A Collection of Problems on a Course of Mathematical Analysis

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 Behaviour 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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