Higher mathematics

Problem solving and variants of typical calculations

T. II

Educational textbook

ББК22.1я7

К93

УДК 517.2; 517.3

Автори: Архіпова О.С., Болотіна Л.В., Васильченко В.Ф., Вовк В.М.,

Волкова Н.М., Кашуба Ж.Б., Кірілова Н.О., Корніль, Т.Л., Курпа Л.В., Курпа Л.І., Лінник Г.Б., Столбова Т.В., Щербініна Т.Є., Ярошенко О.Р., Ясницька Н.М.

Higher mathematics. Problem solving and variants of typical calculations. T. II: Educational textbook / Under edition of Dr.Sci.Tech. Kurpa L.V. - Kharkiv: NTU "KhPІ", 2004 - 306 p. - English.

ISBN 966-593-323-X

ISBN 966-593-325-6

Другий том навчального посібника містить теоретичний довідковий матеріал з диференціального та інтегрального числення функцій багатьох змінних, зразки розв'язання типових задач та варіанти типових розрахунків. В цьому томі розв'язано понад 190 прикладів та задач. Типові розрахунки містять від 5 до 20 завдань, кожне з яких складається з 30 варіантів для індивідуального виконання.

Посібник призначається для студентів інженерно-фізичних, машинобудівних та економічних спеціальностей, а також може бути корисним викладачам, аспірантам. науковцям i всім, хто має справу з застосуванням вищої математики для вирішення науково-технічних проблем.

The second volume of the handbook contains fundamentals of differential calculus of functions of several variables, multiple, curvilinear and surface integrals, field theory and sets, examples of solving typical problems and variants of typical calculations. The second part of the textbook contains 190 solved examples. The typical calculations for individual solving include from 5 to 20 typical calculation each of them contain 30 variants.

For students of physical engineering, energomashinebuilding and economic specialties. Can be useful for teachers, post-graduate students, scientists and those, who applies higher mathematics for solving scientific and technical problems.

Iл. 74. Табл. 7. Бібліогр.: 8 назв.

Content Part II

Introduction

Chapter 9. Functions of many variables

9.1. Main concepts

9.2. Partial derivatives and total differential

9.3. Differentiation of a composite function

9.4. Derivatives and differentials of the higher orders

9.5. Extremum of a function of many variables Conditional extremum

9.5.1. Investigation of a function of many variables on (unconditional) local extremum

9.5.2. Investigation of a function of many variables on conditional Extremum

9.6. Change of variables in differential equations

9.6.1. Functions of one variable

9.6.2. Functions of several variables

9.7. Geometrical applications of differential calculus of functions of many variables

The control tasks to the chapter 9

Chapter 10. Differential equations

10.1. Differential equations of the I^st order

10.1.1 Differential equations solved relatively derivative

10.1.2 Equations with separable variables

10.1.3 Homogeneous differential equations

10.1.4 Linear differential equations

10.1.5 Bernoulli's relation

10.2. Envelope of a one-parametric family of plane curves

10.2.1 Special solutions of differential equations

10.2.2 Equations not solved relatively derivative

10.2.3 Equations by Lagrange and Clairaut

10.3 Equations allowing deflating

10.4 Linear homogeneous differential equations

10.4.1 Solving LHDE with constant coefficients

10.5 Linear inhomogeneous differentia] equations of the second order

10.6 Linear inhomogeneous differential equations of the n* order with constant сoefficients and special right part. The method of undefined coefficients

10.7 Systems of differential equations

10.7.1 Systems of linear homogeneous equations with constant Coefficients

10.7.2 Systems of linear inhomogeneous equations with constant Coefficients

10.7.3 Method of integrating factors

10.7.4 Method of elimination

The control tasks to the chapter 10

Chapter 11. Numeric and functional series

11.1 Numeric series. Basic concepts Necessary condition of convergence

11.2 Sufficient conditions of series convergence with nonnegative Members

11.2.1 Signs of comparison

11.2.2 The sign by d'Alembert

11.2.3 The radical sign by Cauchy

11.2.4 Integral sign of convergence by Cauchy

11.3 Alternating series. Absolute and conditional convergence

11.4 Functional series

11.4.1 Power series

11.4.2 Taylor series. Application of series in approximate calculations

11.5 Fourier series

11.5.1 Expanding periodical function in a Fourier series

11.5.2 Fourier series for even and odd periodical functions

11.5.3 Periodical continuation and expanding a non-periodical function in a Fourier series

11.5.4 Expanding in a Fourier series functions defined on the segment [0,l]

11.5.5 Complex form of a Fourier series

11.6 Fourier integral

11.6.1 Representation of a function as a Fourier integral

11.6.2 Fourier integral in a complex form

The control tasks to the chapter 11

Chapter 12. Multiple integrals

12.1 Double integrals and their calculus in the Cartesian coordinate system

12.2 Change of variables in double integral.Calculation of double integral in the polar coordinates

12.3 Applications of double integrals

12.4 Triple integrals and their calculus in the Cartesian coordinate system

12.5 Triple integral in cylindrical and spherical coordinate systems

The control tasks to the chapter 12

Chapter 13. Contour and surface integrals

13.1 Contour integrals of the I-st kind

13.1.1 Calculus of contour integrals of the I-st kind

13.1.2 Application of contour integrals of the I-st kind

13.1.3 Examples of calculus of contour integrals of the I-st kind

13.2 Contour integrals of the П-nd kind

13.2.1 Examples of calculus of contour integrals of the П-nd kind

13.3 Surface integrals

133.1 Surface integrals of the I-st kind

133.2 Surface integrals of the II-nd kind

The control tasks to the chapter 13

Chapter 14. Elements of a field theory

14.1 Scalar field. Basic characteristics

14.2 Derivative on a direction and gradient of a scalar field

14.3 Vector field

14.4 Flow of vector field through a surface. Definition. Ways of calculus

14.5 Linear integral in a vector field. Circulation

The control tasks to the chapter 14

References