A Collection of Problems on a Course of Mathematical Analysis

International Series of Monographs in Pure and Applied Mathematics

1st Edition - January 1, 1965

Author: G. N. Berman

Editors: I. N. Sneddon, M. Stark, S. Ulam

eBook ISBN:

9 7 8 - 1 - 4 8 3 1 - 3 7 3 4 - 6

A Collection of Problems on a Course of Mathematical Analysis is a collection of systematically selected problems and exercises (with corresponding solutions) in mathematical… Read more

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A Collection of Problems on a Course of Mathematical Analysis is a collection of systematically selected problems and exercises (with corresponding solutions) in mathematical analysis. A common instruction precedes a group of problems of the same type. Problems with a physics content are preceded by the necessary physical laws. In the case of more or less difficult problems, hints are given in the answers.

This book is comprised of 15 chapters and begins with an overview of functions and methods of specifying them; notation for and classification of functions; elementary investigation of functions; and trigonometric and inverse trigonometric functions. The following chapters deal with limits and tests for their existence; differential calculus, with emphasis on derivatives and differentials; functions and curves; definite and indefinite integrals; and methods of evaluating definite integrals. Some applications of the integral in geometry, statics, and physics are also considered; along with functions of several variables; multiple integrals and iterated integration; line and surface integrals; and differential equations. The final chapter is devoted to trigonometric series.

This monograph is intended for students studying mathematical analysis within the framework of a technical college course.

Foreword to the Tenth Russian Edition

I. Functions

1. Functions and Methods of Specifying Them

2. Notation for and Classification of Functions

3. Elementary Investigation of Functions

4. Elementary Functions

5. The Inverse Functions. Power, Exponential and Logarithmic Functions

6. The Trigonometric and Inverse Trigonometric Functions

7. Numerical Problems

II. Limits

1. Basic Definitions

2. Orders of Magnitude. Tests for the Existence of a Limit

3. Continuous Functions

4. Finding Limits. Comparison of Infinitesimals

III. Derivatives and Differentials. Differential Calculus

1. Derivatives. The Rate of Change of a Function

2. Differentiation of Functions

3. Differentials. Differentiability of a Function

4. Derivative as Rate of Change (Further Examples)

5. Repeated Differentiation

IV. The Investigation of Functions and Curves

1. The Behavior of a Function "at a Point"

2. Applications of the First Derivative

3. Applications of the Second Derivative

4. Auxiliary Problems. Solution of Equations

5. Taylor's Formula and its Applications

6. Curvature

7. Numerical Problems

V. The Definite Integral

1. The Definite Integral and its Elementary Properties

2. Fundamental Properties of the Definite Integral

VI. The Indefinite Integral. Integral Calculus

1. Elementary Examples of Integration

2. Basic Methods of Integration

3. Basic Classes of Integrable Functions

VII. Methods of Evaluating Definite Integrals. Improper Integrals

1. Methods of Exact Evaluation of Integrals

2. Approximation Methods

3. Improper Integrals

VIII. Applications of the Integral

1. Some Problems of Geometry and Statics

2. Some Problems of Physics

IX. Series

1. Numerical Series

2. Functional Series

3. Power Series

4. Some Applications of Taylor's Series

5. Numerical Problems

X. Functions of Several Variables. Differential Calculus

1. Functions of Several Variables

2. Elementary Investigation of a Function

3. Derivatives and Differentials of Functions of Several Variables

4. Differentiation of Functions

5. Repeated Differentiation

XI. Applications of the Differential Calculus for Functions of Several Variables

1. Taylor's Formula. Extrema of Functions of Several Variables

2. Plane Curves

3. Vector Functions of a Scalar Argument. Curves in Space. Surfaces

4. Scalar Field. Gradient. Directional Derivative

XII. Multiple Integrals and Iterated Integration

1. Double and Triple Integrals

2. Iterated Integration

3. Integrals in Polar, Cylindrical and Spherical Coordinates

4. Applications of Double and Triple Integrals

5. Improper Integrals. Integrals Depending on a Parameter

XIII. Line and Surface Integrals

1. Line Integrals

2. Coordinate Line Integrals

3. Surface Integrals

XIV. Differential equations

1. Equations of the First Order

2. Equations of the First Order (Continued)

3. Equations of the Second and Higher Orders

4. Linear Equations

5. Systems of Differential Equations

6. Numerical Problems

XV. Trigonometric Series

1. Trigonometric Polynomials

2. Fourier Series

3. Krylov's Method. Harmonic Analysis

XVI. Elements of the Theory of Fields

Answers

Chapter I

Chapter II

Chapter III

Chapter IV

Chapter V

Chapter VI

Chapter VII

Chapter VIII

Chapter IX

Chapter X

Chapter XI

Chapter XII

Chapter XIII

Chapter XIV

Chapter XV

Chapter XVI

Appendix. Tables

1. Trigonometric Functions

2. Hyperbolic Functions

3. Reciprocals, Square and Cube Roots, Logarithms, Exponential Functions

Index

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