MULTIPLE INTEGRALS Field Theory and Series, Budak, Fomin MIR PUBLISHER, in English, 640 pag. 1978

PASTA DURA Contents CHAPTER 1. DOUBLE INTEGRALS 19 § 1. Auxiliary Notions. Area of a Plane Figure 21 1. Interior and Boundary Points. Domain 21 2. Distante Between Two Sets 22 3. Area of a Plane Figure 23 4. Basic Properties of Area 27 5. The Concept of Measure 28 § 2. Definition and Basic Properties of Double Integral 29 1. Definition of Double Integral 29 2. Conditions for Existente of Double Integral. Upper and Lower Darboux Sums 31 3. Some Important Classes of Integrable Functions 38 4. Properties of Double Integral 39 § 3. Additive Set Functions. Derivative of a Set Function with Respect to Area 41 1. Point Functions and Set Functions 41 2. Double Integral as an Additive Function of lts Domain of Integration 42 3. Derivative of a Set Function with Respect to Arca . . . 42 4. Derivative of a Double Integral with Respect to the Arca of Its Domain of Integration 43 5. Reconstruction of an Additive Set Function from lts Derivative 44 6. Definite Integral of a Function of One Argument as a Function of Its Interval of Integration 46 7. Extension of Additive Set Functions 46 § 4. Some Physical and Geometrical Applications of the Double Integral 47 1. Evaluating Volumes 47 2. Computing Areas 48 3. Mass of a Plate 48 4. Coordinates of the Centre of Gravity of a Plate 49 5. Moments of Inertia of a Plate 50 8 CONTENTS 6. Luminous Flux Incident on a Plate 51 7. Flux of a Fluid Through the Cross Section of a Channel . . 51 § 5. Reducing Double Integral to a Twofold Iterated Integral . 52 1. Heuristic Considerations 52 2. The Case of a Rectangular Domain of Integration . . . 54 3. The Case of a Curvilinear Domain 56 § 6. Change of Variables in Double Integral 61 1. Mapping of Plane Figures 61 2. Curvilinear Coordinates 63 3. Polar Coordinates 64 4. Statement of the Problem of Changing Variables in the Double Integral 66 5. Computing Area in Curvilinear Coordinates 66 6. Change of Variables in Double Integral 74 7. Comparison with One-Dimensional Case. Integral Over an Oriented Domain 77 CHAPTER 2. TRIPLE INTEGRALS AND MULTIPLE INTEGRALS OF HIGHER ORDER 79 § I. Definition and Basic Properties of Triple Integral 79 1. Preliminary Observations. Volume of a Space Figure . . 79 2. Definition of Triple Integral 81 3. Conditions for Existente of Triple Integral. Integrability of Continuous Functions 82 4. Properties of Triple Integral 83 5. Triple Integral as an Additive Set Function 84 § 2. Some Applications of Triple Integral in Physics and Geometry . 85 1. Computing Volumes 85 2. Finding the Mass of a Solid from Its Density 85 3. Moment of Inertia 85 4. Determining the Coordinates of the Centre of Gravity . 86 5. Gravitational Attraction of a Material Point by a Solid . . 86 3. Evaluating Triple Integral 87 1. Reducing Triple Integral Over a Rectangular Parallelepiped to an Iterated Integral 88 2. Reducing Triplo Integral Over a Curvilinear Domain to au Iterated Integral 90 § 4. Change of Variables in Triple Integral 93 1. Mapping of Space Figures 94 2. Curvilinear Coordinates in Space 94 CONTENTS 9 3. Cylindrical and Spherical Coordinates 95 4. Element of Volume in Curvilinear Coordinates 97 5. Change of Variables in Triple Integral. Geometric Meaning of the Jacobian 98 § 5. Multiple Integrals of Higher Order 102 1. General Remarks 102 2. Examples 103 CHAPTER 3. ELEMENTS OF DIFFERENTIAL GEOMETR Y . . 107 § 1. Vector Function of a Scalar Argument 107 1. Definition of a Vector Function. Limit. Continuity . . . 107 2. Differentiation of a Vector Function 108 3. Hodograph. Singular Points 110 4. Taylor's Formula 111 5. Integral of a Vector Function with Respect to Scalar Argument 111 6. Vector Function of Several Scalar Arguments 112 § 2. Space Curves 113 1. Vector Equation of a Curve 113 2. Moving Trihedron 115 3. Frenet-Serret Formulas 116 4. Evaluating Curvature and Torsion 117 5. Coordinate System Connected with Moving Trihedron . 119 6. The Shape of a Curve in the Vicinity of Its Point . . . 121 7. Curvature of a Plane Curve ' 123 8. Intrinsic Equations of a Curve 124 9. Some Applications to Mechanics 120 § 3. Parametric Equations of a Surface 128 1. The Concept of a Surface 128 2. Parametrization of a Surface 130 3. Parametric Equations of a Surface 132 4. Curves on a Surface 133 5. Tangent Plane 133 6. Normal to a Surface 135 7. Coordinate Systems in Tangent Planes 135 § 4. Determining Lengths, A ngles and Areas on a Curvilinear Surface First Fundamental Quadratic Form of a Surface 137 1. Affine Coordinate System in the Plane 137 2. Are Length of a Curve on a Surface. First Fundamental Quadratic Form 139 3. Angle Between Two Curves 141 4. Definition of Area of a Surface. The Schwarz Example 142 5. Computing Area of a Smooth Surface 14410 CONTENTS § 5. Curvature of Curves on a Surface. Second Fundamental Quadratic Form of a Surface 149 1. Normal Sections of a Surface and Their Curvature . . . 149 2. Second Fundamental Quadratic Form of a Surface . . 151 3. Dupin Indicatrix 153 4. Principal Directions and Principal Curvatures of a Surface Equation of Euler 154 5. Determining Principal Curvatures 156 6. Total Curvature and Mean Curvature 157 7. Classification of Points on a Surface 157 8. The First and the Second Fundamental Quadratic Forms as Invariants of a Surface 159 § 6. Intrinsic Properties of a Surface 160 1. Applicable Surfaces. Necessary and Sufficient Condition for Applicability 160 2. Intrinsic Properties of a Surface 161 3. Surfaces of Constant Curvature 163 CHAPTER 4. LINE INTEGRALS 165 § 1. Line Integrals of the First Type 165 1. Definition of Line Integral of the First Type 165 2. Properties of Line Integrals 169 3. Some Applications of Line Integrals of the First Type . . 170 4. Line Integrals of the First Type in Space 172 § 2. Line Integrals of the Second Type 173 1. Statement of the Problem. Work of a Field of Force . . 173 2. Definition of Line Integral of the' Second Type . , 174 3. Connection Between Line Integrals of the First and the Second Types 175 4. Evaluating Line Integral of the Second Type 177 5. Dependence of Line Integral of the Second Type on the Orientation of the Path of Integration 180 6. Line Integrals Along Self-Intersecting and Closed Paths 180 7. Line Integral of the Second Type Over a Space Curve . . . 181 § 3. Green's Formula 183 1. Derivation of Green's Formula 184 2. Application of Green's Formula to Computing Areas . . 189 § 4. Conditions for a Line Integral of the Second Type Being PathIndependent. Integrating Total Differentials 190 1. Statement of the Problem 190 2. The Case of a Simply Connected Domain 190 3. Reconstructing a Function from Its Total Differential . . 193 4. Line Integrals in a Multiply Connected Domain 195 CONTENTS 11 CHAPTER 5. SURFACE INTEGRALS 199 § 1. Surface Integral of the First Type 199 1. Definition of Surface Integral of a Scalar Function . . 199 2. Reducing Surface Integral to Double Integral 200 3. Some Applications of Surface Integrals to Mechanics . . 204 4. Surface Integral of a Vector Function. General Concept of Surface Integral of the First Type 205 § 2. Surface Integral of the Second Type 207 1. One-Sided and Two-Sided Surfaces 207 2. Definition of Surface Integral of the Second Type . . 211 3. Reducing Surface Integral of the Second Type to Double Integral 215 § 3. Ostrogradsky Theorem 218 1. Derivation of Ostrogradsky Theorem 218 2. Application of Ostrogradsky Theorem to Evaluating Surface Integrals. Expressing Volume of a Space Figure in the Form of a Surface Integral 222 § 4. Stokes' Theorem 224 1. Derivation of Stokes' Formula 224 2. Application of Stokes' Theorem to Investigating Line Integrals in Space 227 CHAPTER 6. FIELD THEORY 230 § 1. Scalar Field 230 1. Definition and Examples of Scalar Field 230 2. Level Surfaces and Level Linos 231 3. Various Types of Symmetry of Field 232 4. Directional Derivative 233 5. Gradient of Scalar Field 234 § 2. Vector Field 236 1. Definition and Examples of Vector Field 236 2. Vector Lines and Vector Surfaces 237 3. Types of Symmetry of Vector Field 238 4. Field of Gradients. Potential Field 238 § 3. Flux of Vector Field. Dicergence 240 1. Flux of Vector Field Across a Surface 240 2. Divergente' 242 3. Physical Meaning of Divergence for Various Types of Field Examples 244 4. Solenoidal Field 24614 CONTENTS 1. Continuity and Uniform Convergence 331 2. Passage to Limit Under the Sign of Integration and Termwise Integration of a Series 334 3. Passage to Limit Under the Sign of Differentiation and Termwise Differentiation of a Series 337 4. Term-by-Term Passage to Limit in Functional Sequences and Series 339 § 3. Power Series 341 1. Interval of Convergence of Power Series. Radius of Convergence 341 2. On Uniform Convergence of a Power Series and Continuity of Its Sum 348 3. Differentiation and Integration of Power Series 351 4. Arithmetical Operations on Power Series 352 § 4. Expanding Functions in Power Series 354 1. Key Theorerns on Expanding Functions in Power Series. Expanding Elementary Functions ....... . . . . 354 2. Some Applications of Power Series 359 • § 5. Power Series in Complex Argument 362 § 6. Convergence in the Mean 366 1. Mean Square Deviation and Convergence in the Mean . . . 366 2. Cauchy-Bunyakovsky Inequality 367 3. Integration of Sequences and Series Convergent in the Mean 369 4. Connection Between Convergence in the Mean and Term-byTerm Differentiation of Sequences and Series 371 5. Connection Between Convergence in the Mean and Other Types of Convergence 372 Appendix 1 to Chapter 8. Criterion for Compactness of a Family of Functions 374 Appendix 2 lo Chapter 8. Weak Convergence and Delta Function 378 CHAPTER 9. IMPROPER INTEGRALS 383 § 1. Integrals with Infinite Limits of Integration 383 1. Definitions. Examples 383 2. Reducing Improper Integral of the Form f (x) dx to Nume- rical Sequence and Numerical Series 386 3. Cauchy Criterion for Improper Integrals 389 4. Absoluto Convergence. Tests for Absolute Convergence 390 5. Conditional Convergence 397 CONTENTS 15 6. Extending Methods of Evaluating Integrals to the Case of Improper Integrals 399 § 2. Integrals of Unbounded Functions with Finite and Infinite Limits of Integration 400 § 3. Cauchy's Principal Value of a Divergent Improper Integral 408 § 4. Improper Multiple Integrals 411 1-. Integral of an Unbounded Function Over a Finite Domain 412 2. Integrals of Nonnegative Functions 414 3. Absolute Convergence 417 4. Tests for Absolute Convergence 419 5. Equivalence of Convergence and Absolute Convergence in the Case of Improper Multiple Integral 421 6. Improper Integrals with Infinita Domain of Integration 424 7. Methods of Computing Improper Multiple Integrals . . . 425 CHAPTER 10. INTEGRALS DEPENDENT ON PARAMETER . . . 427 § 1. Proper and Simplest Improper Integrals Dependent on Parameter 427 1. Proper Integrals Dependent on Parameter 427 2. Simplest Improper Integrals Dependent on Parameter . . 432 § 2. Improper Integrals Dependent on Parameter 435 1. Uniform Convergence 436 2. Reducing Improper Integral Dependent on Parameter to a Functional Sequence 438 3. Properties of Uniformly Convergent Improper Integrals Dependent on Parameter 441 4. Tests for Uniform Convergence of Improper Integrals Dependent on Parameter 448 5. Examples of Evaluating Improper Integrals Dependent on Parameter by Means of Differentiation and Integration with Respect to Parameter 453 § 3. Euler's Integrals 460 1. Properties of Gamma Function 460 2. Properties of Beta Function 464 § 4. Multiple Integrals Dependent on Parameter 468 CHAPTER 11. FOURIER SERIES AND FOURIER INTEGRAL 476 § 1. Properties of Periodic Functions. Statement of the Key Problem 476 1. Periods of a Periodic Function 476 2. Periodic Extension of a Nonperiodic Function 477 3. Integral of a Periodic Function 478 4. Arithmetical Operations on Periodic Functions 478CONTENTS 5. Superposition of Harmonics with Multiple Frequencies 479 6. Statement of the Key Problem 480 7. Orthogonality of Trigonometric System. Fourier Coefficients and Fourier Series 381. 8. Expanding Even and Odd Functions in Fourier Series . . 484 9. Expanding Functions in Fourier Series on the Interval [-n, 485 2. Fundamental Theorem on Convergence of Fourier Series . . . 486 1. Class of Piecewise Smooth Functions 486 2. Formulation of Fundamental Theorem on Convergence of Fourier Series 488 3. Key Lemma 488 4. Proof of Convergence Theorem 490 5. Examples 495 6. Fourier Sine and Cosine Series for Functions Defined on Interval [0, 1] 498 3. Fourier Series with Respect to General Orthogonal Systems. Bes- sel's Inequality 501 1. Orthogonal Systems of Functions 501 2. Fourier Coefficients and Fourier Series of a Function f (x) with Respect to an Orthogonal System 503 3. Least Square Deviation. Bessel's Inequality 504 § 4. Speed of Convergence of Fourier Series. Acceleration of Conver- gence of Fourier Series 508 1. Conditions for Uniform Convergence of Fourier Series . . 508 2. Connection Between the Degree of Smoothness of a Function and the Speed of Convergence of Its Fourier Series . . . . 512 3. Acceleration of Convergence of Fourier Series 516 § 5. Uniform Approximation of Continuous Function by Trigonometric and Algebraic Polynomials. Weierstrass' Approximation Theorem 518 § 6. Complete and Closed Orthogonal Systems 523 1. Complete Orthogonal System 524 2. Parseval Relation as a Necessary and Sufficient Condition for an Orthogonal System Being Complete 524 3. Properties of Complete Systems 525 4. Completeness of Trigonometric System 527 5. Completeness of Some Other Classical Orthogonal Systems 530 § 7. Fourier Series in Orthogonal Systems of Complex Functions . . . 531 § 8. Fourier Series for Functions of Several Independent Variables 535 CONTENTS 17 § 9. Fourier Integral 538 1. Formal Derivation of Fourier Integral Formula 538 2. Proof of Fourier Integral Theorem 540 3. Fourier Integral as an Expansion into a Sum of Harmonics 544 4. Fourier Integral in Complex Form 545 5. Fourier Transformation 546 6. Fourier Integral. for Functions of Several Independent Variables 550 Appendix 1 to Chapter 11. On Legendre's Polynomials 556 Appendix 2 to Chapter 11. Orthogonality with Weight Function and Orthogonalization Process 558 Appendix 3 to Chapter 11. Functional Space.and Geometric Analogy 565 Appendix 4 to Chapter 11. Some Applications of Fourier. Transforms 568 Appendix 5 to Chapter 11. Expanding Delta Function in Fourier Series and Fourier Integral 574 Appendix 6 to Chapter .11. Uniform Approximation of Functions with Polynomials 576 Appendix 7 to Chapter 11. On Stable Summation of Fourier Series with Perturbed Coefficients 581 SUPPLEMENT 1. ASYMPTOTIC EXPANSIONS 586 § 1. Examples of Asymptotic Expanstons 586 1. Asymptotic Expansions in the Neighbourhood of the Origin 586 2. Asymptotic Expansions in the Neighbourbood of the Point at Infinity 587 § 2. General Definitions and Theorems 590 1. Order of Smallness. Asymptotic Equivalence 590 2. Asymptotic Expansions of Functions 592 § 3. Laplace Method for Deriving Asymptotic Expansions of Some Integrals 598 SUPPLEMENT 2. ON UNIVERSAL DIGITAL COMPUTERS . 603 § 1. Computers 603 1. Introduction 603 2. Basic Types of Computer 603 3. Principal Components of a Computer and Their Functions 604 4. Number Systems Used in Computers 606 5. Representing Numbers Within a Computer 608 -082418 CONTENTS § 2. Basic Operations E xecuted by a Computer. Instructions . . . . 608 1. Types of Operation 608 2. Arithmetical Operations 609 3. Additional Computational Operations 610 4. Logical Operations 610 5. Input and Output Operations 611 6. Transfer of Control 612 7. Realization of Operations Within a Computer 613 § 3. Elements of Programming 614 1. General Notions 614 2. Formula Programming 615 3. Cyclic Processes 617 4. Flow-Chart. Subroutines 621 5. Instruction Codes. Operations on Instructions 621 6. Automatic Programming 623 § 4. Organization of Computer Work 624 1. Conditions for Effective Use of a Computer 624 2. Basic Stages of Solving a Problem on a Computer . . 624 3. Checking Computer Operation. Error Detection 625 Bibliography 627 N ame I ndex 628 Subject Índex 629