r❡ tr②✐♥❣ t♦ ❤✐t ❛ t❛r❣❡t ✕ ❛t ❛ ♣❛rt✐❝✉❧❛r ♠♦♠❡♥t ♦❢ t✐♠❡✳ ❙✉♣♣♦s❡ ✇❡ ✇❛♥t t❤❡ ❝❛♥♥♦♥ ❜❛❧❧ t♦ ❜❡ ❛t (200, 500) ❛❢t❡r 2 s❡❝♦♥❞s✳ ■♥ t❤❛t ❝❛s❡ t❤❡ ♠✉③③❧❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ❝❛♥♥♦♥ ✐s ✉♥s♣❡❝✐✜❡❞ ❛♥❞ ✐t ✐s ✇❤❛t ✇❡ ❛r❡ s✉♣♣♦s❡❞ t♦ ✜♥❞✳ ❲❡ t❤✉s ♥❡❡❞ t♦ ✜♥❞ ❛ s♦❧✉t✐♦♥ ✭t❤❡r❡ ❝♦✉❧❞ ❜❡ ♦♥❡✱ ♠❛♥②✱ ♦r ♥♦♥❡✮ t❤❛t s❛t✐s✜❡s t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✿ • t❤❡ ✜rst ❜♦✉♥❞❛r② ❧♦❝❛t✐♦♥✿ X(0) = (x(0), y(0)) = (0, 200)❀ • t❤❡ s❡❝♦♥❞ ❜♦✉♥❞❛r② ❧♦❝❛t✐♦♥✿ X(2) = (x(2), y(2)) = (500, 100)✳ ❚❤✐s ✐s ❝❛❧❧❡❞ ❛ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠✳ ▲❡t✬s s♦❧✈❡ ✐t✳

❲❤❡r❡ ❞♦ t❤❡ s♦❧✉t✐♦♥s ❝♦♠❡ ❢r♦♠❄ ❚❤❡ ❖❉❊ ❤❛s ❜❡❡♥ s♦❧✈❡❞ ❛♥❞ t❤❡ ✜rst ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ❣✐✈❡s ✉s t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥✿   x = x0 +ut x =0 +ut =⇒ . y = y0 +vt −16t2 y = 200 +vt −16t2 ❲❡ ♥♦✇ ✜♥❞ u, v ❢r♦♠ t❤❡ s❡❝♦♥❞ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥✳ ❖♥❡ ❞✐♠❡♥s✐♦♥ ❛t ❛ t✐♠❡✿ ❢♦r x✱ ❛♥❞ ❢♦r y ✱

x(2) = v · 2 = 500 =⇒ u = 250;

y(2) = 200 + v · 2 − 16 · 22 = 100 =⇒ 2v = 100 − (200 − 16 · 22 ) =⇒ v = (−100 + 16 · 4)/2 = −18 .

◆♦t❡ t❤❛t ❛ ♠♦r❡ ♣r❛❝t✐❝❛❧ s✐t✉❛t✐♦♥ ✐s ✇❤❡♥ t❤❡ ♠✉③③❧❡ ✈❡❧♦❝✐t②✱ ♠❛t❤❡♠❛t✐❝❛❧❧② t❤❡ s♣❡❡❞✱ r❡♠❛✐♥s t❤❡ s❛♠❡ ❛♥❞ ✐t ✐s t❤❡ ❛♥❣❧❡ ♦❢ t❤❡ ❜❛rr❡❧ t❤❛t ✇❡ ♥❡❡❞ t♦ ✜♥❞✳

✺✳✾✳

❇♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s

✸✹✷

❊①❛♠♣❧❡ ✺✳✾✳✺✿ s♣r✐♥❣

❈♦♥s✐❞❡r t❤❡ ❢❛♠✐❧✐❛r ❖❉❊ ❢♦r t❤❡ ♦s❝✐❧❧❛t✐♦♥ ♦❢ ❛♥ ♦❜❥❡❝t ♦♥ ❛ s♣r✐♥❣✿ y ′′ (x) + y(x) = 0 .

❇✉t t❤✐s t✐♠❡ ✇❡ ❛r❡♥✬t tr②✐♥❣ t♦ ♣r❡❞✐❝t ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ♦❜❥❡❝t ✇✐t❤ ❦♥♦✇♥ ♣♦s✐t✐♦♥ ❛♥❞ ✈❡❧♦❝✐t②✳ ❲❡ ❛s❦ ♦✉rs❡❧✈❡s ✐❢ t❤❡ s②st❡♠ ❝❛♥ ❜r✐♥❣ t❤❡ ♦❜❥❡❝t ❢r♦♠ ❛ ♣❛rt✐❝✉❧❛r ♣♦s✐t✐♦♥ t♦ ❛♥♦t❤❡r ✐♥ ❛ s♣❡❝✐✜❝ ❛♠♦✉♥t ♦❢ t✐♠❡✱ ❢♦r ❡①❛♠♣❧❡✿ y(0) = 0, y(π/2) = 2 .

❚❤❡ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ t♦ ♦✉r ❖❉❊ ✐s y(x) = A sin x + B cos x .

◆♦✇✱ ❢r♦♠ t❤❡ ✜rst ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✇❡ ♦❜t❛✐♥✿ 0 = A · 0 + B · 1 =⇒ B = 0 .

❋r♦♠ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✇❡ ♦❜t❛✐♥✿ 2 = A · 1 =⇒ A = 2 .

❚❤✉s ✐♠♣♦s✐♥❣ t❤❡s❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ♣r♦❞✉❝❡s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥✿ y(x) = 2 sin x.

❲❤❡♥ t❤❡ ❢✉♥❝t✐♦♥s ❤❛✈❡ t✇♦ ♦r ♠♦r❡ ✈❛r✐❛❜❧❡s✱ r❡❣✐♦♥s ❛r❡ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ❛♥❞ s♦ ❛r❡ t❤❡✐r ❜♦✉♥❞❛r✐❡s✳ ❙✉❝❤ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❛r❡ ❜❡②♦♥❞ t❤❡ s❝♦♣❡ ♦❢ t❤✐s ❜♦♦❦✳

❈❤❛♣t❡r ✻✿ P❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s

❈♦♥t❡♥ts

✻✳✶ ❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✷ ❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ ♣❡r♠❡❛❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✸ ❍❡❛t tr❛♥s❢❡r ✐s ❝❛✉s❡❞ ❜② t❡♠♣❡r❛t✉r❡ ❞✐✛❡r❡♥❝❡s ✳ ✳ ✻✳✹ ❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ t❤❡ ❣❡♦♠❡tr② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✺ ❚❤❡ ❤❡❛t P❉❊ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✻ ❈❡❧❧s ❛♥❞ ❢♦r♠s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✼ ❍❡❛t tr❛♥s❢❡r ✐♥ ❞✐♠❡♥s✐♦♥ 2✿ ❛ ♣❧❛t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✽ ❚❤❡ ❤❡❛t P❉❊ ❢♦r ❞✐♠❡♥s✐♦♥ 2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✾ ❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥ 1✿ s♣r✐♥❣s ❛♥❞ str✐♥❣s ✻✳✶✵ ❚❤❡ ✇❛✈❡ P❉❊ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✶✶ ❲❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥ 2✿ ❛ ♠❡♠❜r❛♥❡ ✳ ✳ ✳

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✸✹✸ ✸✺✸ ✸✻✵ ✸✻✺ ✸✼✵ ✸✼✺ ✸✽✸ ✸✾✶ ✸✾✸ ✹✵✵ ✹✵✺

✻✳✶✳ ❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜ ❥❡❝ts

▼♦t✐♦♥ ❤❛♣♣❡♥s ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡✳ ❍♦✇❡✈❡r✱ ②♦✉ ❝❛♥✬t ❜❡ ❛t t✇♦ ♣❧❛❝❡s ❛t t❤❡ s❛♠❡ t✐♠❡✦ ■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✇✐❧❧ ✐♥✈❡st✐❣❛t❡ ♣r♦❝❡ss❡s t❤❛t ❤❛♣♣❡♥ ✕ s❡♣❛r❛t❡❧② ❜✉t ♥♦t ✐♥❞❡♣❡♥❞❡♥t❧② ✕ ❛t ❡✈❡r② ❧♦❝❛t✐♦♥ ♦❢ ❛ r❡❣✐♦♥✳ ❆ ❝✉♣ ♠❛② ❜❡ ❣r❛❞✉❛❧❧② ❝♦♦❧✐♥❣ ❢r♦♠ t❤❡ ♦✉ts✐❞❡✿

❚❤❡ t❡♠♣❡r❛t✉r❡ ✈❛r✐❡s ✇✐t❤ t✐♠❡ ❛t ❡✈❡r② ❧♦❝❛t✐♦♥ ❜✉t ❛❧s♦ ❢r♦♠ ❧♦❝❛t✐♦♥ t♦ ❧♦❝❛t✐♦♥ ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡✳ ❖r ✐t ♠❛② ❜❡ ✇❛r♠✐♥❣ ✉♣ ♦♥ ❛ st♦✈❡✿

✻✳✶✳

❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts

✸✹✹

❚❤❡ ♣❛tt❡r♥s ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❡♠♣❡r❛t✉r❡ ✐♥ t❤❡ ❝✉♣ ❞✐✛❡r ✐♥ t❤❡s❡ t✇♦ s❝❡♥❛r✐♦s✳ ❲❡ st❛rt ✇✐t❤ ❛ r❡✈✐❡✇ ❛♥❞ t❤❡♥ ♣r♦❝❡❡❞ t♦ t❤❡

1✲❞✐♠❡♥s✐♦♥❛❧

❝❛s❡✳

▲❡t✬s r❡❝❛❧❧ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ❈♦♦❧✐♥❣ ✭❈❤❛♣t❡r ✶✮✿

◮ ✏❚❤❡ r❛t❡ ♦❢ ❝♦♦❧✐♥❣ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❝✉rr❡♥t t❡♠♣❡r❛t✉r❡ ❛♥❞ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡✑✳ ❆♥ ❡①❛♠♣❧❡ ✐s ❝♦♦❧✐♥❣ ♦❢ ❛ ❝✉♣ ♦❢ ❝♦✛❡❡ ♦r ✇❛r♠✐♥❣ ✉♣ ❛ ❝❛♥ ♦❢ s♦❞❛✳ ❲❡ ✐♥tr♦❞✉❝❡ ✈❛r✐❛❜❧❡s✿ ✶✳

t

✷✳

u

✐s t❤❡ t✐♠❡✳ ✐s t❤❡ t❡♠♣❡r❛t✉r❡✳

❲❡ t❤❡♥ r❡✇r✐t❡ t❤❡ ❞❡s❝r✐♣t✐♦♥ ✐♥ t❡r♠s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿

∆u = k(r − u) , ∆t ✇❤❡r❡

r

✐s t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡✱ ❢♦r s♦♠❡ ❝♦♥st❛♥t

k > 0✳

❲❡ ❝❛♥ s❡❡ t❤❛t ✇❤❡♥ t❤❡ t❡♠♣❡r❛t✉r❡ ✐s ❤✐❣❤❡r

t❤❛♥ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡✱ ✐t ❞❡❝r❡❛s❡s ❛♥❞ ✇❤❡♥ t❤❡ t❡♠♣❡r❛t✉r❡ ✐s ❧♦✇❡r t❤❛♥ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡✱ ✐t ✐♥❝r❡❛s❡s✳ ❚❤❡ ❡q✉❛t✐♦♥ ❣✐✈❡s ✉s ❛ s♦❧✉t✐♦♥ ✈✐❛ t❤✐s r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ✭❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✮✿

u(tn+1 ) = u(tn ) + K(r − u(tn )) · ∆t , ❝♦♠❜✐♥❡❞ ✇✐t❤

tn+1 = tn + ∆t . ❚❤❡s❡ ❢♦r♠✉❧❛s ❛r❡ ✉s❡❞ ❢♦r s✐♠✉❧❛t✐♦♥s✳ ❲❡ ❝❛♥ s❡❡ ❜❡❧♦✇ s❡✈❡r❛❧ ♣♦ss✐❜❧❡ s♦❧✉t✐♦♥s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❛ ❝✉♣ ♦❢ ❝♦✛❡❡ ✐s ❝♦♦❧✐♥❣ ❞♦✇♥ ❛♥❞ ❛ ❝❛♥ ♦❢ s♦❞❛ ✐s ✇❛r♠✐♥❣ ✉♣✿

❇♦t❤ t❡♠♣❡r❛t✉r❡s ❛r❡ ❛♣♣r♦❛❝❤✐♥❣ t❤❛t ♦❢ t❤❡ r♦♦♠ ❛s t❤❡② ❡①❝❤❛♥❣❡ t❤❡ ❤❡❛t ✇✐t❤ t❤❡ s✉rr♦✉♥❞✐♥❣ ❛✐r ❛❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠ ✐♥ ❈❤❛♣t❡r ✶✳ ❚❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡ r❡♠❛✐♥s ❝♦♥st❛♥t ❜❡❝❛✉s❡ t❤❡ ❛♠♦✉♥t ♦❢ ❤❡❛t ✐s t♦♦ ❧❛r❣❡ ✐♥ ❝♦♠♣❛r✐s♦♥✳ ❲❡ ♣r♦❝❡❡❞ ❛s ❢♦❧❧♦✇s✳

✻✳✶✳

❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts

❲❡ ✜rst t❛❦❡ t❤✐s ♠♦❞❡❧ ❛s t❤❛t ♦❢

✸✹✺

t✇♦ ♦❜❥❡❝ts ✐♥ t❤❡ r♦♦♠ ❛t t❤❡ s❛♠❡ t✐♠❡✿ s♦❞❛

❛✐r

❝♦✛❡❡

❲❡ ❤❛✈❡ t✇♦ ❢✉♥❝t✐♦♥s✿

u

✐s t❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ s♦❞❛ ❛♥❞

v

✐s t❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ❝♦✛❡❡✳

❚❤❡♥✱ ✇❡ ❤❛✈❡ t✇♦ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❡❛❝❤ ♦t❤❡r✿

u(tn+1 ) = u(tn ) + K(r − u(tn )) · ∆t ◆❡①t st❡♣✿ ❲❤❛t ✐❢ t❤❡ t✇♦ ♦❜❥❡❝ts ❛r❡

❛❞❥❛❝❡♥t

❛♥❞

v(tn+1 ) = v(tn ) + K(r − v(tn )) · ∆t .

t♦ ❡❛❝❤ ♦t❤❡r❄ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ t✇♦ t♦✉❝❤ ❡❛❝❤ ♦t❤❡r

✇❤✐❧❡ ✐♥s✉❧❛t❡❞ ❡❧s❡✇❤❡r❡ ✭♥♦ ❛✐r✮✿ s♦❞❛

❝♦✛❡❡

❚❤❡♥ t❤❡ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ♦❜❥❡❝ts ❡①❝❤❛♥❣❡s t❤❡ ❤❡❛t ✇✐t❤ t❤❡ ♦t❤❡r ❛♥❞ ♥♦t❤✐♥❣ ❡❧s❡✳ ❚❤❡ ❧❛✇ st✐❧❧ ❛♣♣❧✐❡s✿



✏❚❤❡ r❛t❡ ♦❢ ❤❡❛t ❡①❝❤❛♥❣❡ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❝✉rr❡♥t t❡♠♣❡r❛t✉r❡ ❛♥❞

t❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ❛❞❥❛❝❡♥t ♦❜❥❡❝t✑✳

❆t ❡❛❝❤ st❡♣✱ t❤❡r❡❢♦r❡✱ t❤❡ ❞②♥❛♠✐❝s r❡♠❛✐♥s t❤❡ s❛♠❡ ❛s ❜❡❢♦r❡ ❜✉t t❤❡ r❡❢❡r❡♥❝❡ ✐s t♦ t❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ♦t❤❡r ♦❜❥❡❝t r❛t❤❡r t❤❛♥ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡✳ ❲❡ t❛❦❡ t❤❡ t✇♦ ❢♦r♠✉❧❛s ❢♦r♠✉❧❛s ❛❜♦✈❡ ❛♥❞ ❥✉st r❡♣❧❛❝❡



❢♦r s♦❞❛



❢♦r ❝♦✛❡❡

u✿ r

r❡♣❧❛❝❡❞ ✇✐t❤ ❝♦✛❡❡

v✿ r

r❡♣❧❛❝❡❞ ✇✐t❤ s♦❞❛

r

✭❛✐r✮ ✇✐t❤ t❤❡ ♦t❤❡r ♦❜❥❡❝t✬s t❡♠♣❡r❛t✉r❡✿

v✱ u✳

❚❤✐s ✐s t❤❡ r❡s✉❧t✿

 u(tn+1 ) = u(tn ) + k v(tn ) − u(tn ) · ∆t

❛♥❞

 v(tn+1 ) = v(tn ) + k u(tn ) − v(tn ) · ∆t .

❲❡ ❝❛♥ s❡❡ t❤❛t t❤❡ ❤❡❛t ❛❞❞❡❞ t♦ ♦♥❡ ✐s t❤❡ ❤❡❛t t❛❦❡♥ ❢r♦♠ t❤❡ ♦t❤❡r❀ t❤❛t

❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❊♥❡r❣② ✦

❊①❛♠♣❧❡ ✻✳✶✳✶✿ s♦❞❛✲❝♦✛❡❡ ▲❡t✬s ❝❛rr② ♦✉t t❤✐s ♣❧❛♥ ✇✐t❤ ❛

s♣r❡❛❞s❤❡❡t✳

❲❡ ❝❤♦♦s❡ t❤❡ ✐♥✐t✐❛❧ t❡♠♣❡r❛t✉r❡s✿ s♦❞❛

k = .1

40 ❛♥❞ ❝♦✛❡❡ 100 ❞❡❣r❡❡s❀ t❤❡② ❛r❡ ✐♥ t❤❡ ✜rst r♦✇✳

❲❡ ❝❤♦s❡

❢♦r ❛ s❧♦✇ ❡①❝❤❛♥❣❡✳ ❚❤❡ t✇♦ ❢♦r♠✉❧❛s ❛r❡ s✐♠✐❧❛r✿

❂❘❬✲✶❪❈✰✵✳✶✯✭❘❬✲✶❪❈❬✶❪✲❘❬✲✶❪❈✮✯❘✷❈

❛♥❞

❂❘❬✲✶❪❈✰✵✳✶✯✭❘❬✲✶❪❈❬✲✶❪✲❘❬✲✶❪❈✮✯❘✷❈

❇❡❧♦✇ ✐s t❤❡ ♣r❡❝❡❞❡♥❝❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡s❡ ❢♦r♠✉❧❛s✿

✻✳✶✳

❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts

✸✹✻

❆s ❡①♣❡❝t❡❞✱ t❤❡ ❞②♥❛♠✐❝s ✐s ❛❧♠♦st ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ♦r✐❣✐♥❛❧✿

❚❤❡ ❞✐✛❡r❡♥❝❡ ✐s t❤❛t t❤❡ t❡♠♣❡r❛t✉r❡s ❝♦♥✈❡r❣❡ t♦ t❤❡ ♠✐❞❞❧❡ ✐♥st❡❛❞ t♦ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡✿ 40 + 100 = 70 . 2

❊①❡r❝✐s❡ ✻✳✶✳✷ Pr♦✈❡ t❤❡ ❧✐♠✐t✳ ❖♥❡ st❡♣ ❢✉rt❤❡r✳ ❲❤❛t ✐❢ t❤❡r❡ ❛❧s♦ ❛

❝✉♣ ♦❢ t❡❛ ❄

❲❡ ❛ss✉♠❡ ❛❣❛✐♥ t❤❛t t❤❡ t❤r❡❡ t♦✉❝❤ ❡❛❝❤ ♦t❤❡r ✇❤✐❧❡ ✐♥s✉❧❛t❡❞ ❡❧s❡✇❤❡r❡✿ s♦❞❛ ❝♦✛❡❡ t❡❛ ❚❤❡ ❢✉♥❝t✐♦♥s ❛r❡ r❡s♣❡❝t✐✈❡❧②✿ u v

w

❚❤❡♥ t❤❡ s♦❞❛ ❡①❝❤❛♥❣❡s ❤❡❛t ✇✐t❤ t❤❡ ❝♦✛❡❡ ❥✉st ❛s ❜❡❢♦r❡ ❛♥❞ t❤❡ t❡❛ ❛❧s♦ ❡①❝❤❛♥❣❡s ❤❡❛t ✇✐t❤ t❤❡ ❝♦✛❡❡ ❥✉st ❛s ❜❡❢♦r❡✳ ❊✐t❤❡r ❤❛s ♦♥❧② ♦♥❡ ♥❡✐❣❤❜♦r✦ ❲❡ ✇r✐t❡ ❢♦r t❤❡s❡ t✇♦✿  u(tn+1 ) = u(tn ) + k v(tn ) − u(tn ) · ∆t

 w(tn+1 ) = w(tn ) + k v(tn ) − w(tn ) · ∆t .

❚❤❡ ❝♦✛❡❡ ✐s ♥♦✇ ✐♥ ❛ ♥❡✇ ♣♦s✐t✐♦♥✳ ■t ❤❛s t✇♦ ♥❡✐❣❤❜♦rs✦ ❙✐♥❝❡ ✇❡ ❡①♣❡❝t t❤❡ ❤❡❛t ❛❞❞❡❞ t♦ t❤❡ ❝♦✛❡❡ ✐s t❤❡ ❤❡❛t t❛❦❡♥ ❢r♦♠ t❤❡ t✇♦ ♦t❤❡rs✱ ✇❡ s✐♠♣❧② ❝♦♣② t❤❡ t❡r♠s ❢r♦♠ t❤❡ ❛❜♦✈❡ ❢♦r♠✉❧❛s✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿   v(tn+1 ) = v(tn ) + k u(tn ) − v(tn ) · ∆t + k w(tn ) − v(tn ) · ∆t .

❊①❛♠♣❧❡ ✻✳✶✳✸✿ s♦❞❛✲❝♦✛❡❡✲t❡❛

❲❡ ❝❤♦♦s❡ t❤❡ ✐♥✐t✐❛❧ t❡♠♣❡r❛t✉r❡s✿ s♦❞❛ 40✱ ❝♦✛❡❡ 100✱ ❛♥❞ t❡❛ 85 ❞❡❣r❡❡s✳ ❚❤❡ t✇♦ ❢♦r♠✉❧❛s t❤❡ s♦❞❛ ❛♥❞ t❤❡ t❡❛ ❛r❡ t❤❡ s❛♠❡ ❛s ❜❡❢♦r❡✿ ❂❘❬✲✶❪❈✰✵✳✶✯✭❘❬✲✶❪❈❬✶❪✲❘❬✲✶❪❈✮✯❘✷❈ ❛♥❞ ❂❘❬✲✶❪❈✰✵✳✶✯✭❘❬✲✶❪❈❬✲✶❪✲❘❬✲✶❪❈✮✯❘✷❈

❚❤❡② t❛❦❡ ❢r♦♠ t❤❡✐r r✐❣❤t ❛♥❞ ❧❡❢t ♥❡✐❣❤❜♦rs r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ❢♦r♠✉❧❛s t❤❡ ❝♦✛❡❡ ✐s ♥❡✇✿ ❂❘❬✲✶❪❈✰✵✳✶✯✭❘❬✲✶❪❈✲❘❬✲✶❪❈❬✲✶❪✮✯❘✷❈✶✰✵✳✶✯✭❘❬✲✶❪❈✲❘❬✲✶❪❈❬✶❪✮✯❘✷❈✶

✻✳✶✳

❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts

✸✹✼

■t t❛❦❡s ❢r♦♠ t❤❡✐r ❧❡❢t ❛♥❞ r✐❣❤t ♥❡✐❣❤❜♦rs ❛t t❤❡ s❛♠❡ t✐♠❡✳ ❚❤❡s❡ ❛r❡ t❤❡ ✜rst r♦✇s✿

❚❤❡ ❞②♥❛♠✐❝s ✐s s✐♠✐❧❛r t♦ t❤❡ ❧❛st✿

❚❤❡ s♦❞❛ ✐s ✇❛r♠✐♥❣ ✉♣✱ t❤❡ ❝♦✛❡❡ ✐s ❝♦♦❧✐♥❣ ❞♦✇♥✱ ✇❤✐❧❡ t❤❡ t❡❛ ✜rst ✐s ✇❛r♠✐♥❣ ✉♣ ✭❜❡❝❛✉s❡ t❤❡ ❛❞❥❛❝❡♥t ❝♦✛❡❡ ✐s ❤♦tt❡r✮ ❛♥❞ t❤❡♥ ❝♦♦❧✐♥❣ ❞♦✇♥ ✭❛s t❤❡ ❝♦✛❡❡ ✐s✮✳ ❚❤❡② ❛r❡ ❛❧❧ ❝♦♥✈❡r❣✐♥❣ ♦♥ t❤❡ ❛✈❡r❛❣❡ t❡♠♣❡r❛t✉r❡✿ 40 + 100 + 85 = 75 . 3

◆❡①t st❡♣✿ ❲❡ ✐♠❛❣✐♥❡ t❤❛t t❤❡r❡ ✐s ❛ s❡r✐❡s ♦❢ ♦❜❥❡❝ts t❤❛t ❡①❝❤❛♥❣❡ t❤❡✐r ❤❡❛t ✇✐t❤ t❤❡✐r ♥❡✐❣❤❜♦rs✳ ❊①❛♠♣❧❡ ✻✳✶✳✹✿ t❡♥ ❝❛♥s

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ 10 ♦❜❥❡❝ts ✭♦r ❝♦♥t❛✐♥❡rs✮ ✇✐t❤ ❝♦❧❞ ❡♥❞s ❛♥❞ ❤♦t ❝❡♥t❡r✳ ❚❤❡ ❧❛st ❢♦r♠✉❧❛ ✐s r❡♣❡❛t❡❞ ✭❡✐❣❤t t✐♠❡s✮ ✐♥ ❛ s♣r❡❛❞s❤❡❡t ✇✐t❤ 10 ❝♦❧✉♠♥s✳ ❲❡ ❝♦❧♦r t❤❡ ❝❡❧❧s t♦ s❡❡ t❤❡ ❞②♥❛♠✐❝s✿

❚❤❡ ❞②♥❛♠✐❝s ✐s ❛s ❡①♣❡❝t❡❞✿ ❛✈❡r❛❣✐♥❣ ♦❢ t❤❡ t❡♠♣❡r❛t✉r❡✳ ❲❡ ❛❧s♦ r❡❛❧✐③❡ t❤❛t t❤✐s ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✦ ■t ❝❛♥ ❛❧s♦ ❜❡ ✈✐s✉❛❧✐③❡❞ ❜② ✐ts ❣r❛♣❤✿

✻✳✶✳

❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts

✸✹✽

❆♥♦t❤❡r ❡①♣❡r✐♠❡♥t✱ ✇✐t❤ ❤♦t ❛♥❞ ❝♦❧❞ ❡♥❞s✿

❖♥❝❡ ❛❣❛✐♥✱ ❝♦♥✈❡r❣❡♥❝❡ t♦✇❛r❞ t❤❡ ❛✈❡r❛❣❡ t❡♠♣❡r❛t✉r❡✦ ❊①❡r❝✐s❡ ✻✳✶✳✺

■♠♣❧❡♠❡♥t ❤❡❛t tr❛♥s❢❡r ✐♥ ❛ ❝✐r❝✉❧❛r ❛rr❛♥❣❡♠❡♥t✳ ❚❤❡ ♠♦❞❡❧ ✐s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❛ ♠❡t❛❧

r♦❞ ✿

❚❤✐s ✐s ❛♥ ✐♥s✉❧❛t❡❞ r♦❞ ❛♥❞ ♠✐❣❤t ❜❡ ❤❡❛t❡❞ ❛t ♦♥❡ ❡♥❞ ❛s ❛❜♦✈❡✳ ❚❤❡ t❡♠♣❡r❛t✉r❡ t❤❡♥ ✈❛r✐❡s ❛❧♦♥❣ ✐ts ❧❡♥❣t❤ ❡✈❡♥ ✇❤❡♥ t❤❡ t✐♠❡ ✐s ✜①❡❞✳ ❲❡ ♥❡❡❞ ❛♥♦t❤❡r ✈❛r✐❛❜❧❡✦ ❲❡ r❡✲✐♥tr♦❞✉❝❡ t❤❡ ✈❛r✐❛❜❧❡s✿ ✶✳ t ✐s t❤❡ t✐♠❡✳ ✷✳ x ✐s t❤❡ ❧♦❝❛t✐♦♥✳ ✸✳ u ✐s t❤❡ t❡♠♣❡r❛t✉r❡✳ ❆❧❧ t❤r❡❡ ❛r❡ ❥✉st ♥✉♠❜❡rs✳ ❍♦✇ ❞♦ t❤❡② ❞❡♣❡♥❞ ♦♥ ❡❛❝❤ ♦t❤❡r❄ ❚❤❡r❡ ✐s ♥♦ ♠♦t✐♦♥✦ ❚❤❡r❡❢♦r❡✱ ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡ t ♦♥❡ ❝❛♥ ❜❡ ❛t ❡✈❡r② ❧♦❝❛t✐♦♥ x ❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ❚❤✐s ♠❡❛♥s t❤❛t t ❛♥❞ x ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❡❛❝❤ ♦t❤❡r✳ ◆❡①t✱ ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡ t ❛♥❞ ❛t ❡✈❡r② ❧♦❝❛t✐♦♥ x ✇❡ ❝❛♥ ♠❡❛s✉r❡ t❤❡ t❡♠♣❡r❛t✉r❡ u✳ ❚❤✐s ♠❡❛♥s t❤❛t u ❞❡♣❡♥❞s ♦♥ t ❛♥❞ x✳ ❙♦✱ ✇❡ ❤❛✈❡ t❤❡

✻✳✶✳

❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts

✸✹✾

❢♦❧❧♦✇✐♥❣ ❞❡♣❡♥❞❡♥❝❡ ❞✐❛❣r❛♠✿

t ց x ❲❡ s✐♠♣❧② ❤❛✈❡ ❛

u

ր

❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✿ u = f (t, x) .

❇❡❢♦r❡ ❞❡✈❡❧♦♣✐♥❣ ❛ ♠♦❞❡❧ ❛♥❞ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❢♦r t❤✐s ❢✉♥❝t✐♦♥✱ ✇❤❛t ✐s ❛ r❡❛s♦♥❛❜❧❡ s♦❧✉t✐♦♥ t❤❛t ✇❡ ❝❛♥ ❡♥✈✐s✐♦♥❄ ▲❡t✬s s❡t ✉♣ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳ ❙✉♣♣♦s❡ t❤❡ r♦❞ ✐s ❝♦♦❧ ✕ ♥❡❣❛t✐✈❡ ✕ ❛t ♦♥❡ ❡♥❞ ❛♥❞ ❤♦t ✕ ♣♦s✐t✐✈❡ ✕ ❛t t❤❡ ♦t❤❡r✳ ❋✐rst ✇❡ ✐♠❛❣✐♥❡ t❤❛t ✇❡ s✐t ♥❡①t t♦ ❛ ♣❛rt✐❝✉❧❛r ❧♦❝❛t✐♦♥ ♦♥ t❤❡ r♦❞ ❛♥❞ ♦❜s❡r✈❡ t❤❡ ❞②♥❛♠✐❝s ♦❢ t❡♠♣❡r❛t✉r❡ ♦✈❡r

t✐♠❡✳

❏✉❞❣✐♥❣ ❜② ◆❡✇t♦♥✬s ▲❛✇✱ t❤❡ t❡♠♣❡r❛t✉r❡ ✇✐❧❧ ❣r❛❞✉❛❧❧② ❝♦♥✈❡r❣❡ t♦ ❛ ♣❛rt✐❝✉❧❛r ✈❛❧✉❡ ♦✈❡r

t✐♠❡✳ ❙✉♣♣♦s❡ t❤✐s ✈❛❧✉❡ ✐s

0✿

◆♦✇ ✇❡ ✐♠❛❣✐♥❡ t❤❛t ✇❡ ✏❢r❡❡③❡✑ t✐♠❡ ❛♥❞ r✉♥ ❛❧♦♥❣ t❤❡ r♦❞ ♦❜s❡r✈✐♥❣ ❤♦✇ t❤❡ t❡♠♣❡r❛t✉r❡ ✐s ❝❤❛♥❣✐♥❣ ♦✈❡r

s♣❛❝❡✳

❙✉♣♣♦s❡ t❤❡ t❡♠♣❡r❛t✉r❡ ✈❛r✐❡s ❧✐♥❡❛r❧② ❢r♦♠ ♦♥❡ ❡♥❞ t♦ t❤❡ ♦t❤❡r✳ ❏✉❞❣✐♥❣ ❜② t❤❡ ❈♦♥s❡r✈❛t✐♦♥ ♦❢

❊♥❡r❣② ▲❛✇✱ t❤❡ t❡♠♣❡r❛t✉r❡ ✇✐❧❧ ❣r❛❞✉❛❧❧② ❛✈❡r❛❣❡ ✐ts❡❧❢ ✇✐t❤ t✐♠❡✳ ■t ✐s ❝♦♥❝❡✐✈❛❜❧❡ t❤❛t t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ t❡♠♣❡r❛t✉r❡ ✇✐❧❧ r❡♠❛✐♥ ❧✐♥❡❛r✿

❲❡ ❝❛♥ tr② t♦ ✐♠❛❣✐♥❡ ✇❤❛t t❤❡ ❢✉♥❝t✐♦♥

f

✐s ❧✐❦❡✳ ❚❤❡ ❖❉❊ ❤❛s t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛② s♦❧✉t✐♦♥s✿

u = r + Ce−kt . ❚♦ ❣❡t t❤❡ ❣❡♥❡r❛❧ s❤❛♣❡✱ ✇❡ ❝❛♥ tr②✿

u = xe−t

♦r

u = x/t .

✻✳✶✳

❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts

✸✺✵

▲❡t✬s ❝♦❧❧❡❝t t❤❡ ❞❛t❛ ❢♦r t❤✐s ❢✉♥❝t✐♦♥✿ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts✳ ❚❤❡ ✐♥♣✉ts ❜❡✐♥❣ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❡❛❝❤ ♦t❤❡r ❢♦r♠ ❛♥

❛rr❛②✱ s❛② x ✐♥ r♦✇s ❛♥❞ t ✐♥ ❝♦❧✉♠♥s✿

t\x 0.0 1.5 2.0 2.5 0 1 2 ■♥ ♦t❤❡r ✇♦r❞s✱ ♦✉r ❢✉♥❝t✐♦♥

f

❤❛s t❤❡

(t, x)✲♣❧❛♥❡

♦r ✐ts s✉❜s❡t ❢♦r ❛ ❞♦♠❛✐♥✳

◆♦✇ t❤❡ ♦✉t♣✉ts✳ ❲❡ ❥✉st ♣❧❛❝❡ t❤❡♠ ❛t ❡❛❝❤ ♦❢ t❤❡ ❝❡❧❧s ♦❢ t❤❡ ❛❜♦✈❡ t❛❜❧❡✿

t\x 0 1 2

0 1 3 0

1.5 2 4 0

2 2.5 0 0 −1

P❧♦tt✐♥❣ t❤❡ s♦❧✉t✐♦♥s t♦ ♦✉r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ♠❛t❝❤❡s t❤✐s ❞❡s❝r✐♣t✐♦♥✿

❚❤✐s t✐♠❡✱ ❛❧❧ ❜✉t t❤❡ ♦❜❥❡❝ts ❛t t❤❡ ❡♥❞s ❢♦❧❧♦✇ t❤❡ s❛♠❡ ❢♦r♠✉❧❛ t❤❛t r❡❢❡rs t♦ ✐ts

t✇♦ ♥❡✐❣❤❜♦rs✿

  v(ti+1 ) = v(ti ) + k u(ti ) − v(ti ) · ∆t + k w(ti ) − v(ti ) · ∆t .

❲❡ r❡✇r✐t❡ t❤❡ ❢♦r♠✉❧❛ ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳ ❙♦✱ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ ♦❢

t

❛♥❞

x✳

❚❤❡ t✐♠❡ ✐s ❛s ❜❡❢♦r❡✳ ❚❤❡s❡ ❛r❡ t❤❡ ♥♦❞❡s ♦❢ ♦✉r ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡

t✲❛①✐s✿

..., ti−1 , ti , ti+1 , ..., ✇✐t❤ t❤❡ ✐♥❝r❡♠❡♥t✿

∆t = ti+1 − ti . ❚❤❡ s♣❛❝❡ ✐s s✐♠✐❧❛r✳ ❚❤❡s❡ ❛r❡ t❤❡ ♥♦❞❡s ♦❢ ♦✉r ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡

x✲❛①✐s

✭t❤❡ r♦❞✮✿

..., xn−1 , xn , xn+1 , ... ❲❡ ❛r❡ ♠❛❦✐♥❣ ❛ s✇✐t❝❤ ❢r♦♠ t❤r❡❡ ❢✉♥❝t✐♦♥s ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡s ♦♥❡ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❤❛✈❡ t❤✐s ❝♦rr❡s♣♦♥❞❡♥❝❡ t♦ t❤❡ ♣r❡✈✐♦✉s ♥♦t❛t✐♦♥✿

u(·) v(·) w(·) ... xn−1 xn xn+1 ... ... u(·, xn−1 ) u(·, xn ) u(·, xn+1 ) ... ... −•− −•− −•− ... ❚❤❡ ❤❡❛t ✐s ❡①❝❤❛♥❣❡❞ ❜❡t✇❡❡♥ t❤❡s❡ ✏❝♦♥t❛✐♥❡rs✑ t❤r♦✉❣❤ t❤❡s❡ ✏♣✐♣❡s✑✳

✻✳✶✳ ❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts ❚❤❡♥ ♦✉r ❢♦r♠✉❧❛✱ ❜❡❝♦♠❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿

✸✺✶

  v(ti+1 ) = v(ti ) + k u(ti ) − v(ti ) · ∆t + k w(ti ) − v(ti ) · ∆t

  u(ti+1 , xn ) = u(ti , xn ) + k u(ti , xn−1 ) − u(ti , xn ) · ∆t + k u(ti , xn+1 ) − u(ti , xn ) · ∆t .

❚❤❡ ❢♦r♠✉❧❛ ❝❛♥ ❜❡ ✉s❡❞ ❢♦r s✐♠✉❧❛t✐♦♥ ❛s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✳ ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❢♦r♠✉❧❛❄ ❋✐rst✱ ✇❡ ♥♦t✐❝❡ t❤❡ ❞✐✛❡r❡♥❝❡ ✇✐t❤ r❡s♣❡❝t t♦ t✿

∆t u(ti , xn ) = u(ti+1 , xn ) − u(ti , xn ) .

❲❡ ❛❞❞ ❛ s✉❜s❝r✐♣t t t♦ ✐♥❞✐❝❛t❡ t❤❡ ✈❛r✐❛❜❧❡ ❥✉st ❛s ✇✐t❤ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s✦ ❚❤❡♥✱ ♦✉r ❢♦r♠✉❧❛ ❜❡❝♦♠❡s✿   ∆t u(ti , xn ) = k u(ti , xn−1 ) − u(ti , xn ) · ∆t + k u(tn , xi+1 ) − u(ti , xn ) · ∆t .

❚❤❡ t✇♦ t❡r♠s ❛r❡ t❤❡ ❛♠♦✉♥ts ♦❢ ❤❡❛t ❡①❝❤❛♥❣❡❞ ✇✐t❤ t❤❡ t✇♦ ♥❡✐❣❤❜♦rs✳ ❉✐✈✐❞✐♥❣ ❜② ∆t s✐♠♣❧✐✜❡s t❤✐♥❣s✿

  ∆t u(ti , xn ) = k u(ti , xn−1 ) − u(ti , xn ) + u(ti , xn+1 ) − u(ti , xn ) . ∆t

❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ t❡r♠ ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡❄ ■s t❤✐s t❤❡ s✉♠ ♦❢ t✇♦ ❞✐✛❡r❡♥❝❡s❄✦ ◆♦✳ ❋✐rst✱ t❤❡s❡ ❛r❡ ❞✐✛❡r❡♥❝❡s ❜✉t ♥♦t ❞✐✛❡r❡♥❝❡s ∆✳ ❚❤❡ t❡♠♣❡r❛t✉r❡ ❛t t❤❡ ❧♦❝❛t✐♦♥ ✐s s✉❜tr❛❝t❡❞ ❢r♦♠ t❤❛t ♦❢ ❛ ♥❡✐❣❤❜♦r✱ ♥♦ ♠❛tt❡r ❧❡❢t ♦r r✐❣❤t✱ ❛ s♠❛❧❧❡r ♦r ❧❛r❣❡r ♣♦s✐t✐♦♥ ✇✐t❤✐♥ t❤❡ x✲❛①✐s✳ ❚❤✐s ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ✇✐t❤ r❡s♣❡❝t t♦ x✿ ∆x u(ti , xn ) = u(ti , xn+1 ) − u(ti , xn ) .

❘✐❣❤t ♠✐♥✉s ❧❡❢t✦ ▲❡t✬s ✜♥❞ t❤❡ ❞✐✛❡r❡♥❝❡s ✇✐t❤ r❡s♣❡❝t t♦ s♣❛❝❡ ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✭· st❛♥❞s ❢♦r ti ✮✿   u ·, xn−1 ) − u(·, xn + u ·, xn+1 ) − u(·, xn    = − u(·, xn − u(·, xn−1 )] + u(·, xn+1 ) − u(·, xn )     = − ∆x u(·, xn ) + ∆x u(·, xn+1 ) .

❚❤✐s ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡✦

■t ♠❛❦❡s s❡♥s❡✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t❤r❡❡ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts ✇✐t❤ t❤❡s❡ t❡♠♣❡r❛t✉r❡s✿ 4

7

9

❚❤❡ ❤❡❛t ✢♦✇s ❢r♦♠ r✐❣❤t t♦ ❧❡❢t✿ 4 ← ← 7 ← ← 9

• ❲✐❧❧ t❤❡ t❡♠♣❡r❛t✉r❡ ❣♦ ✉♣ ♦r ❞♦✇♥ ✐♥ t❤❡ ❧❡❢t ❝❡❧❧❄ ❯♣✦ ❲❤②❄ ❇❡❝❛✉s❡ t❤❡ ♥❡✐❣❤❜♦r ✐s ✇❛r♠❡r✳

• ❲✐❧❧ t❤❡ t❡♠♣❡r❛t✉r❡ ❣♦ ✉♣ ♦r ❞♦✇♥ ✐♥ t❤❡ r✐❣❤t ❝❡❧❧❄ ❉♦✇♥✦ ❲❤②❄ ❇❡❝❛✉s❡ t❤❡ ♥❡✐❣❤❜♦r ✐s ❝♦♦❧❡r✳

♠✐❞❞❧❡ ❝❡❧❧❄ ❉♦✇♥✦ ❲❤②❄ ❚❤❡r❡ ❛r❡ t✇♦ ♥❡✐❣❤❜♦rs✱ ♦♥❡ ❝♦♦❧❡r ❛♥❞ t❤❡ ♦t❤❡r ✇❛r♠❡r✳✳✳ ❚❤❡② ❤❛✈❡ ♦♣♣♦s✐t❡ ❡✛❡❝ts ♦♥ t❤✐s ❝❡❧❧✦ ❙♦✱ ✇❤② ❞♦✇♥❄

• ❲✐❧❧ t❤❡ t❡♠♣❡r❛t✉r❡ ❣♦ ✉♣ ♦r ❞♦✇♥ ✐♥ t❤❡

❚❤✐s ✐s ✇❤②✿

↑ 4 ↑ ↓ 7 ↓ ↓ 9 ↓

✻✳✶✳

❍❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝ts ◮

❇❡❝❛✉s❡

7

✐s ❝❧♦s❡r t♦

9

t❤❛♥ t♦

✸✺✷

4✳

❚❤❡ ❤❡❛t ✢♦✇s ❢r♦♠ r✐❣❤t t♦ ❧❡❢t ❜✉t t❤❡ ♠✐❞❞❧❡ ♦♥❡ ❧♦s❡s ♠♦r❡ t❤❛♥ ✐t ❣✐✈❡s✿

4 ⇔ ⇔ 7 ← ← 9 ▲❡t✬s t❛❦❡ t❤✐s ❛♣❛rt✳ ✏❈❧♦s❡✑ ✐s ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡s✳ ✏❈❧♦s❡r✑ ✐s✱ t❤❡r❡❢♦r❡✱ ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥❝❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡s✦ ❚❤✐s ✐❞❡❛ ✐s♥✬t ♥❡✇❀ ✐t✬s ❛❧❧ ❛❜♦✉t

❝♦♥❝❛✈✐t②

✭❛♥❞✱ ❡✈❡♥t✉❛❧❧②✱ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✱ ❈❤❛♣t❡r ✷❉❈✲✹✮ ♦❢ t❤❡

❢✉♥❝t✐♦♥✿

▲❡t✬s ✜♥❞ t❤❡ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ✇✐t❤ r❡s♣❡❝t t♦ s♣❛❝❡ ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ♦✉r ❡q✉❛t✐♦♥✿

−∆x u(·, xn ) + ∆x u(·, xn+1 ) = ∆x ∆x u(·, xn ) . ❲❡ ❤❛✈❡ t❤❡r❡❢♦r❡✿

∆t u(ti , xn ) = k∆x ∆x u(ti , xn ) . ∆t ❲❛r♥✐♥❣✦

❚❤❡ s✐♠♣❧✐✜❝❛t✐♦♥ ♦♥❧② ✇♦r❦s ✇❤❡♥ k ✐s ❝♦♥st❛♥t✳ ❙♦✱ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❝♦♥❝❛✈✐t② ❞❡t❡r♠✐♥❡s ❜② ❤♦✇ ♠✉❝❤ t❤❡ ✈❛❧✉❡ ✐s ♣✉s❤❡❞ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✿

❚❤❡ r❡s✉❧t ✐s ❛✈❡r❛❣✐♥❣✳ ❚❤✐s ✐s t❤❡ s❤♦rt❡♥❡❞ ✈❡rs✐♦♥✿

∆t u = k∆2x u ∆t ■t ✐s ❝❛❧❧❡❞ t❤❡

❤❡❛t ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✳

❚❤❡ r❡❛❧✐t② ✐s ♠♦r❡ ❝♦♠♣❧❡①✿



❚❤❡ tr❛♥s❢❡r r❛t❡ ♠✐❣❤t ❜❡ ❞✐✛❡r❡♥t ❛t ❞✐✛❡r❡♥t ❧♦❝❛t✐♦♥s✳

✻✳✷✳ ❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ ♣❡r♠❡❛❜✐❧✐t②

✸✺✸

✻✳✷✳ ❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ ♣❡r♠❡❛❜✐❧✐t②

❲❡ st❛rt ✐♥ t❤✐s s❡❝t✐♦♥ ✇✐t❤ ❤❡❛t tr❛♥s❢❡r ✇✐t❤✐♥ ❛ ♠❡t❛❧ r♦❞✳ ■t✬s ❤❡❛t✲tr❛♥s❢❡rr✐♥❣ ♣r♦♣❡rt✐❡s ♠✐❣❤t ❜❡ ♥♦♥✲✉♥✐❢♦r♠✿

❚❤❡ r♦❞ ✐s s❡❡♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤②♣♦t❤❡t✐❝❛❧ ✇❛②✿ ❛s ✐❢ ✐t✬s s♣❧✐t ✐♥t♦ ♣✐❡❝❡s✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ❤❡❛t ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ str✐♥❣ ♦❢ ❝♦♥t❛✐♥❡rs ❛♥❞ ❡❛❝❤ ❝♦♥t❛✐♥❡r ❡①❝❤❛♥❣❡s t❤❡ ❤❡❛t ✇✐t❤ ✐ts t✇♦ ♥❡✐❣❤❜♦rs t❤r♦✉❣❤ t❤❡ ♣✐♣❡s✿

❍❡r❡ ✇❡ ❤❛✈❡✿ • x, y, z ❛r❡ s♦♠❡ ♦❢ t❤❡ r♦♦♠s✳ • a, b ❛r❡ s♦♠❡ ♦❢ t❤❡ ♣✐♣❡s✳

❖❢ ❝♦✉rs❡✱ ✇❡ r❡♣❧❛❝❡ t❤❡ ❝♦♥t❛✐♥❡rs ✇✐t❤ ♥♦❞❡s ❛♥❞ ♣✐♣❡s ✇✐t❤ ❡❞❣❡s✳ ❆ ❝❛r❡❢✉❧ ❧♦♦❦ r❡✈❡❛❧s t❤❛t t♦ ♠♦❞❡❧ ❤❡❛t tr❛♥s❢❡r✱ ✇❡ ♥❡❡❞ t♦ s❡♣❛r❛t❡❧② r❡❝♦r❞ t❤❡ ❡①❝❤❛♥❣❡ ♦❢ ❤❡❛t ♦❢ ❡❛❝❤ ❝♦♥t❛✐♥❡r ✇✐t❤ ❡❛❝❤ ♦❢ t❤❡ ❛❞❥❛❝❡♥t ❝♦♥t❛✐♥❡rs✳ ❚❤❡ ♣r♦❝❡ss ✇❡ st✉❞② ♦❜❡②s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛♠✐❧✐❛r ❧❛✇ ♦❢ ♣❤②s✐❝s✳ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ❈♦♦❧✐♥❣✿ ❚❤❡ r❛t❡ ♦❢ ❝♦♦❧✐♥❣ ♦❢ ❛♥ ♦❜❥❡❝t ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ✐ts t❡♠♣❡r❛t✉r❡ ❛♥❞ t❤❡ t❡♠♣❡r❛t✉r❡ ❛♥ ❛❞❥❛❝❡♥t ♦❜❥❡❝t✳ ❖❢ ❝♦✉rs❡✱ ❝♦♦❧✐♥❣ ♠❡❛♥s ❤❡❛t✐♥❣ ✇❤❡♥ t❤❡ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ♦t❤❡r ♦❜❥❡❝t ✐s ❤✐❣❤❡r✳ ❖✉r ✐♥✐t✐❛❧ ❛ss✉♠♣t✐♦♥ ✐s t❤❛t ❛❧❧ ❝♦♥t❛✐♥❡rs ❤❛✈❡ ❡q✉❛❧ s✐③❡✳ ■t ❢♦❧❧♦✇s t❤❛t t❤❡ ❛♠♦✉♥t ♦❢ ❤❡❛t ✐♥ t❤❡ ❝♦♥t❛✐♥❡rs ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡✐r r❡s♣❡❝t✐✈❡ t❡♠♣❡r❛t✉r❡s✳ ❚❤✐s ✐s ✇❤② ✇❡ ❝❛♥ ✉♥❞❡rst❛♥❞ t❤❡ ❧❛✇ ♦❢ ❝♦♦❧✐♥❣ ❛s ❢♦❧❧♦✇s✿ ◮ ❚❤❡ r❛t❡ ♦❢ ❤❡❛t tr❛♥s❢❡r ❜❡t✇❡❡♥ ❡✈❡r② t✇♦ ❛❞❥❛❝❡♥t ❝♦♥t❛✐♥❡rs ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❞✐✛❡r❡♥❝❡

❜❡t✇❡❡♥ t❤❡ ❛♠♦✉♥ts ♦❢ ❤❡❛t ✐♥ t❤❡♠✳

✻✳✷✳

❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ ♣❡r♠❡❛❜✐❧✐t②

✸✺✹

❚❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ ❝♦♥s❡r✈❛t✐♦♥ ♦❢ ❡♥❡r❣② ✐♥ ❝♦♥t❛✐♥❡r x ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ ❛♠♦✉♥t ♦❢ t❤❡ ❤❡❛t ✐♥ ❝♦♥t❛✐♥❡rs x ♦✈❡r t❤❡ t✐♠❡ ✐♥❝r❡♠❡♥t ❢r♦♠ ti t♦ ti+1 ✐s ❡q✉❛❧ t♦ u(ti+1 , x) − u(ti , x) = −





s✉♠ ♦❢ t❤❡ ♦✉t✢♦✇ g t❤r♦✉❣❤ t❤❡ ♣✐♣❡s ♦❢ x .

❚❤❡ ♦✉t✢♦✇ ❣✐✈❡s t❤❡ ❛♠♦✉♥t ♦❢ ✢♦✇ ❛❝r♦ss ❛♥ ❡❞❣❡ ✭❢r♦♠ t❤❡ ❝♦♥t❛✐♥❡r t♦ ✐ts ♥❡✐❣❤❜♦r✮ ♣❡r ✉♥✐t ♦❢ t✐♠❡✳ ❙♣❡❝✐✜❝❛❧❧②✱ t❤❡ ✢♦✇ ✐s ♣♦s✐t✐✈❡ ❛t x ✐❢ ✐t ✐s ❢r♦♠ ❧❡❢t t♦ r✐❣❤t ❛♥❞ t❤❡ ♦♣♣♦s✐t❡ ❢♦r y ❀ t❤❡♥✿  u(ti+1 , x) − u(ti , x) = − g(ti , a) − g(ti , b) = g(ti , a) − g(ti , b) . ❲❛r♥✐♥❣✦ ❲❤❛t ✐s t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤✐s r❛t❡❄

■t ❝❛♥♥♦t

❜❡ s♦ ❢❛st t❤❛t t❤❡ ✉♣❞❛t❡❞ ✈❛❧✉❡ ♦❢ t❤❡ ❤❡❛t ♦❢ t❤❡ ❝♦♥t❛✐♥❡r s✉r♣❛ss❡s t❤❛t ♦❢ t❤❡ ♦t❤❡r✦

■♥ ♦r✲

❞❡r t♦ ❛✈♦✐❞ t❤✐s ✏♦✈❡rs❤♦♦t✑✱ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ ❤❡❛t s❤♦✉❧❞♥✬t ❜❡ ♠♦r❡ t❤❛t ❤❛❧❢✲❞✐✛❡r❡♥❝❡ ❢r♦♠ t❤❡ ❤❡❛t ♦❢ t❤❡ ♦t❤❡r ❝♦♥t❛✐♥❡r✳

◆♦✇✱ ✇❡ ♥❡❡❞ t♦ ❡①♣r❡ss g ✐♥ t❡r♠s ♦❢ u✳ ❚❤❡ ✢♦✇ g(t, a) t❤r♦✉❣❤ ♣✐♣❡ a ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❛♠♦✉♥ts ♦❢ ❤❡❛t ✐♥ a ❛♥❞ t❤❡ ♦t❤❡r ❝♦♥t❛✐♥❡r ❛❞❥❛❝❡♥t t♦ a✳ ❙♦✱ ♣✐♣❡ a : g(ti , a) = −K(a) u(ti , x) − u(ti , y) ∆t ♣✐♣❡ b : g(ti , b) = −K(b) u(ti , z) − u(ti , x) ∆t 

❍❡r❡✱ K(a) ≥ 0 r❡♣r❡s❡♥ts t❤❡ ♣❡r♠❡❛❜✐❧✐t② ♦❢ t❤❡ ♣✐♣❡ a ❛t ❛ ❣✐✈❡♥ t✐♠❡ ♦✈❡r t❤❡ s❛♠❡ t✐♠❡ ♣❡r✐♦❞✳

❚❤❡s❡ ♥✉♠❜❡rs K(a) ❛r❡ s✐♠♣❧② t❤❡ ♣r♦♣♦rt✐♦♥❛❧✐t② ❝♦❡✣❝✐❡♥ts✳ ❚❤❡②✱ t❤❡r❡❢♦r❡✱ ♣r♦❞✉❝❡ ❛ ❞✐s❝r❡t❡ 1✲❢♦r♠✳ ❚❤❡ ❜❡♥❡✜t ♦❢ t❤✐s ❛♣♣r♦❛❝❤ ✐s t❤❛t ✇❡ ✇♦♥✬t ♥❡❡❞ t♦ ❤❛✈❡ s❡♣❛r❛t❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ❡♥❞s ♦❢ t❤❡ r♦❞✦ ❊①❡r❝✐s❡ ✻✳✷✳✶

❈❛♥ K ❛❧s♦ ❞❡♣❡♥❞ ♦♥ t❄ ❚❤❡ r❡s✉❧t ♦❢ t❤❡ s✉❜st✐t✉t✐♦♥ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥✿ h h i i u(ti+1 , x) = u(ti , x) + − K(a) u(ti , x) − u(ti , y) ∆t − − K(b) u(ti , z) − u(ti , x) ∆t

▲❡t✬s r❡✈✐❡✇ t❤❡ s❡t✉♣ ❢♦r ❞✐s❝r❡t❡ ❢♦r♠s ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡✳

❲❡ ❤❛✈❡ ❛ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b]✳ ❲❡ ❞❡❝♦♠♣♦s❡ ✐t ✐♥t♦ n ❡❞❣❡s ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡ ♥♦❞❡s✿ a = x0 , x1 , x2 , ..., xn−1 , xn = b .

❚❤❡s❡ ❛r❡ t❤❡ ❡❞❣❡s✿ c1 = [x0 , x1 ], c2 = [x1 , x2 ], ..., cn = [xn−1 , xn ] .

❆ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ ♥♦❞❡s ✐s ❛ 0✲❢♦r♠ ❛♥❞ ✐ts ❞✐✛❡r❡♥❝❡ ♦r ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ❛ 1✲❢♦r♠✿

✻✳✷✳

❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ ♣❡r♠❡❛❜✐❧✐t②

✸✺✺

❚❤❡ 0✲❢♦r♠s ❤❛✈❡ ♥♦❞❡s ❛s ✐♥♣✉ts ❛♥❞ t❤❡ 1✲❢♦r♠s ❤❛✈❡ ❡❞❣❡s ❛s ✐♥♣✉ts ❜✉t t❤❡ ♦✉t♣✉ts ❛r❡ r❡❛❧ ♥✉♠❜❡rs✿

❲❤❛t ♠❛❦❡s t❤✐s ❞✐✛❡r❡♥t ❢r♦♠ ❖❉❊s ✐s t❤❛t t❤❡ ❢✉♥❝t✐♦♥s ✇✐❧❧ ❤❛✈❡ t✇♦ ✈❛r✐❛❜❧❡s ✕ ♦♥❡ ❢♦r ❧♦❝❛t✐♦♥ ❛♥❞ ♦♥❡ ❢♦r t✐♠❡✳ ❚❤❡ ❛♠♦✉♥t ♦❢ ❤❡❛t u = u(t, x) ✐s s✐♠♣❧② ❛ ♥✉♠❜❡r ❛ss✐❣♥❡❞ t♦ ❡❛❝❤ ❝♦♥t❛✐♥❡r x✳ ❚❤✐s ✐s ✶✳ ❛ ❞✐s❝r❡t❡ 0✲❢♦r♠ ✇✐t❤ r❡s♣❡❝t t♦ ❧♦❝❛t✐♦♥ x ❛♥❞ ✷✳ ❛ ❞✐s❝r❡t❡ 0✲❢♦r♠ ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡ t✳ ▲❡t✬s ❝♦♥s✐❞❡r t❤❡ s❡t✉♣ ❢♦r ❞✐s❝r❡t❡ ❢♦r♠s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳ ❲❡ st❛rt ✇✐t❤ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥s ✐♥ t❤❡ x✲ ❛♥❞ t❤❡ t✲❛①❡s✿

❚❤❡② ❞❡✈❡❧♦♣ ✐♥ ♣❛r❛❧❧❡❧✿

• ❆ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ✐♥t❡r✈❛❧ [a, b] ✐♥ t❤❡ x✲❛①✐s ❝♦♥s✐sts ♦❢ n + 1 ♥♦❞❡s✿ a = x0 , x1 , x2 , ..., xn−1 , xn = b , ❛♥❞ n ❡❞❣❡s✿

s1 , s2 , ..., sn−1 , sn , ✇✐t❤ t❤❡ ✐♥❝r❡♠❡♥ts✿

∆xk = xk − xk−1 , k = 1, 2, ..., n . • ❆ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ t❤❡ ✐♥t❡r✈❛❧ [c, d] ✐♥ t❤❡ t✲❛①✐s ❝♦♥s✐sts ♦❢ m + 1 ♥♦❞❡s✿ c = t0 , t1 , t2 , ..., tm−1 , tm = d , ❛♥❞ m ❡❞❣❡s✿

q1 , q2 , ..., qm−1 , qm , ✇✐t❤ t❤❡ ✐♥❝r❡♠❡♥ts✿

∆ti = ti − ti−1 , i = 1, 2, ..., m . ❲❡ ✉s❡ t❤❡s❡ ❞❡❝♦♠♣♦s✐t✐♦♥s ♦❢ t❤❡ ✐♥t❡r✈❛❧s t♦ ❝♦♥str✉❝t ❛ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ P ♦❢ ❛ r❡❝t❛♥❣❧❡ R = [a, b] × [c, d] ✐♥ t❤❡ xt✲♣❧❛♥❡✳ ❚❤❡ ❧✐♥❡s x = xk ❛♥❞ t = ti ❝✉t t❤❡ r❡❝t❛♥❣❧❡ [a, b] × [c, d] ✐♥t♦ s♠❛❧❧❡r r❡❝t❛♥❣❧❡s [xk , xk+1 ] × [ti , ti+1 ]✳

✻✳✷✳

❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ ♣❡r♠❡❛❜✐❧✐t②

✸✺✻

❲❡ t❛❦❡ ♦✉r ❡q✉❛t✐♦♥✿

h

i

h

i

u(ti+1 , x) − u(ti , x) = − K(a) u(ti , x) − u(ti , y) ∆t − − K(b) u(ti , z) − u(ti , x) ∆t , ❛♥❞ r❡✇r✐t❡ ✐t✿

h i h i u(ti+1 , xk ) − u(ti , xk ) = − K(sk ) u(ti , xk ) − u(ti , xk−1 ) ∆t − − K(sk+1 ) u(ti , xk+1 ) − u(ti , xk ) ∆t . ❲❡ ❤❛✈❡ ❛ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ♥❡①t t❡♠♣❡r❛t✉r❡ ❛t ❛ ❧♦❝❛t✐♦♥ t❤❛t ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❝✉rr❡♥t t❡♠♣❡r❛t✉r❡s ♦❢ t❤❡ ❧♦❝❛t✐♦♥ ❛♥❞ t❤❡ ❛❞❥❛❝❡♥t ❧♦❝❛t✐♦♥s✿

  u(ti+1 , xk ) = u(ti , xk ) − K(sk ) u(ti , xk ) − u(ti , xk−1 ) ∆t + K(sk+1 ) u(ti , xk+1 ) − u(ti , xk ) ∆t ■t ❝❛♥ ❜❡ ✉s❡❞ ❢♦r s✐♠✉❧❛t✐♦♥s✦ ❙✉♣♣♦s❡ ♦✉r r♦❞ ✐s s♣❧✐t ✐♥t♦ t❡♥ ♣✐❡❝❡s ❛s ✐❢ t❤❡② ❛r❡ s❡♣❛r❛t❡ ❝♦♥t❛✐♥❡rs ❛s ❡①♣❧❛✐♥❡❞ ❛❜♦✈❡✳ ❊❛❝❤ ♦❢ t❤❡♠ ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ ✭✇✐❞❡r✮ ❝❡❧❧ ✐♥ t❤❡ s♣r❡❛❞s❤❡❡t✳ ❚❤✐s ✐s ✇❤❛t ②♦✉ s❡❡ ❛t t❤❡ t♦♣ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t ❜❡❧♦✇✿

❚❤❡ ♣✐♣❡s ❛r❡ s❤♦✇♥ ❛s ✭♥❛rr♦✇❡r✮ ❝❡❧❧s✳ ❚❤❡ t✐♠❡ ❛①✐s ✐s ✈❡rt✐❝❛❧✳ ◆♦✇✱ ❡❛❝❤ ♦❢ t❤❡ ❝❡❧❧s r❡♣r❡s❡♥t✐♥❣ t❤❡ ❝♦♥t❛✐♥❡rs ❝♦♥t❛✐♥s ❛ ♥✉♠❜❡r ✇❤✐❝❤ ✐s t❤❡ ❛♠♦✉♥t ♦❢ ❤❡❛t t❤❛t ❧♦❝❛t✐♦♥ ❛t t❤❛t t✐♠❡✳

u

❛t

◆❡①t✱ ❡❛❝❤ ♦❢ t❤❡ ❝❡❧❧s r❡♣r❡s❡♥t✐♥❣ t❤❡ ♣✐♣❡s ❝♦♥t❛✐♥s ❛ ♥✉♠❜❡r ✇❤✐❝❤ ✐s t❤❡

✻✳✷✳

❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ ♣❡r♠❡❛❜✐❧✐t②

♣❡r♠❡❛❜✐❧✐t②

k

✸✺✼

❛t t❤❛t ❧♦❝❛t✐♦♥ ❛t t❤❛t t✐♠❡✳ ❚❤❡ ❤❡❛t ✢♦✇ t❤r♦✉❣❤ t❤❡ ♣✐♣❡s ✐s ❝♦♠♣✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡

❢♦r♠✉❧❛ ❞✐s❝✉ss❡❞ ❛❜♦✈❡ ❛♥❞ t❤❡ ♥❡✇ ✈❛❧✉❡ ♦❢

u

✐s ❝♦♠♣✉t❡❞ ❡✈❡r② t✐♠❡ ✇❡ ♠♦✈❡ t♦ t❤❡ ♥❡①t r♦✇✳

❚❤❡ s♣r❡❛❞s❤❡❡t ❢♦r♠✉❧❛ ✐s ❛s ❢♦❧❧♦✇s✿

❂❘❬✲✶❪❈✲❘❈❬✲✶❪✯✭❘❬✲✶❪❈✲❘❬✲✶❪❈❬✲✷❪✮✰❘❈❬✶❪✯✭❘❬✲✶❪❈❬✷❪✲❘❬✲✶❪❈✮ ❚❤✐s ✐s t❤❡ ❞❡♣❡♥❞❡♥❝❡ ❞✐❛❣r❛♠ ♦❢ t❤❡ ❢♦r♠✉❧❛✿

❊①❛♠♣❧❡ ✻✳✷✳✷✿ ✐♥s✉❧❛t❡❞ r♦❞

❙✉♣♣♦s❡ ✇❡ ♣✉t ✐♥s✉❧❛t♦rs t♦ t❤❡ ❡♥❞s ♦❢ ♦✉r r♦❞ ❛♥❞ s✉♣♣♦s❡ t❤❡② ❤❛✈❡ ✐♥✐t✐❛❧❧② ❛ ❧♦✇❡r t❡♠♣❡r❛t✉r❡ t❤❛♥ t❤❛t ♦❢ t❤❡ r♦❞✳ ❚❤❡ ❛r❡❛s ❛❞❥❛❝❡♥t t♦ t❤❡ ❡♥❞s q✉✐❝❦❧② ❝♦♦❧s ❞♦✇♥✳ ❆t t❤✐s ♣♦✐♥t ✇❡ st❛rt ♦✉r s✐♠✉❧❛t✐♦♥✳ ❍❡❛t tr❛♥s❢❡r ❝♦♥t✐♥✉❡s ✇✐t❤✐♥ t❤❡ ❜♦❞② ♦❢ t❤❡ r♦❞ ✇✐t❤ ✈✐rt✉❛❧❧② ♥♦ tr❛♥s❢❡r t❤r♦✉❣❤ t❤❡ ✐♥s✉❧❛t♦rs✳ ❲❡ s❡t t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢

u

t♦ ❣♦ ❢r♦♠

0

t♦

1

❛♥❞ t❤❡♥ ❜❛❝❦ t♦

0

❛t t❤❡ ♦t❤❡r ❡♥❞✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡

♠❛② ❤❛✈❡✿

u(0, x) = 10 sin(πx) . ❚❤❡ ♣❡r♠❡❛❜✐❧✐t② ✐s ③❡r♦ ❛t t❤❡ ❡♥❞s ❛♥❞ ♥♦♥✲③❡r♦ t❤r♦✉❣❤♦✉t t❤❡ r❡st ♦❢ t❤❡ r♦❞✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ♠❛② ❤❛✈❡✿

K(0) = K(1) = 0

❛♥❞

K(x) = .45

❢♦r

0 < x < 1.

❚❤✐s ✐s t❤❡ r❡s✉❧t ♦❢ ♦✉r s✐♠✉❧❛t✐♦♥ ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❣r❛♣❤s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❡❛❝❤

t

✭t❤❡ ❣r❛♣❤ ♦❢

u

✐s ♦❢ ❝♦✉rs❡ ❛ s✉r❢❛❝❡✮✿

u(t, ·)

♦❢ ♦♥❡ ✈❛r✐❛❜❧❡ ❢♦r

❆❜♦✈❡ ✇❡ s❡❡ ❡✐t❤❡r t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ❤❡❛t t❤r♦✉❣❤♦✉t t❤❡ r♦❞ ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡ ♦r t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ t❡♠♣❡r❛t✉r❡ ❛t ❡✈❡r② ❧♦❝❛t✐♦♥✳ ❈♦♥❝❧✉s✐♦♥✿ t❤❡ t❡♠♣❡r❛t✉r❡ ❡✈❡♥s ♦✉t✦ ◆♦t❡ t❤❛t t❤❡ t♦t❛❧ ❛♠♦✉♥t ♦❢ ❤❡❛t ✐♥ t❤❡ r♦❞ r❡♠❛✐♥s t❤❡ s❛♠❡ ✭❣✐✈❡♥ ✉♥❞❡r ✏t♦t❛❧✑ ✐♥ t❤❡ s♣r❡❛❞s❤❡❡t✮✳ ❆s ❛ ✇❛r♥✐♥❣✱ ✐❢ ✇❡ ❝❤♦♦s❡ ♦✉r ❝♦❡✣❝✐❡♥t ♦❢ ♣❡r♠❡❛❜✐❧✐t② ❧❛r❣❡r t❤❛♥

1/2✱

t❤❡ s✐♠✉❧❛t✐♦♥ ❢❛✐❧s q✉✐❝❦❧②✿

✻✳✷✳

❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ ♣❡r♠❡❛❜✐❧✐t②

✸✺✽

❊①❛♠♣❧❡ ✻✳✷✳✸✿ ❤❡❛t❡❞ r♦❞ ❙✉♣♣♦s❡ ♦✉r r♦❞ ✐s ✐♥s❡rt❡❞ t♦ ✜① ❛ ❣❛♣ ✐♥ ❛ ❧♦♥❣ ♣✐♣❡ ✇✐t❤ ✇❛r♠❡r t❡♠♣❡r❛t✉r❡✳ ❖♥❝❡ ✐♥✱ t❤❡ ❛r❡❛s ❛❞❥❛❝❡♥t t♦ t❤❡ ❡♥❞s ♦❢ t❤❡ r♦❞ st❛rt t♦ ❡①❝❤❛♥❣❡ ❤❡❛t ✇✐t❤ t❤❡ ♣✐♣❡✳ ❍♦✇❡✈❡r✱ t❤❡ ❡✛❡❝t ♦♥ t❤❡ ♣✐♣❡✬s t❡♠♣❡r❛t✉r❡ ✐s ♥❡❣❧✐❣✐❜❧❡✳ ❆t t❤✐s ♣♦✐♥t ✇❡ st❛rt ♦✉r s✐♠✉❧❛t✐♦♥✳ ❲❡ s❡t t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢ u t♦ ❜❡ 0 t❤r♦✉❣❤♦✉t t❤❡ r♦❞ ✇❤✐❧❡ t❤❡ ♦✉ts✐❞❡ t❡♠♣❡r❛t✉r❡ ✐s 5✳ ❚❤❡ ♣❡r♠❡❛❜✐❧✐t② ✐s .45 t❤r♦✉❣❤♦✉t t❤❡ r♦❞✳ ❚❤✐s ✐s t❤❡ r❡s✉❧t ♦❢ ♦✉r s✐♠✉❧❛t✐♦♥✿

❚❤❡ t❡♠♣❡r❛t✉r❡ ❡✈❡♥s ♦✉t✦ ■❢ ✇❡ ❝❤♦♦s❡ ♦✉r ❝♦❡✣❝✐❡♥t ♦❢ ♣❡r♠❡❛❜✐❧✐t② ❧❛r❣❡r t❤❛♥ 1/2✱ t❤❡ s✐♠✉❧❛t✐♦♥ ❢❛✐❧s q✉✐❝❦❧②✿

❊①❡r❝✐s❡ ✻✳✷✳✹ ❲❤❛t ❦✐♥❞ ♦❢ ❝✉r✈❡ ❞♦ ✇❡ s❡❡ ❛t t❤❡ ❡❞❣❡ ✭x = 0✮ ♦❢ t❤✐s s✉r❢❛❝❡❄

❊①❛♠♣❧❡ ✻✳✷✳✺✿ s✐♠♣❧✐✜❡❞ ❆ s✐♠♣❧✐✜❡❞ ✈❡rs✐♦♥ ♠❛❦❡s k ✐♥❞❡♣❡♥❞❡♥t ♦❢ t✐♠❡ ❛♥❞ ♣✉ts ✐t ❛t t❤❡ t♦♣ r♦✇✳ ❚❤✐s ✐s t❤❡ ❞❡♣❡♥❞❡♥❝❡✿

✻✳✷✳ ❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ ♣❡r♠❡❛❜✐❧✐t②

✸✺✾

❚❤❡ r❡s✉❧t ✐s ♦♥ t❤❡ ❧❡❢t✿

❊①❡r❝✐s❡ ✻✳✷✳✻

❲❤❛t ❤❛♣♣❡♥❡❞ ✐♥ t❤❡ s❡❝♦♥❞ s♣r❡❛❞s❤❡❡t❄ ❊①❛♠♣❧❡ ✻✳✷✳✼✿ ❤❡❛t❡❞ r♦❞

❲❤❛t ✇✐❧❧ ❤❛♣♣❡♥ t♦ t❤❡ r♦❞ ❤❡❛t❡❞ ❛t t❤❡ ❡♥❞s ✇❤❡♥ t❤❡ ♣❡r♠❡❛❜✐❧✐t② ✈❛r✐❡s ❄ ❚❤❡ ✐♥✐t✐❛❧ t❡♠♣❡r❛t✉r❡ ✐s st✐❧❧ 0 ✇❤✐❧❡ t❤❡ ♦✉ts✐❞❡ t❡♠♣❡r❛t✉r❡ ✐s 5✳ ❚❤❡ ♣❡r♠❡❛❜✐❧✐t② ✇✐❧❧ r❡♠❛✐♥ .45 t❤r♦✉❣❤♦✉t t❤❡ ❤❛❧❢ ♦❢ t❤❡ r♦❞ ❛♥❞ .1 ♦♥ t❤❡ ❧❡❢t✳ ❚❤✐s ✐s t❤❡ r❡s✉❧t ♦❢ ♦✉r s✐♠✉❧❛t✐♦♥✿

❚❤❡ t❡♠♣❡r❛t✉r❡ ❡✈❡♥s ♦✉t ❛❣❛✐♥ ❜✉t✱ ♣r❡❞✐❝t❛❜❧②✱ t❤❡ ❧❡❢t ❡♥❞ ✐s ❢❛❧❧✐♥❣ ❜❡❤✐♥❞✦ ❲❛r♥✐♥❣✦ ❚❤❡ r♦❞ ✐s ✐ts ♦✇♥ ✉♥✐✈❡rs❡✳

❲❡ ✇✐❧❧ ♥♦✇ ✐♥t❡r♣r❡t t❤❡ ❞✐✛❡r❡♥❝❡s ✇❡ s❡❡ ✐♥ ♦✉r r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿   u(ti+1 , xk ) = u(ti , xk ) − K(sk ) u(ti , xk ) − u(ti , xk−1 ) ∆t + K(sk+1 ) u(ti , xk+1 ) − u(ti , xk ) ∆t .

■❢ ❛ ❢✉♥❝t✐♦♥ y = f (x) ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s xk , k = 0, 1, 2, ..., n✱ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ f ✐s ❞❡✜♥❡❞ ❛t t❤❡ ❡❞❣❡s ❜②✿ ∆f (ck ) = f (xk+1 ) − f (xk )

❢♦r ❡❛❝❤ k = 1, 2, ..., n ✳

✻✳✸✳

❍❡❛t tr❛♥s❢❡r ✐s ❝❛✉s❡❞ ❜② t❡♠♣❡r❛t✉r❡ ❞✐✛❡r❡♥❝❡s

❚❤❡ ♣❛rt✐❛❧

✸✻✵

❞✐✛❡r❡♥❝❡s ♦❢ u ✇✐t❤ r❡s♣❡❝t t♦ x ❛♥❞ t ❛r❡ ❞❡✜♥❡❞ ❛t t❤❡s❡ ❡❞❣❡s ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ ∆x u (ti , sk ) = u(ti , xk+1 ) − u(ti , xk )

❛♥❞

∆t u (qi , xk ) = u(ti+1 , xk ) − u(ti , xk )

❲❡ t❛❦❡ ♦✉r r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❛♥❞ r❡✇r✐t❡ ✐t ✐♥ t❡r♠s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿

∆u (qi , xk ) = K(sk+1 )∆x u(ti , sk+1 ) − K(sk )∆x u(ti , sk ) ∆t ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡❄

✻✳✸✳ ❍❡❛t tr❛♥s❢❡r ✐s ❝❛✉s❡❞ ❜② t❡♠♣❡r❛t✉r❡ ❞✐✛❡r❡♥❝❡s ❊①❛♠♣❧❡ ✻✳✸✳✶✿ ❧❛r❣❡ t❡❛ ❝✉♣

❲❤❛t ✐❢ t❤❡ t❡❛ ❝✉♣ ✐t

3

t✐♠❡s ❧❛r❣❡r t❤❛♥ t❤❡ ♦t❤❡r t✇♦❄ s♦❞❛ ❝♦✛❡❡

❲❡ ❛❞❥✉st t❤❡ ❢♦r♠✉❧❛ ❜② ❞✐✈✐❞✐♥❣ ❜②

3

t❡❛

t❤❡ ❛♠♦✉♥t t❤❛t ✐s ❛❞❞❡❞ t♦

w

✭t❡❛✮✿

❂❘❬✲✶❪❈✲✵✳✶✯❘✷❈✯✭❘❬✲✶❪❈✲❘❬✲✶❪❈❬✲✶❪✮✯❘✷❈✶✲✵✳✶✯❘✷❈❬✶❪✯✭❘❬✲✶❪❈✲❘❬✲✶❪❈❬✶❪✮✯❘✷❈✶✴✸ ❚❤❡ ✐❞❡❛ ✐s t❤❛t ✐s t❤❡ ❡✛❡❝t ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ t❡♠♣❡r❛t✉r❡s ♦♥ t❤❡ t❡❛ ✐s ♦♥❡ t❤✐r❞✿

❚❤❡ ♣❛tt❡r♥ ♦❢ ❛✈❡r❛❣✐♥❣ ❝♦♥t✐♥✉❡s ❜✉t t❤❡ ❛✈❡r❛❣❡ ❤❛s ❝❤❛♥❣❡❞✿

40 + 100 + 85 · 3 = 79 . 5 ❆ ✏q✉✐❝❦✲❛♥❞✲❞✐rt②✑ ✜① ❤❛s ♣r♦❞✉❝❡❞ t❤❡ ❡①♣❡❝t❡❞ r❡s✉❧t✦

❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤ ❛ ♠♦r❡ ❣❡♥❡r❛❧ ♠♦❞❡❧ ❢♦r ❤❡❛t tr❛♥s❢❡r ✐♥ ❛ r♦❞✳ ❚❤✐s t✐♠❡✱ t❤❡ ❝♦♥t❛✐♥❡rs✬ s✐③❡s ♠❛② ✈❛r② ❛♥❞ s♦ ❞♦ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ✐♥t❡r✈❛❧s ♦❢ t✐♠❡✳ ❲❤❛t ♠❛tt❡rs ♥♦✇ ✐s t❤❡ ❥✉①t❛♣♦s✐t✐♦♥✿



❤❡❛t ✈s✳ t❡♠♣❡r❛t✉r❡✱



✇❡✐❣❤t ✈s✳ ❞❡♥s✐t②✳

❚❤❡ ❧❛tt❡r ✐s t❤❡ ❛✈❡r❛❣❡ ❛♠♦✉♥t ♦❢ t❤❡ ❢♦r♠❡r✿

t❡♠♣❡r❛t✉r❡

=

❤❡❛t s✐③❡

✻✳✸✳ ❍❡❛t tr❛♥s❢❡r ✐s ❝❛✉s❡❞ ❜② t❡♠♣❡r❛t✉r❡ ❞✐✛❡r❡♥❝❡s ❛♥❞

❞❡♥s✐t② =

✸✻✶ ✇❡✐❣❤t . s✐③❡

❚❤❡ s✐③❡s ❛r❡ ♥♦t❤✐♥❣ ❜✉t ❛ s❡q✉❡♥❝❡ ♦❢ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs ❛ss✐❣♥❡❞ t♦ ❡❛❝❤ ♥♦❞❡ xk ✳ ▲❡t✬s ❝❛❧❧ t❤❡♠ ∆sk ✳ ✭❚❤❡ ❧❛tt❡r ❢♦r♠✉❧❛ ♠❛t❝❤❡s t❤❡ ♦♥❡ ✐♥ ❈❤❛♣t❡r ✸■❈✲✸✳✮ ■❢ u ✐s t❤❡ t❡♠♣❡r❛t✉r❡✱ t❤❡♥ t❤❡ ❛♠♦✉♥t ♦❢ ❤❡❛t ✐♥ t❤❡ r♦♦♠ x ✐s✿ u(·, xk ) · ∆sk .

■t ✐s ❤❡❛t t❤❛t ✐s ❜❡✐♥❣ ❡①❝❤❛♥❣❡❞✳ ❇✉t ✐t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ t❡♠♣❡r❛t✉r❡s t❤❛t ❞r✐✈❡s t❤❡ ❡①❝❤❛♥❣❡✳ ❖✉r r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❜❡❝♦♠❡s✿ u(ti+1 , xk ) · ∆sk = u(ti , xk ) · ∆sk + K(sk+1 )∆x u(ti , sk+1 )∆t − K(sk )∆x u(ti , sk )∆t . ❲❛r♥✐♥❣✦ ❚❤❡ s✐③❡

∆sk

✈❛r✐❡s ❢r♦♠ ❧♦❝❛t✐♦♥ t♦ ❧♦❝❛t✐♦♥✳

❚❤❡ ♥❡✇ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ t❡♠♣❡r❛t✉r❡ ✐s✿   u(ti+1 , xk ) = u(ti , xk ) + K(sk+1 )∆x u(ti , sk+1 ) − K(sk )∆x u(ti , sk ) ∆t/∆sk

❲❡ ✉s❡ ✐t ❢♦r s✐♠✉❧❛t✐♦♥s✳ ❊①❛♠♣❧❡ ✻✳✸✳✷✿ r♦❞

❲❡ ♣❧❛❝❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❛❧✉❡ ♦❢ ∆s ❛t t❤❡ t♦♣ ♦❢ ❡❛❝❤ ❝♦❧✉♠♥✿

❚❤❡ ❢♦r♠✉❧❛s ✐s ❛s ❢♦❧❧♦✇s✿ ❂❘❬✲✶❪❈✲✵✳✶✯❘✷❈✯✭❘❬✲✶❪❈✲❘❬✲✶❪❈❬✲✶❪✮✯❘✷❈✶✴❘✸❈✲✵✳✶✯❘✷❈❬✶❪✯✭❘❬✲✶❪❈✲❘❬✲✶❪❈❬✶❪✮✯❘✷❈✶✴❘✸❈

❚❤✐s ✐s t❤❡ ♥❡✇ ❞❡♣❡♥❞❡♥❝❡✿

✻✳✸✳ ❍❡❛t tr❛♥s❢❡r ✐s ❝❛✉s❡❞ ❜② t❡♠♣❡r❛t✉r❡ ❞✐✛❡r❡♥❝❡s

✸✻✷

❲❡ ❛❧s♦ ❝♦♠♣✉t❡ t❤❡ t♦t❛❧ ❤❡❛t ❛♥❞ t❤❡ ❛✈❡r❛❣❡ t❡♠♣❡r❛t✉r❡ ❛t ❡✈❡r② ♠♦♠❡♥t ♦❢ t✐♠❡❀ t❤❡ ❧❛st t✇♦ ❝♦❧✉♠♥s✳ ❚❤❡ ❝♦♥s❡r✈❛t✐♦♥ ♦❢ ❡♥❡r❣② ✐s ❝♦♥✜r♠❡❞✦ ❚❤❡ ❛✈❡r❛❣✐♥❣ ❧♦♦❦s ❛ ❜✐t ❞✐✛❡r❡♥t✿

❊①❛♠♣❧❡ ✻✳✸✳✸✿ ❡①❝❤❛♥❣❡ ✇✐t❤ ♦✉ts✐❞❡

◆♦✇✱ s✉♣♣♦s❡ t❤❡ r♦❞ ✐s ♥♦t ✐ts ♦✇♥ ✉♥✐✈❡rs❡ t❤✐s t✐♠❡ ❛♥❞ ✐♥t❡r❛❝ts ✇✐t❤ t❤❡ ♦✉ts✐❞❡✳ ❚❤❡♥✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ❡①❝❤❛♥❣❡ ❜❡t✇❡❡♥ ✐ts ❝❡❧❧s✱ t❤❡r❡ ✐s ❡①❝❤❛♥❣❡ ✇✐t❤ t❤❡ ❛✐r ❛t r♦♦♠ t❡♠♣❡r❛t✉r❡✳ ❚❤❡ ❧❛tt❡r ✐s ❡①❡❝✉t❡❞ ❥✉st ❛s ✐♥ t❤❡ ♦r✐❣✐♥❛❧ ◆❡✇t♦♥✬s ▲❛✇ ♠♦❞❡❧ ❜✉t ❢♦r ❡❛❝❤ ❝♦♥t❛✐♥❡r ✐♥❞❡♣❡♥❞❡♥t❧②✦ ❲❡ ❥✉st ❛❞❞ ❛♥ ❡①tr❛ t❡r♠ t♦ t❤❡ ❢♦r♠✉❧❛ ✭0 r♦♦♠ t❡♠♣❡r❛t✉r❡✮✿ ✲✵✳✶✯❘❬✲✶❪❈

❖❢ ❝♦✉rs❡✱ t❤❡r❡ ✐s ♥♦ ❝♦♥s❡r✈❛t✐♦♥ ♦❢ ❡♥❡r❣② t❤✐s t✐♠❡✳ ❲❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡ t❡♠♣❡r❛t✉r❡ st❛❜✐❧✐③❡s t♦✇❛r❞ t❤❡ r♦♦♠ t❡♠♣❡r❛t✉r❡ ✭t❤❡ st❡❛❞② st❛t❡✮✿

❚❤✐s ♦❜s❡r✈❛t✐♦♥ ❝♦♥✜r♠s t❤❡ ♦r✐❣✐♥❛❧ ◆❡✇t♦♥✬s ▲❛✇ ♠♦❞❡❧✳ ❆♥ ❛❧t❡r♥❛t✐✈❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦♥ t❤❡ r♦❞ ❛♥❞ t❤❡ ❤❡❛t ❡①❝❤❛♥❣❡ ✐s ❛s ❢♦❧❧♦✇s✳ ❚❤❡ r♦❞ ✐s s❡❡♥ ❛s s♣❧✐t ✐♥t♦ ♣✐❡❝❡s ❛s ✐❢ t❤❡② ❛r❡ s❡♣❛r❛t❡ ♣✐❡❝❡s✿ t❤❡ ❤❡❛t ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ r♦✇ ♦❢ r♦♦♠s ❛♥❞ ❡❛❝❤ r♦♦♠ ❡①❝❤❛♥❣❡s t❤❡ ♠❛t❡r✐❛❧ ✇✐t❤ ✐ts t✇♦ ♥❡✐❣❤❜♦rs t❤r♦✉❣❤ ✐ts ✇❛❧❧s✿

✻✳✸✳ ❍❡❛t tr❛♥s❢❡r ✐s ❝❛✉s❡❞ ❜② t❡♠♣❡r❛t✉r❡ ❞✐✛❡r❡♥❝❡s

✸✻✸

❇❡❧♦✇ ✇❡ ❤❛✈❡✿ • a = AB ✐s ♦♥❡ ♦❢ t❤❡ r♦♦♠s✳

• b, c ❛r❡ t❤❡ t✇♦ ❛❞❥❛❝❡♥t r♦♦♠s✱ ❧❡❢t ❛♥❞ r✐❣❤t✳ • A, B ❛r❡ t❤❡ ✇❛❧❧s ♦❢ a✱ ❧❡❢t ❛♥❞ r✐❣❤t✳

❖❢ ❝♦✉rs❡✱ ✇❡ r❡♣❧❛❝❡ t❤❡ r♦♦♠s ✇✐t❤ ❡❞❣❡s ❛♥❞ ✇❛❧❧s ✇✐t❤ ♥♦❞❡s✿

❚❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ✐s ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ♦♥❡ ❛❜♦✈❡❀ ✐♥st❡❛❞ ♦❢ r♦♦♠s ❛♥❞ ✇❛❧❧s ❜❡t✇❡❡♥ t❤❡♠✱ ✇❡ s♣♦❦❡ ♦❢ ❝♦♥t❛✐♥❡rs ❛♥❞ ♣✐♣❡s ❜❡t✇❡❡♥ t❤❡♠✳ ❘❡❝❛❧❧✿ • a ✐s ♦♥❡ ♦❢ t❤❡ ❝♦♥t❛✐♥❡rs✳

• b, c ❛r❡ t❤❡ t✇♦ ❛❞❥❛❝❡♥t ❝♦♥t❛✐♥❡rs✱ ❧❡❢t ❛♥❞ r✐❣❤t✳ • A, B ❛r❡ t❤❡ ♣✐♣❡s t❤❛t st❛rt ❛t a✱ ❧❡❢t ❛♥❞ r✐❣❤t✳

❚❤❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

❖❢ ❝♦✉rs❡✱ ✇❡ r❡♣❧❛❝❡ t❤❡ ❝♦♥t❛✐♥❡rs ✇✐t❤ ♥♦❞❡s ❛♥❞ t❤❡ ♣✐♣❡s ✇✐t❤ ❡❞❣❡s✿

✻✳✸✳ ❍❡❛t tr❛♥s❢❡r ✐s ❝❛✉s❡❞ ❜② t❡♠♣❡r❛t✉r❡ ❞✐✛❡r❡♥❝❡s ❚❤✐s ✐s ❞✉❛❧✐t② ✿ ❚✇♦ ✐❞❡♥t✐❝❛❧ st❡♣s ❛♥❞ ✇❡ ❛r❡ ❜❛❝❦ ✇❤❡r❡ ✇❡ st❛rt❡❞✳ ▲❡t✬s r❡✈✐❡✇✳ ❲❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛t❝❤ ❜❡t✇❡❡♥ t❤❡ ❝❡❧❧s✿

❚❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ❝❡❧❧s ❛r❡ ✐♥❞✐❝❛t❡❞ ✐♥ t❤❡ s❡❝♦♥❞ r♦✇✿ ∆xk = xk+1 − xk ❛♥❞ ∆sk = sk+1 − sk .

❚❤❡ ❢♦r♠❡r ♦♥❡s ❤❛✈❡♥✬t ❛♣♣❡❛r❡❞ ②❡t✳ ❯♥❞❡r t❤❡ st❛r ♦♣❡r❛t♦r✱ t❤❡ 0✲❝❡❧❧s ♦❢ t❤❡ ❢♦r♠❡r ❝♦rr❡s♣♦♥❞ t♦ t❤❡ 1✲❝❡❧❧s ♦❢ t❤❡ ❧❛tt❡r ❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ❚❤❡ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥s ♦❢ ♦✉r r❡❝t❛♥❣❧❡ ❞✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r ❛r❡ s❤♦✇♥ ❜❡❧♦✇✿

❲❡ t✉r♥ t♦ ♦✉r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✿

♦r✱ ♦r✱

  u(ti+1 , xk ) = u(ti , xk ) + K(sk+1 )∆x u(ti , sk+1 ) − K(sk )∆x u(ti , sk ) ∆t/∆sk , ∆u K(sk+1 )∆x u(ti , sk+1 ) − K(sk )∆x u(ti , sk ) (qi , xk ) = , ∆t ∆sk   ∆ K∆x u ∆u (qi , xk ) = (ti , xk ) . ∆t ∆sk

❲❤❡♥ K ✐s ❝♦♥st❛♥t✱ ✇❡ ❤❛✈❡ ❛♥ ❛❜❜r❡✈✐❛t❡❞ ❢♦r♠✿ ∆(∆x u) ∆u (qi , xk ) = K (ti , xk ) ∆t ∆sk

❚❤✐s ✐s ✇❤❡r❡ t❤❡ ❞✐s❝r❡t❡ ❢♦r♠s ✐♥ t❤❡ ❧❡❢t✲ ❛♥❞ r✐❣❤t✲❤❛♥❞ t❡r♠s ♦❢ t❤❡ ❡q✉❛t✐♦♥ r❡s✐❞❡✱ r❡s♣❡❝t✐✈❡❧②✿

✸✻✹

✻✳✹✳ ❍❡❛t tr❛♥s❢❡r ❞❡♣❡♥❞s ♦♥ t❤❡ ❣❡♦♠❡tr②