Mathematical Analysis for Engineers

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