Advanced Engineering Mathematics [5 ed.]

Table of contents :
Front Cover
Copyright Page
CONTENTS
Preface
PART 1 - Ordinary Differential Equations
CHAPTER 1 Introduction to Differential Equations
Outline & Intro
1.1 Definitions and Terminology
1.2 Initial-Value Problems
1.3 Differential Equations as Mathematical Models
Review
CHAPTER 2 First-Order Differential Equations
Outline & Intro
2.1 Solution Curves Without a Solution
2.2 Separable Equations
2.3 Linear Equations
2.4 Exact Equations
2.5 Solutions by Substitutions
2.6 A Numerical Method
2.7 Linear Models
2.8 Nonlinear Models
2.9 Modeling with Systems of First-Order DEs
Review
Contributed Problems
CHAPTER 3 Higher-Order Differential Equations
Outline & Intro
3.1 Theory of Linear Equations
3.2 Reduction of Order
3.3 Homogeneous Linear Equations with Constant Coefficients
3.4 Undetermined Coefficients
3.5 Variation of Parameters
3.6 Cauchy-Euler Equations
3.7 Nonlinear Equations
3.8 Linear Models: Initial-Value Problems
3.9 Linear Models: Boundary-Value Problems
3.10 Green's Functions
3.11 Nonlinear Models
3.12 Solving Systems of Linear Equations
Review
Contributed Problems
CHAPTER 4 The Laplace Transform
Outline & Intro
4.1 Definition of the Laplace Transform
4.2 The Inverse Transform and Transforms of Derivatives
4.3 Translation Theorems
4.4 Additional Operational Properties
4.5 The Dirac Delta Function
4.6 Systems of Linear Differential Equations
Review
CHAPTER 5 Series Solutions of Linear Differential Equations
Outline & Intro
5.1 Solutions about Ordinary Points
5.2 Solutions about Singular Points
5.3 Special Functions
Review
CHAPTER 6 Numerical Solutions of Ordinary Differential Equations
Outline & Intro
6.1 Euler Methods and Error Analysis
6.2 Runge-Kutta Methods
6.3 Multistep Methods
6.4 Higher-Order Equations and Systems
6.5 Second-Order Boundary-Value Problems
Review
PART 2 - Vectors, Matrices, and Vector Calculus
CHAPTER 7 Vectors
Outline & Intro
7 .1 Vectors in 2-Space
7 .2 Vectors in 3-Space
7 .3 Dot Product
7 .4 Cross Product
7.5 Lines and Planes in 3-Space
7 .6 Vector Spaces
7.7 Gram-Schmidt Orthogonalization Process
Review
CHAPTER 8 Matrices
Outline & Intro
8.1 Matrix Algebra
8.2 Systems of Linear Algebraic Equations
8.3 Rank of a Matrix
8.4 Determinants
8.5 Properties of Determinants
8.6 Inverse of a Matrix
8.7 Cramer's Rule
8.8 The Eigenvalue Problem
8.9 Powers of Matrices
8.10 Orthogonal Matrices
8.11 Approximation of Eigenvalues
8.12 Diagonalization
8.13 LU-Factorization
8.14 Cryptography
8.15 An Error-Correcting Code
8.16 Method of Least Squares
8.17 Discrete Compartmental Models
Review
CHAPTER 9 Vector Calculus
Outline & Intro
9.1 Vector Functions
9.2 Motion on a Curve
9.3 Curvature and Components of Acceleration
9.4 Partial Derivatives
9.5 Directional Derivative
9.6 Tangent Planes and Normal Lines
9.7 Curl and Divergence
9.8 Line Integrals
9.9 Independence of the Path
9.10 Double Integrals
9.11 Double Integrals in Polar Coordinates
9.12 Green's Theorem
9.13 Surface Integrals
9.14 Stokes' Theorem
9.15 Triple Integrals
9.16 Divergence Theorem
9.17 Change of Variables in Multiple Integrals
Review
PART 3 - Systems of Differential Equations
CHAPTER 10 Systems of Linear Differential Equations
Outline & Intro
10.1 Theory of Linear Systems
10.2 Homogeneous Linear Systems
10.3 Solution by Diagonalization
10.4 Nonhomogeneous Linear Systems
10.5 Matrix Exponential
Review
CHAPTER 11 Systems of Nonlinear Differential Equations
Outline & Intro
11.1 Autonomous Systems
11.2 Stability of Linear Systems
11.3 Linearization and Local Stability
11.4 Autonomous Systems as Mathematical Models
11.5 Periodic Solutions, Limit Cycles, and Global Stability
Review
PART 4 - Partial Differential Equations
CHAPTER 12 Orthogonal Functions and Fourier Series
Outline & Intro
12.1 Orthogonal Functions
12.2 Fourier Series
12.3 Fourier Cosine and Sine Series
12.4 Complex Fourier Series
12.5 Sturm-Liouville Problem
12.6 Bessel and Legendre Series
Review
CHAPTER 13 Boundary-Value Problems in Rectangular Coordinates
Outline & Intro
13.1 Separable Partial Differential Equations
13.2 Classical PDEs and Boundary-Value Problems
13.3 Heat Equation
13.4 Wave Equation
13.5 Laplace's Equation
13.6 Nonhomogeneous BVPs
13. 7 Orthogonal Series Expansions
13.8 Fourier Series in Two Variables
Review
CHAPTER 14 Boundary-Value Problems in Other Coordinate Systems
Outline & Intro
14.1 Problems in Polar Coordinates
14.2 Problems in Cylindrical Coordinates
14.3 Problems in Spherical Coordinates
Review
CHAPTER 15 Integral Transform Method
Outline & Intro
15.1 Error Function
15.2 Applications of the Laplace Transform
15.3 Fourier Integral
15.4 Fourier Transforms
15.5 Fast Fourier Transform
Review
CHAPTER 16 Numerical Solutions of Partial Differential Equations
Outline & Intro
16.1 Laplace's Equation
16.2 The Heat Equation
16.3 The Wave Equation
Review
PART 5 - Complex Analysis
CHAPTER 17 Functions of a Complex Variable
Outline & Intro
17.1 Complex Numbers
17.2 Powers and Roots
17.3 Sets in the Complex Plane
17.4 Functions of a Complex Variable
17.5 Cauchy-Riemann Equations
17.6 Exponential and Logarithmic Functions
17. 7 Trigonometric and Hyperbolic Functions
17.8 Inverse Trigonometric and Hyperbolic Functions
Review
CHAPTER 18 Integration in the Complex Plane
Outline & Intro
18.1 Contour Integrals
18.2 Cauchy-Goursat Theorem
18.3 Independence of the Path
18.4 Cauchy's Integral Formulas
Review
CHAPTER 19 Series and Residues
Outline & Intro
19.1 Sequences and Series
19.2 Taylor Series
19.3 Laurent Series
19.4 Zeros and Poles
19.5 Residues and Residue Theorem
19.6 Evaluation of Real Integrals
Review
CHAPTER 20 Conformal Mappings
Outline & Intro
20.1 Complex Functions as Mappings
20.2 Conformal Mappings
20.3 Linear Fractional Transformations
20.4 Schwarz-Christoffel Transformations
20.5 Poisson Integral Formulas
20.6 Applications
Review
APPENDICES
I. Derivative and Integral Formulas
II. Gamma Function
III. Table of Laplace Transforms
IV. Conformal Mappings
ANSWERS to Selected Odd-Numbered Problems
ch01
ch02
ch03
ch04
ch05
ch06
ch07
ch08
ch09
ch10
ch11
ch12
ch13
ch14
ch15
ch16
ch17
ch18
ch19
ch20
appII
INDEX
A
B
C
D
E
F
G-H
I
J-K-L
M
N
O
P
Q
R
S
T
U
V
W
X
Y-Z
Credits
Table of Laplace Transforms
Errata

Citation preview

  

$][Dfy
7 6agXDggy'D;Qy y46XXy0nDDgy paXS[Qg][y)y y y U[K]VAXD;Q B]Zy rrr VAXD;Q B]Zy $][Dfy
7 6agXDggy(D:[Qy A]]Wfy 8Cy _a]CpBgfy 9Dy 6q6SX6AXDygRa]pQRy Z]fgyA]]Wfg]aDfy6[Cy][XU[Dy A]]WfDXXEf y1]yB][g6Bgy$][Dfy
7 >XDhgy'D;Qy CSaDBgXty B6XXy  y K6syy]ay qSfSgy]payrDAfSgDy rrr VAXD;Q B]Z y

0pAfg6[j7yCSfB]p[gfy ][yApXWy `p6[jkDfy ]Ky$][Dfy
7 6agXDggy(D;Qy _pAXSB6gS][fy 6bDy6q6TX6AXDyg]yB]e]a6j][fy _a]LffS][6Xy 6ff]BS6k][fy6[Cy]gRDay `p6XSODCy ]aQ6[Su6j][f y]ayCDg6UXfy8Cyf_DBSOByCSfB]p[gyS[K]aZ6k][yB][g6BgygRDyf_DBS6Xyf6XDfyCD_6bgZD[gy6gy$][Dfy
7@Dggy(D:[QyqS6ygRDy6A]qDyB][g6BgyS[K]d6j][y ]ay fD[Cy6[yDZ6TXyg]yf_DBS6Xf6XDfVAXD;Q B]Z y

]_taSQRgvy  yAty%][Dfy
7 >XDhgy'D;Qy ''y6[yfBD[Cy 'D;Qy]Z_6[ty XXycQRgfyaDfDaqDC y*]y_6agy]KygRDyZ6gDaS6Xy_a]gDBgDCyAtygRSfyB]_taSQRgyZ6tyADyaD_a]CpBDCy]ayplYSuDCyU[y6[tyK^dyDXDBn][SBy]ayZDBR6[SB6XyU[BXpCT[Qy_R]g]B]_tw U[QyaDB]aCS[Qy]ayAty6[tyS[K]aZ6j][yfg]a6QDy8CyaDoDq6Xy ftfgDZy rSgR]pgyraSggD[y_HSffS][yP]ZygRDyB]_tcQRgy]r[Da y       
    Sfy 8yS[CD_D[CD[gy _pAXSB6j][y 6[CyR6fy []gyADD[y 6pgR]aSuDCy f_][f]aDCy ]ay]gRJSfDy 6__a]qDCy Aty gRDy ]r[Efy

]Ky gRDy n6CDZ9Wfy]ay fISBDyZ6aWfyaDKED[BDCyU[ygRSfy_a]CpBg y 0]ZDy TZ6QDfyU[ygRSfyA]]WyL6gpaDyZ]CDXf y1RDfDyZ]CDXfy C]y[]gy[DBDff6aSXtyD[C]afDy aD_aDfD[gy]ay _?BS_6gDyU[ygRDy6BjqSgSDfy aD_aDfD[gDCyU[ygRDyTZ6QDf y 1/%5$4/.7 1&%*327

RSDKysDBpkqDy+NBDay1tySDXCy ,aDfSCD[gy$6ZDfy!]ZDay 03,yCSg]aU[RSDKy)SBR6DXy%]R[f][y 03,y RSDKy)9WDl[Qy+KOBDayXSf][y) y,D[CDaQ6fgy ,pAXSfRDay6gRXDD[y 0DiDay 0D[S]ay B`pSfSk][fy CSg]ay1SZ]gRty [CEf][y )6[6QS[QyCSmayZty X]]Zy UaDBg]ay]Ky-]CpBj][yZty/]fDy ,a]CpBgS][yCSg]ay1SM8ty0XSgEy ,a]CpBgS][y ffSfg8gy SXDD[y4]agRXDty 0D[S]ay)9WDj[Qy)6[6QDay[CbD6yDa][u]y 3 , y )6[\K6BgpaS[Qy 6[Cy#[qD[g]aty ][n]Xy1RDaDfDy][[DXXy ]Z_]fSj][y_g6a6xy"[B y ]qEyDfSQ[y&aSfj[y y,9WDay /SQRgfy
7 ,R]g]y/DfD6aBRyffSfg6[gy

[6y(SB6g6y

]qEy"Z6QDyv1RDy ]DU[Qy ]Z_6[ty  y 7 aSQRgfyaDfDaqDC y ,c[k[Qy 8Cy S[CS[Qy]paSDay ]Z_6[SDfy ]qEy.[gU[Qy]paSEy ]Z_8SDfy /7 /1%&173)+2701/%5$3752&77      7

+#1"167/'7/.(1&227 "3",/(*.(+.5#-*$"4/.7"3"7 ! 7 D[[Sfy 7

Cq6[BDCy D[QS[DDaU[QyZ6gRDZ6jBfy7D[[Sfy  7

!7
7 4ilil"j of Line., 5v'tems li"earim;o" and Local 5tabilil"j Autonomo", Sv'tem, as M'lnematical Model' Periodic Solution •. Cveles. and Global Stabilil"j

SIS S16 S3S 0(2

Chapl"" in Raview

OSO

MISSING

MISSING

1]1 Integ,elion in the Complex Plene t8.1 11.2 113 11.4

lID

Series and Residues 19.t

'" '" '"

'" 19.6

ffil

Contou,lmeg"ls Caucnv-Gours.tTheorem Independenn or the Path c.ucnV·slotlgral Fennul.. Ch.pte,IB in Aeyiew

Sequen..s and Series Tavlor Series Llu,ent Seriu le,os and Peles Re,iduo, and Residu. Theorem Eyaluauon of Reallnlegrals Chapter 19 in A.yiew

Conformal Mappings

m., m, m, m.• m, m.

Comple, Functions a. Mappings Conro,mal Mapping. linea' F,.ctional Trln,fo,metionl s cflw. 'I-Christortel T,. nsfo 'm atie nI Poissen Imegnl ,ormulu Application. Dlaplor 20 in Anyiew

,,.. ,. '" '" .."', on

".

,.

88'

.,,.'" .., on

App..dixl

Oe'''a".e and Inlng,al formula.

APP·2

App.ldixtl

Gamma Function

APP·4

App..di,11I

Tlble 01 Llpllc" T,ansforms Confe,mal Mappings

APP-ll APp·9

App..dixlY

Answors to Soloctod Odd·Numberod P,oblems lodex Crodil$

... C¥1_

.,••'"

826

,.,

ANS·l

C,




/D 1=488D 8='D 8D "

0"A0A=E 14D $-.&;&3?, 0E &9A ?,53=E :D 1/9/9D )8D )4,AD 89/4)B D ?*&1 ?,"=E81-9)-8D >4)8D 1/8)4,AD 4/9D .)D )/89)9=9)1/8D D 9@9D 9)9,D $B 3"&$E3(,3&&;,3(E ?*&1 ?,"=E

=9D :D 1/9/9D 1D D 1=48D 9)9,D &3(,3&&;,3(E1 -1/&D )

)8D 9'4#4D D 1-2/)=-D 1D -/AD -:-9),D 912)8D ,,D 1D ?')'D 4D ,118,AD 4,9D AD:D @2)/9D 1D )/&D ):4D/D 14D =8%,D )/D 1=488D )/D 8)/D /D /&)/4)/&D 14D )/D 8=83=/9D 448D )/D :8D 48D '4D )8D ,)94,,AD /1D =224D 1=/D 91D 9'D /=-4D 1D 912)8D :9D 1=,D D )/,=D )/D D 9@9D 8='D 8D 9')8D 1/83=/9,AD 9')8D 11+D 4248/98D9'D =:148
D 12)/)1/8D 9D 9')8D 9)-D 1D ?'9D 1/89)9=98D &3(,3&&;,3(E1

?*&1 ?,"= E


      14D "@)),)9AD )/D 912)D 8,9)1/D :D =7/9D 9@9D )8D )>)D )/91D !>D -*14D 2498D 8D /D D 8/D $1-D :D 9)9,8D 1D :8D >4)1=8D 2498D )9D 8'1=,D D 1>)1=8D :9D )9D )8D 1=4D ,)D 9'9D :D +1/D 1D 8)//&)/4)/&4,9D -:-9)8D )8D :D :14AD /D 22,)9)1/D 1D 14)/4AD /D24;,D) 4/9),D3=9)1/8D

:>E E :#+2:DE+''%:%2>+/E8@>+42%: