Higher mathematics. Problem solving and variants of typical
calculations
T. I
Educational
textbook
ББК
22.1я7
К 93
УДК
517.2; 517.3
Автори: Архіпова O.C., Болотіна Л.В., Кашуба Ж.Б., Корніль Т.Л.,
Курпа Л.І., Курпа Л.В.,
Лемешева Л.П., Лінник
Г.Б.,
Протопопова
В.П., Ярошенко
А.Р., Ясницька
Н.М.
Higher
mathematics. Problem solving and
variants of typical calculations. T. I: Educational textbook / Under edition of Dr. Sci. Tech. Kurpa
L.V. - Kharkiv: NTU "KhPI",
2004. - 320 p. - English.
ISBN
966-593-323-X
ISBN 966-593-324-8
Навчальний посібник містить теоретичний довідковий матеріал з основних розділів вищої математики: лінійної алгебри, аналітичної геометрії та математичного аналізу, зразки розв'язання типових задач та варіанти типових розрахунків. В першому томі посібника розглянуто 226 прикладів. Типові завдання для індивідуального виконання складаються з 30 варіантів.
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 educational textbook includes
theoretical material of main sections of higher mathematics: linear algebra,
analytical geometry and calculus, examples of solving typical problems and
variants of typical calculations. The first part of the textbook contains 226
solved examples. The typical calculations for individual solving include 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.
Іл.
74. Табл. 7. Бібліогр.:
16 назв.
Part 1
Introduction
Chapter
1. Matrixes,
determinants, solutions of systems of linear equations
1.1.
Matrixes and operations on them
1.2. Determinants
1.3. Inverse matrix. Rank of matrix
1.4. Solution of system of linear
equations
1.4.1. General
concepts
1.4.2. The rale by Cramer
1.4.3. Solving of system of equations
using inverse matrix
1.4.4. Systems of т linear equations in n unknown variables
The theorem by Kronecker
– Kapelly
1.4.5. Solution of a homogeneous SLAE
1.4.6. The
method by Gauss (or the method of sequential elimination
1.4.7. The
method by Jordan – Gauss
Control
tasks to the chapter 1
Chapter
2. Vector algebra
2.1. Main
concepts
2.2. Scalar
product
2.3. Vector
product
2.4. Mixed
product of vectors
2.5. Linear
n-dimensional space
2.6. Linear
dependence of vectors
2.7. Vector
decomposition on the given basis
2.8. Transition
to a new basis
2.9. Euclidean
space
2.10.
The linear operators
2.11.
The eigenvectors and eigenvalues of the linear
operator
2.12.
Quadratic forms
Control
tasks to the chapter 2
Chapter
3. Surfaces
and curves of the 1st and the 2nd order
3.1.
Surfaces and curves of the first order. Plane and straight line
3.2.
Curves and surfaces of the second order
3.2.1.
Curves of the second order
3.2.2.
Investigation of general equation of the second order
3.2.3.
Surfaces of the second order
Control
tasks to the chapter 3
Chapter
4. Limits
and continuity of a function of one variable
Basic elementary functions
4.1. Limit,
continuity of function
4.1.1. Limit
of a numeric sequence
4.1.2. Infinitesimal
and their main properties
Comparison
of two infinitesimal values
4.1.3. Indefinitely
large values
Comparison
of two indefinitely large values
4.1.4. Basic
theorems concerning math operations
4.1.5. Limit
of a function. I and II remarkable limits and
4.1.6. Techniques
of the limits calculus
4.1.7. Comparison
of two infinitesimal
4.2. Continuity of a function
4.3. Points
of discontinuity and their classification
4.4.
Arithmetic operations on continuous functions Continuity of composite and inverse functions
4.5.
Properties of
the functions continuous on closed intervals
4.6. Concept of uniformly continuous function
Control tasks
to the chapter 4
Chapter
5. The fundamentals of the differential
calculus for functions
5.1.
Derivative
5.2.
Differential of a function
5.3.
Derivatives and differentials of the higher orders
5.4.
The applications of derivatives to functions investigating plots constructing
and limits calculating
5.4.1. The
averaging theorems
5.4.2. Disclosing indeterminate forms by L'Hospital's rale
5.4.3.
5.4.4. Requirement of the monotonicity of the function. The extremums
5.4.5. Convexity and concavity of a
curve. Points of inflection
5.4.6. Asymptotes of curves
5.4.7. General plan for
investigating a function and constructing its plot
5.4.8. Minimum and maximum of a function
in an interval
Control tasks to the chapter
5
Chapter 6. An
indefinite integral. Methods of integrating
6.1. Antiderivative,
properties of an indefinite integral
6.2. Methods of integrating
6.2.1. Integrating by substitution
(Change of variable)
6.2.2. Method of integrating in parts
6.2.3. Integration of rational fractions
Control tasks to the
chapter 6
Chapter 7. Definite integral and
its applications
7.1. Definition, properties, geometrical meaning of a
definite integral Methods of their calculus
7.2. Geometric applications of the
definite integral, computing volumes,
7.3. Physical and mechanical
applications of a definite integral
Control tasks to the
chapter 7
Chapter 8. Improper integrals, problems
of their convergence
8.1. Improper integrals with infinite
limits of integration (I kind)
8.2. Improper integrals of the 2nd
kind - integrals of unlimited functions
8.3. Main values of improper integrals
Control tasks to the
chapter 8
References