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CONTENTS ix

7.2.12. Stieltjes Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
7.2.13. Square Integrable Functions .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 314
7.2.14. Approximate (Numerical) Methods for Computation of Definite Integrals . . . . 315
7.3. Double and Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
7.3.1. Definition and Properties of the Double Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 317
7.3.2. Computation of the Double Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
7.3.3. Geometric and Physical Applications of the Double Integral . . . . . . . . . . . . . . . . 323
7.3.4. Definition and Properties of the Triple Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
7.3.5. Computation of the Triple Integral. Some Applications. Iterated Integrals and
Asymptotic Formulas .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 325
7.4. Line and Surface Integrals . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 329
7.4.1. Line Integral of theFirst Kind .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 329
7.4.2. Line Integral of theSecond Kind .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 330
7.4.3. Surface Integral of the First Kind .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 332
7.4.4. Surface Integral of the Second Kind .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 333
7.4.5. Integral Formulas of Vector Calculus . . . . . . . . . . . . . . . ... .. .. .. .. .. .. .. .. 334
References for Chapter 7 . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 335
8. Series ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 337
8.1. Numerical Series and Infinite Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
8.1.1. Convergent Numerical Series and Their Properties. Cauchy’s Criterion . . . . . . . 337
8.1.2. Convergence Criteria for Series with Positive (Nonnegative) Terms . . . . . . . . . . 338
8.1.3. Convergence Criteria for Arbitrary Numerical Series. Absolute and Conditional
Convergence . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 341
8.1.4. Multiplication of Series. Some Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
8.1.5. Summation Methods. Convergence Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . 344
8.1.6. Infinite Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
8.2. Functional Series . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 348
8.2.1. Pointwise and Uniform Convergence of Functional Series . . . . . . . . . . . . . . . . . . 348
8.2.2. Basic Criteria of Uniform Convergence. Properties of Uniformly Convergent
Series .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 349
8.3. Power Series . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 350
8.3.1. Radius of Convergence of Power Series. Properties of Power Series . . . . . . . . . . 350
8.3.2. Taylor and Maclaurin Power Series . . . . . . . . . . . . . . . . . ... .. .. .. .. .. .. .. .. 352
8.3.3. Operations with Power Series. Summation Formulas for Power Series . . . . . . . . 354
8.4. Fourier Series . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 357
8.4.1. Representation of 2π-Periodic Functions by Fourier Series. Main Results . . . . . 357
8.4.2. Fourier Expansions of Periodic, Nonperiodic, Odd, and Even Functions . . . . . . . 359
8.4.3. Criteria of Uniform and Mean-Square Convergence of Fourier Series . . . . . . . . . 361
8.4.4. Summation Formulas for Trigonometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
8.5. Asymptotic Series .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 363
8.5.1. Asymptotic Series of Poincar´ e Type. Formulas for the Coefficients .. .. .. .. .. . 363
8.5.2. Operations with Asymptotic Series . . . . . . . . . . . . . . . . . ... .. .. .. .. .. .. .. .. 364
References for Chapter 8 . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 366
9. Differential Geometry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 367
9.1. Theory of Curves .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 367
9.1.1. PlaneCurves . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 367
9.1.2. Space Curves .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 379