This two-volume book was written for students of technical colleges
who have had the usual mathematical training. It contains just enough information to continue with a wide variety of engineering disciplines. It covers analytic geometry and linear algebra, differential and integral calculus for functions of one and more variables, vector analysis, numerical and functional series (including Fourier series), ordinary differential equations, functions of a complex variable, Laplace and Fourier transforms, and equations of mathematical physics. This list itself demonstrates that the book covers the material for both a basic course in higher mathematics and several specialist sections that are important for applied problems. Hence, it may be used by a wide range of readers. Besides students in technical colleges and those starting a mathematics course, it may be found useful by engineers and scientists who wish to refresh their knowledge of some aspects of mathematics.
Author(s): M. Krasnov, A. Kiselev, G. Makarenko, E. Shikin
Publisher: Mir Publishers
Year: 1990
Language: English
Pages: 601
Preface
1. An Introduction to Analytic Geometry
1.1 Cartesian Coordinates
1.2 Elementary Problems of Analytic Geometry
1.3 Polar Coordinates
1.4 2nd and 3rd-Order Determinants
2. Elements of Vector Algebra
2.1 Fixed Vectors and Free Vectors
2.2 Linear Operations on Vectors
2.3 Coordinates and Components of a Vector
2.4 Projection of a Vector onto an Axis
2.5 Scalar Product of Two Vectors
2.6 Vector Product of Two Vectors
2.7 Mixed Products of Three Vectors
Exercises
Answers
3. The Line and the Plane
3.1 The Plane
3.2 Straight Line in a Plane
3.3 Straight Line in Three-Dimensional Space
Exercises
Answers
4. Curves and Surfaces of the 2nd Order
4.1 Changing the Axes of Coordinates in a Plane
4.2 Curves of the 2nd Order
4.3 The Ellipse
4.4 The Hyperbola
4.5 The Parabola
4.6 Optical Properties of Curves of the 2nd Order
4.7 Classification of Curves of the 2nd Order
4.8 Surfaces of the 2nd Order
4.9 Classification of Surfaces
4.10 Standard Equations of Surfaces of the 2nd Order
Exercises
Answers
5. Matrices. Determinants. Systems of Linear Equations
5.1 Matrices
5.2 Determinants
5.3 Inverse Matrices
5.4 Rank of a Matrix
5.5 Systems of Linear Equations
Exercises
Answers
6. Linear Spaces and Linear Operators
6.1 The Concept of Linear Space
6.2 Linear Subspaces
6.3 Linearly Dependent Vectors
6.4 Basis and Dimension
6.5 Changing a Basis
6.6 Euclidean Spaces
6.7 Orthogonalization
6.8 Orthocompliments of Linear Subspaces
6.9 Unitary Spaces
6.10 Linear Mappings
6.11 Linear Operators
6.12 Matrices of Linear Operators
6.13 Eigenvalues and Eigenvectors
6.14 Adjoint Operators
6.15 Symmetric Operators
6.16 Quadratic Forms
6.17 Classification of Curves and Surfaces of the 2nd Order
Exercises
Answers
7. An Introduction to Analysis
7.1 Basic Concepts
7.2 Sequences of Numbers
7.3 Functions of One Variable and Limits
7.4 Infinitesimals and Infinities
7.5 Operations on Limits
7.6 Continuous Functions. Continuity at a Point
7.7 Continuity on a Closed Interval
7.8 Comparison of Infinitesimals
7.9 Complex Numbers
Exercises
Answers
8. Differential Calculus. Functions of One Variable
8.1 Derivatives and Differentials
8.2 Differentiation Rules
8.3 Differentiation of Composite and Inverse Functions
8.4 Derivatives and Differentials of Higher Orders
8.5 Mean Value Theorems
8.6 L'Hospital's Rule
8.7 Tests for Increase and Decrease of a Function on a Closed Interval and at a Point
8.8 Extrema of a Function. Maximum and Minimum of a Function on a Closed Interval
8.9 Investigating the Shape of a Curve. Points of Inflection
8.10 Asymptotes of a Curve
8.11 Curve Sketching
8.12 Approximate Solution of Equations
8.13 Taylor's Theorem
8.14 Vector Function of a Scalar Argument
Exercises
Answers
9. Integral Calculus. The Indefinite Integral
9.1 Basic Concepts and Definitions
9.2 Methods of Integration
9.3 Integrating Rational Function
9.4 Integrals Involving Irrational Functions
9.5 Integrals Involving Trigonometric Functions
Exercises
Answers
10. Integral Calculus. The Definite Integral
10.1 Basic Concepts and Definitions
10.2 Properties of the Definite Integral
10.3 Fundamental Theorems for Definite Integrals
10.4 Evaluating Definite Integrals
10.5 Computing Areas and Volumes by Integration
10.6 Computing Arc Lengths by Integration
10.7 Applications of the Definite Integral
10.8 Numerical Integration
Exercises
Answers
11. Improper Integrals
11.1 Integrals with Infinite Limits of Integration
11.2 Integrals of Nonnegative Functions
11.3 Absolutely Convergent Improper Integrals
11.4 Cauchy Principal Value of the Improper Integrals
11.5 Improper Integrals of Unbounded Functions
11.6 Improper Integrals of Unbounded Nonnegative Functions. Convergence Tests
11.7 Cauchy Principal Value of the Improper Integral Involving Unbounded Functions
Exercises
Answers
12. Functions of Several Variables
12.1 Basic Notions and Notation
12.2 Limits and Continuity
12.3 Partial Derivatives and Differentials
12.4 Derivatives of Composite Functions
12.5 Implicit Functions
12.6 Tangent Planes and Normal Lines to a Surface
12.7 Derivatives and Differentials of Higher Orders
12.8 Taylor's Theorem
12.9 Extrema of a Function of Several Variables
Exercises
Answers
Appendix I. Elementary Functions
Index
