A Problem Book in Mathematical Analysis (Old Edition) 5th Edition  (English, G N Berman)

Table of Contents

Introduction: Mathematical Analysis

1. Function

• Preliminaries
• Simplest Properties of Functions
• Basic Elementary Functions
• Inverse Function, Power Exponential and Logarithmic Functions
• Trignometric and Inverse Trignometric, Functions
• Computational Problems

2. Limit, Continuity

• Basic Definitions
• Infinite Magnitudes, Tests for the Existence of the Limit
• Continuous Functions
• Finding Limits, Comparison of Infinitesimals

 

3. Derivative and Differential. Differential Calculus

• Derivative. The Rate of Change of a Function
• Differentiating Functions
• Differential. Differentiability of a Function
• The Derivative as the Rate of Change
• Repeated Differentiation

4. Investigating Functions and Their Graphs

• Behaviour of a Function
• Application of the First Derivative
• Application of the Second Derivative
• Additional Items. Solving Equations
• Taylor’s Formula and Its Application
• Curvature
• Computational Problems

 

5. The Definite Integral

• The Definite Integral and Its Simplest Properties
• Basic Properties of the Definite Integral

 

6. Indefinite Integral. Integral Calculus

• Simplest Integration Rules
• Basic Methods of Integration
• Basic Classes of Integrable Functions

 

7. Methods for Evaluating Definite Integrals Improper Integrals

• Methods for Exact Evaluation of Integrals
• Approximate Methods
• Improper Integrals

8. Application of Integral Calculus

• Some Problems in Geometry and Statics
• Some Physics Problems

9. Series

• Numerical Series
• Functional Series
• Power Series
• Some Applications of Taylor’s Series

10. Functions of Several Variables. Differential Calculus

• Functions of Several Variables
• Differential Calculus
• Simplest Properties of Functions
• Derivatives and Differentials of Functions of Several Variables
• Differentiating Functions
• Repeated Differentiation

11. Application of Differential Calculus of Functions of Several Variables

• Taylor’s Formula. Extrema of Functions of Several Variables
• Plane Curves
• Vector Function of a Scalar Argument. Space Curves. Surfaces
• Scalar Field. Gradient. Directional Derivative

12. Multiple Integrals

• Double and Triple Integrals
• Multiple Integration
• Integrals in Polar, Cylindrical and Spherical Coordinates
• Application of Double and Triple Integrals
• Improper Integrals. Integrals Dependent on Parameters

 

13. Line Integrals and Surface Integrals

• Line Integrals with Respect to Arc Length
• Line Integrals with Respect to Coordinates
• Surface Integrals

14. Differential Equations

• Equations of the First Order
• General Differential Equations of the First Order
• Equations of the Second and Higher Orders
• Linear Equations
• Systems of Differential Equations
• Computational Problems

15. Trigonometric Series

• Trigonometric Polynomials
• Fourier Series
• Kryloy’s Method. Harmonic Analysis

 

16. Elements of Field Theory

 

Answers

Appendix