Book Summary of A Problem Book in Mathematical Analysis
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