Solving Problems in Mathematical Analysis, Part II: Definite, Improper and Multidimensional Integrals, Functions of Several Variables and Differential Equations (Problem Books in Mathematics)

✍ Scribed by Tomasz Radożycki

Publisher: Springer
Year: 2020
Tongue: English
Leaves: 389
Series: Problem Books in Mathematics
Edition: 1st ed. 2020
Category: Library

No coin nor oath required. For personal study only.

✦ Synopsis

This textbook offers an extensive list of completely solved problems in mathematical analysis. This second of three volumes covers definite, improper and multidimensional integrals, functions of several variables, differential equations, and more. The series contains the material corresponding to the first three or four semesters of a course in Mathematical Analysis.

Based on the author’s years of teaching experience, this work stands out by providing detailed solutions (often several pages long) to the problems. The basic premise of the book is that no topic should be left unexplained, and no question that could realistically arise while studying the solutions should remain unanswered. The style and format are straightforward and accessible. In addition, each chapter includes exercises for students to work on independently. Answers are provided to all problems, allowing students to check their work.

Though chiefly intended for early undergraduate students of Mathematics, Physics and Engineering, the book will also appeal to students from other areas with an interest in Mathematical Analysis, either as supplementary reading or for independent study.

✦ 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