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viii CONTENTS

6. Limits and Derivatives .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 235
6.1. BasicConcepts of Mathematical Analysis . . . . . . . . . . . . . . . . . ... .. .. .. .. .. .. .. .. 235
6.1.1. Number Sets. Functions of Real Variable .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 235
6.1.2. Limit of a Sequence . .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 237
6.1.3. Limitofa Function. Asymptotes .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 240
6.1.4. Infinitely Small and Infinitely Large Functions .. .. .. .. ... .. .. .. .. .. .. .. .. 242
6.1.5. Continuous Functions. Discontinuities of the First and the Second Kind . . . . . . . 243
6.1.6. Convex and Concave Functions .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 245
6.1.7. Functions of Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
6.1.8. Convergence of Functions .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 249
6.2. Differential Calculus for Functions of aSingle Variable . .. .. .. ... .. .. .. .. .. .. .. .. 250
6.2.1. Derivative and Differential, Their Geometrical and Physical Meaning . . . . . . . . . 250
6.2.2. Table of Derivatives and Differentiation Rules .. .. .. .. ... .. .. .. .. .. .. .. .. 252
6.2.3. Theorems about Differentiable Functions. L’Hospital Rule . . . . . . . . . . . . . . . . . 254
6.2.4. Higher-Order Derivatives and Differentials. Taylor’s Formula . . . . . . . . . . . . . . . 255
6.2.5. Extremal Points. Points of Inflection . . . . . . . . . . . . . . . . ... .. .. .. .. .. .. .. .. 257
6.2.6. Qualitative Analysis of Functions and Construction of Graphs . . . . . . . . . . . . . . 259
6.2.7. Approximate Solution of Equations (Root-Finding Algorithms for Continuous
Functions) . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 260
6.3. Functions of Several Variables. Partial Derivatives . . . . . . . . . . . ... .. .. .. .. .. .. .. .. 263
6.3.1. Point Sets. Functions. Limits and Continuity . . . . . . . . . ... .. .. .. .. .. .. .. .. 263
6.3.2. Differentiation of Functions of Several Variables .. .. .. ... .. .. .. .. .. .. .. .. 264
6.3.3. Directional Derivative. Gradient. Geometrical Applications .. .. .. .. .. .. .. .. 267
6.3.4. Extremal Points of Functions of Several Variables . . . . . ... .. .. .. .. .. .. .. .. 269
6.3.5. Differential Operators of the Field Theory . . . . . . . . . . . ... .. .. .. .. .. .. .. .. 272
References for Chapter 6 . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 272
7. Integrals .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 273
7.1. Indefinite Integral .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 273
7.1.1. Antiderivative. Indefinite Integral and Its Properties . . . . ... .. .. .. .. .. .. .. .. 273
7.1.2. Table of Basic Integrals. Properties of the Indefinite Integral. Integration
Examples .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 274
7.1.3. Integration of Rational Functions .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 276
7.1.4. Integration of Irrational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
7.1.5. Integration of Exponential and Trigonometric Functions . . . . . . . . . . . . . . . . . . . 281
7.1.6. Integration of Polynomials Multiplied by Elementary Functions . . . . . . . . . . . . . 283
7.2. Definite Integral .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 286
7.2.1. Basic Definitions. Classes of Integrable Functions. Geometrical Meaning of the
Definite Integral .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 286
7.2.2. Properties of Definite Integrals and Useful Formulas . . . ... .. .. .. .. .. .. .. .. 287
7.2.3. General Reduction Formulas for the Evaluation of Integrals . . . . . . . . . . . . . . . . 289
7.2.4. General Asymptotic Formulas for the Calculation of Integrals . . . . . . . . . . . . . . . 290
7.2.5. Mean Value Theorems. Properties of Integrals in Terms of Inequalities.
Arithmetic Mean and Geometric Mean of Functions . . . ... .. .. .. .. .. .. .. .. 295
7.2.6. Geometric and Physical Applications of the Definite Integral .. .. .. .. .. .. .. . 299
7.2.7. Improper Integrals with Infinite Integration Limit .. .. .. ... .. .. .. .. .. .. .. .. 301
7.2.8. General Reduction Formulas for the Calculation of Improper Integrals . . . . . . . . 304
7.2.9. General Asymptotic Formulas for the Calculation of Improper Integrals . . . . . . . 307
7.2.10. Improper Integrals of Unbounded Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
7.2.11. Cauchy-Type Singular Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310