Solving Problems in Mathematical Analysis, Part III (Problem Books in Mathematics)

✍ Scribed by Radożycki

• Publisher: Springer
• Year: 2020
• Tongue: English
• Leaves: 386
• Category: Library

✦ Synopsis

This textbook offers an extensive list of completely solved problems in mathematical analysis. This third of three volumes covers curves and surfaces, conditional extremes, curvilinear integrals, complex functions, singularities and Fourier series. 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 undergraduatestudents 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 Examining Curves and Surfaces
1.1 Finding Curvature and Torsion of Curves
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
1.2 Examining k-Surfaces in N Dimensions
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
1.3 Examining Ruled Surfaces
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
1.4 Exercises for Independent Work
2 Investigating Conditional Extremes
2.1 Using the Method of the Lagrange Multipliers
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
2.2 Looking for Global Extremes
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
2.3 Exercises for Independent Work
3 Investigating Integrals with Parameters
3.1 Examining Limits and Continuity
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
3.2 Differentiating with Respect to Parameters
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
Problem 5
Solution
Problem 6
Solution
3.3 Integrating over Parameters
Problem 1
Solution
Problem 2
Solution
3.4 Exercises for Independent Work
4 Examining Unoriented Curvilinear Integrals
4.1 Finding Area of Surfaces
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
4.2 Calculating Various Curvilinear Integrals
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
4.3 Exercises for Independent Work
5 Examining Differential Forms
5.1 Studying the Exterior Forms Operating on Vectors
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
5.2 Performing Various Operations on Differential Forms
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
Problem 5
Solution
5.3 Calculating Exterior Derivatives
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
5.4 Looking for Primitive Forms
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
Problem 5
Solution
5.5 Finding Potentials in R3
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
Problem 5
Solution
5.6 Exercises for Independent Work
6 Examining Oriented Curvilinear Integrals
6.1 Calculating Integrals over Curves
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
6.2 Calculating Integrals over Surfaces
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
6.3 Using Stokes' Theorem
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
Problem 5
Solution
Problem 6
Solution
6.4 Exercises for Independent Work
7 Studying Functions of Complex Variable
7.1 Examining the Holomorphicity of Functions
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
7.2 Finding Domains of Convergence of Complex Series
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
7.3 Calculating Contour Integrals
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
Problem 5
Solution
7.4 Using Cauchy's Theorem
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
7.5 Looking for Images of Sets
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
7.6 Exercises for Independent Work
8 Investigating Singularities of Complex Functions
8.1 Identifying the Types of Singularities
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
8.2 Expanding Functions into Laurent Series
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
8.3 Using the Residue Theorem to Calculate Definite Integrals
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
8.4 Using Residue Theorem to Find Sums of Series
Problem 1
Solution
Problem 2
Solution
8.5 Exercises for Independent Work
9 Dealing with Multivalued Functions
9.1 Analytically Continuing Functions
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
9.2 Calculating Integrals Involving Functions with Branch Points
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
Problem 5
Solution
Problem 6
Solution
Problem 7
Solution
Problem 8
Solution
9.3 Exercises for Independent Work
10 Studying Fourier Series
10.1 Examining Expandability of Functions into Fourier Series
Problem 1
Solution
Problem 2
Solution
10.2 Finding Fourier Coefficients
Problem 1
Solution
Problem 2
Solution
Problem 3
Solution
Problem 4
Solution
Problem 5
Solution
10.3 Exercises for Independent Work
Index