Problems In Mathematics With Hints And Solutions by V. Govorov; P. Dybov; N. Miroshin; S. Smirnova

The book contains more than three thousand mathematics problems and covers each topic taught at school. The problems were contributed by 120 of the higher schools of the USSR and all the universities.

The book is divided into, four parts: algebra and trigonometry, fundamentals of analysis, geometry and vector algebra, and the problems and questions set during oral examinations. The authors considered it necessary to include some material relating to complex numbers, combinatorics, the binomial theorem, elementary trigonometric inequalities, and set theory and the method of coordinates. The authors believe that this material will help the readers systematize their knowledge in the principal divisions of mathematics.

In writing the book, the authors have used their experience of examining students in mathematics at higher schools and the preparation of television courses designed to help students revise their knowledge for the entrance examinations to higher educational establishments.

To make it easier for readers to grasp the material, some of the sections have been supplemented with explanatory text. The problems are all answered and some have additional hints or complete solutions.

The more difficult problems are marked with asterisks. Part 4 is entitled “Oral Examination Problems and Questions” and includes samples suggested by the higher schools.

The authors hope that this book will help those who want to enter the various types of higher school, aid the teachers, and be of use to all those who want to deepen and systematize their knowledge of mathematics.

EDITED BY PROF. A.I. PRILEPKO, D.Sc.

Translated from the Russian by Irene Aleksanova

You can get the book here and here.

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Contents

Preface 5
Part 1 Algebra Trigonometry and Elementary Functions 9
1.1 Problems on Integers Criteria for Divisibility 9
1.2 Real Numbers Transformation of Algebraic Expressions 13
1.3 Mathematical Induction Elements of Combinatorics Binomial Theorem
1.4 Equations and Inequalities of the First and the Second Degree
1.5 Equations of Higher Degrees Rational Inequalities
1.6 Irrational Equations and Inequalities
1.7 Systems of Equations and Inequalities
1.8 The Domain of Definition and the Range of a Function
1.9 Exponential and Logarithmic Equations and Inequalities
1.10 Transformations of Trigonometric Expressions Inverse Trigonometric Functions
1.11 Solution of Trigonometric Equations Inequalities and Systems of Equations
1.12 Progressions
1.13 Solution of Problems on Derivation of Equations
1.14 Complex Numbers
Part 2 Fundamentals of Mathematical Analysis
2.1 Sequences and Their Limits An Infinitely Decreasing Geometric Progression Limits of Functions
2.2 The Derivative Investigating the Behaviour of Functions with the Aid of the Derivative
2.3 Graphs of Functions
2.4 The Antiderivative The Integral The Area of a Curvilinear Trapezoid
Part 3 Geometry and Vector Algebra
3.1 Vector Algebra
3.2 Plane Geometry Problems on Proof
3.3 Plane Geometry Construction Problems
3.4 Plane Geometry Calculation Problems
3.5 Solid Geometry Problems on Proof
3.6 Solid Geometry Calculation Problems
Part 4 Oral Examination Problems and Questions 241
4.1 Sample Examination Papers 241
4.2 Problems Set at an Oral Examination 244
Hints and Answers 265
Appendix 386

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This book aims to equip readers with the mathematical physics skills necessary to solve problems in mechanics, heat conduction, and electromagnetism. It covers a wide range of topics, from basic to advanced, and is intended for both students and researchers. The book provides hints for solving problems and includes detailed solutions to selected ones. Readers should have a solid background in applied mathematics to fully benefit from the book, but most problems in the earlier chapters are accessible to those with a basic understanding of mathematical physics methods.

Translated from the Russian by Richard A. Silverman.

Get the book here.

Contents

PART 1

CONTENTS

PROBLEMS, Page 1

DERIVATION OF EQUATIONS AND FORMULATION OF PROBLEMS, Page 3

1. Mechanics, 3
2. Heat Conduction, 9
3. Electricity and Magnetism, 11

SOME SPECIAL METHODS FOR SOLVING HYPERBOLIC AND ELLIPTIC EQUATIONS, Page 19

1. Hyperbolic Equations, 19
2. Elliptic Equations: The Green’s Function Method, 27
3. Elliptic Equations: The Method of Conformal Mapping, 33

STEADY-STATE HARMONIC OSCILLATIONS, Page 42

1. Elastic Bodies: Free Oscillations, 43
2. Elastic Bodies: Forced Oscillations, 46
3. Electromagnetic Oscillations, 49

THE FOURIER METHOD, Page 55

1. Mechanics: Vibrating Systems, Acoustics, 60
2. Mechanics: Statics of Deformable Media, Fluid Dynamics, 73
3. Heat Conduction: Nonstationary Problems, 77
4. Heat Conduction: Stationary Problems, 83
5. Electricity and Magnetism, 91

THE EIGENFUNCTION METHOD FOR SOLVING INHOMOGENEOUS PROBLEMS, Page 103

1. Mechanics: Vibrating Systems, 107
2. Mechanics: Statics of Deformable Media, 114
3. Heat Conduction: Nonstationary Problems, 119
4. Heat Conduction: Stationary Problems, 124
5. Electricity and Magnetism, 131

INTEGRAL TRANSFORMS, Page 143

1. The Fourier Transform, 146
2. The Hankel Transform, 160
3. The Laplace Transform, 169
4. The Mellin Transform, 189
5. Integral Transforms Involving Cylinder Functions of Imaginary Order, 194

CURVILINEAR COORDINATES, Page 203

1. Elliptic Coordinates, 204
2. Parabolic Coordinates, 210
3. Two-Dimensional Bipolar Coordinates, 212
4. Spheroidal Coordinates, 219
5. Paraboloidal Coordinates, 231
6. Toroidal Coordinates, 233
7. Three-Dimensional Bipolar Coordinates, 242
8. Some General Problems on Separation of Variables, 247

INTEGRAL EQUATIONS, Page 253

1. Diffraction Theory, 254
2. Electrostatics, 259

PART 2 SOLUTIONS, Page 273

MATHEMATICAL APPENDIX, Page 381

1. Special Functions Appearing in the Text, 381
2. Expansions in Series of Orthogonal Functions, 384
3. Some Definite Integrals Frequently Encountered in the Applications, 386
4. Expansion of Some Differential Operators in Orthogonal Curvilinear Coordinates, 388

Supplement: VARIATIONAL AND RELATED METHODS, Page 391

1. Variational Methods, 392
   1.1 Formulation of Variational Problems, 392
   1.2 The Ritz Method, 396
   1.3 Kantorovich’s Method, 401
2. Related Methods, 404
   2.1 Galerkin’s Method, 404
   2.2 Collocation, 407
   2.3 Least Squares, 411

References, 412

BIBLIOGRAPHY, Page 415

NAME INDEX, Page 423

SUBJECT INDEX, Page 427

https://mirtitles.org/2024/10/09/worked-problems-in-applied-mathematics-by-n-n-lebedev-i-p-skalskaya-y-s-uflyand/

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