Solving Problems in Mathematical Analysis. Part II Definite, Improper and Multidimensional Integrals, Functions of Several Variables and Differential Equations 9783030368470, 9783030368487

Table of contents :
Preface
Contents
Definitions and Notation
1 Exploring the Riemann and Definite Integral
1.1 Examining the Integrability of Functions
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
1.2 Finding Riemann Integrals by Definition
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
1.3 Finding Limits of Certain Sequences
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
1.4 Calculating Various Interesting Definite Integrals
Problem 1
Solution
Problem 2
Solution
Integral (a)
Integral (b)
Integral (c)
Problem 3
Solution
Problem 4
Solution
Problem 5
Solution
Problem 6
Solution
1.5 Explaining Several Apparent Paradoxes
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
1.6 Using the (Second) Mean Value Theorem
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
1.7 Exercises for Independent Work
2 Examining Improper Integrals
2.1 Investigating the Convergence of Integrals by Definition
Problem 1
Solution
Problem 2
Solution
2.2 Using Different Criteria
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
Problem 5
Solution
2.3 Using Integral Test for Convergence of Series
Problem 1
Solution
Problem 2
Solution
2.4 Exercises for Independent Work
3 Applying One-Dimensional Integrals to Geometry and Physics
3.1 Finding Lengths of Curves
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
3.2 Calculating Areas of Surfaces
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
3.3 Finding Volumes and Surface Areas of Solids of Revolution
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
3.4 Finding Various Physical Quantities
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
Problem 5
Solution
3.5 Exercises for Independent Work
4 Dealing with Functions of Several Variables
4.1 Finding Images and Preimages of Sets
Problem 1
Solution
Problem 2
Solution
4.2 Examining Limits and Continuity of Functions
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
Problem 5
Solution
4.3 Exercises for Independent Work
5 Investigating Derivatives of Multivariable Functions
5.1 Calculating Partial and Directional Derivatives
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
5.2 Examining Differentiability of Functions
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
5.3 Exercises for Independent Work
6 Examining Higher Derivatives, Differential Expressions, and Taylor's Formula
6.1 Verifying the Existence of the Second Derivatives
Problem 1
Solution
Problem 2
Solution
6.2 Transforming Differential Expressions and Operators
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
Problem 5
Solution
6.3 Expanding Functions
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
6.4 Exercises for Independent Work
7 Examining Extremes and Other Important Points
7.1 Looking for Global Maxima and Minima of Functions on Compact Sets
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
7.2 Examining Local Extremes and Saddle Points of Functions
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
7.3 Exercises for Independent Work
8 Examining Implicit and Inverse Functions
8.1 Investigating the Existence and Extremes of Implicit Functions
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
Problem 5
Solution
8.2 Finding Derivatives of Inverse Functions
Problem 1
Solution
Problem 2
Solution
8.3 Exercises for Independent Work
9 Solving Differential Equations of the First Order
9.1 Finding Solutions of Separable Equations
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
9.2 Solving Homogeneous Equations
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
9.3 Solving Several Specific Equations
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
Problem 5
Solution
9.4 Solving Exact Equations
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
Problem 5
Solution
9.5 Exercises for Independent Work
10 Solving Differential Equations of Higher Orders
10.1 Solving Linear Equations with Constant Coefficients
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
10.2 Using Various Specific Methods
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
10.3 Exercises for Independent Work
11 Solving Systems of First-Order Differential Equations
11.1 Using the Method of Elimination of Variables
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
11.2 Solving Systems of Linear Equations with ConstantCoefficients
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
Problem 5
Solution
11.3 Exercises for Independent Work
12 Integrating in Many Dimensions
12.1 Examining the Integrability of Functions
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
12.2 Calculating Integrals on Given Domains
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
12.3 Changing the Order of Integration
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
12.4 Exercises for Independent Work
13 Applying Multidimensional Integrals to Geometry and Physics
13.1 Finding Areas of Flat Surfaces
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
13.2 Calculating Volumes of Solids
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
13.3 Finding Center-of-Mass Locations
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
13.4 Calculating Moments of Inertia
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
13.5 Finding Various Physical Quantities
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
13.6 Exercises for Independent Work
Index

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