1 Vectors and Matrices 1 1.1 Introduction 1 1.2 Vectors in Space 1 1.3 Linear Independence ~ Lines and Planes 6 1.4 Determinants 9 1.5 Simultaneous Linear Equations 13 1.6 Matrices 18 1.7 Addition of Matrices ~ Scalar Times Matrix 19 1.8 Multiplication of Matrices 21 1.9 Inverse of a Square Matrix 26 1.10 Gaussian Elimination 32 1.11 Eigenvalues of a Square Matrix 35 1.12 The Transpose 39 1.13 Orthogonal Matrices 41 1.14 Analytic Geometry and Vectors in n-Dimensional Space 46 1.15 Axioms for Vn 51 1.16 Linear Mappings 55 1.17 Subspaces ~ Rank of a Matrix 62 1.18 Other Vector Spaces 67
2 Differential Calculus of Functions of Several Variables 73 2.1 Functions of Several Variables 73 2.2 Domains and Regions 74 2.3 Functional Notation a Level Curves and Level Surfaces 76 2.4 Limits and Continuity 78 2.5 Partial Derivatives 83 2.6 Total Differential ~ Fundamental Lemma 86 2.7 Differential of Functions of n Variables ~ The Jacobian Matrix 90 2.8 Derivatives and Differentials of Composite Functions 96 2.9 The General Chain Rule 101 2.10 Implicit Functions 105 2.11 Proof of a Case of the Implicit Function Theorem 112 2.12 Inverse Functions ~ Curvilinear Coordinates 118 2.13 Geometrical Applications 122 2.14 The Directional Derivative 131 2.15 Partial Derivatives of Higher Order 135 2.16 Higher Derivatives of Composite Functions 138 2.17 The Laplacian in Polar, Cylindrical, and Spherical Coordinates 140 2.18 Higher Derivatives of Implicit Functions 142 2.19 Maxima and Minima of Functions of Several Variables 145 2.20 Extrema for Functionsf with Side Conditions ~ Lagrange Multipliers 154 2.21 Maxima and Minima of Quadratic Forms on the Unit Sphere 155 2.22 Functional Dependence 161 2.23 Real Variable Theory ~ Theorem on Maximum and Minimum 167
3 Vector Differential Calculus 175 3.1 Introduction 175 3.2 Vector Fields and Scalar Fields 176 3.3 The Gradient Field 179 3.4 The Divergence of a Vector Field 181 3.5 The Curl of a Vector Field 182 3.6 Combined Operations 183 3.7 Curvilinear Coordinates in Space ~ Orthogonal Coordinates 187 3.8 Vector Operations in Orthogonal Curvilinear Coordinates 190 3.9 Tensors 197 3.10 Tensors on a Surface or Hypersurface 208 3.11 Alternating Tensors ~ Exterior Product 209
4 Integral Calculus of Functions of Several Variables 215 4.1 The Definite Integral 215 4.2 Numerical Evaluation of Indefinite Integrals ~ Elliptic Integrals 221 4.3 Double Integrals 225 4.4 Triple Integrals and Multiple Integrals in General 232 4.5 Integrals of Vector Functions 233 4.6 Change of Variables in Integrals 236 4.7 Arc Length and Surface Area 242 4.8 Improper Multiple Integrals 249 4.9 Integrals Depending on a Parameter ~ Leibnitz's Rule 253 4.10 Uniform Continuity ~ Existence of the Riemann Integral 258 4.11 Theory of Double Integrals 261
5 Vector Integral Calculus 267 Two-Dimensional Theory 5.1 Introduction 267 5.2 Line Integrals in the Plane 270 5.3 Integrals with Respect to Arc Length ~ Basic Properties of Line Integrals 276 5.4 Line Integrals as Integrals of Vectors 280 5.5 Green's Theorem 282 5.6 Independence of Path ~ Simply Connected Domains 287 5.7 Extension of Results to Multiply Connected Domains 297 Three-Dimensional Theory and Applications 5.8 Line Integrals in Space 303 5.9 Surfaces in Space ~ Orientability 305 5.10 Surface Integrals 308 5.11 The Divergence Theorem 314 5.12 Stokes's Theorem 321 5.13 Integrals Independent of Path ~ Irrotational and Solenoidal Fields 325 5.14 Change of Variables in a Multiple Integral 331 5.15 Physical Applications 339 5.16 Potential Theory in the Plane 350 5.17 Green's Third Identity 358 5.18 Potential Theory in Space 361 5.19 Differential Forms 364 5.20 Change of Variables in an m-Form and General Stokes's Theorem 368 5.21 Tensor Aspects of Differential Forms 370 5.22 Tensors and Differential Forms without Coordinate8 371
6 Infinite Series 375 6.1 Introduction 375 6.2 Infinite Sequences 376 6.3 Upper and Lower Limits 379 6.4 Further Properties of Sequences 381 6.5 Infinite Series 383 6.6 Tests for Convergence and Divergence 385 6.7 Examples of Applications of Tests for Convergence and Divergence 392 6.8 Extended Ratio Test and Root Test 397 6.9 Computation with Series ~ Estimate of Error 399 6.10 Operations on Series 405 6.11 Sequences and Series of Functions 410 6.12 Uniform Convergence 411 6.13 Weierstrass M-Test for Uniform Convergence 416 6.14 Properties of Uniformly Convergent Series and Sequences 418 6.15 Power Series 422 6.16 Taylor and Maclaurin Series 428 6.17 Taylor's Formula with Remainder 430 6.18 Further Operations on Power Series 433 6.19 Sequences and Series of Complex Numbers 438 6.20 Sequences and Series of Functions of Several Variables 442 6.21 Taylor's Formula for Functions of Several Variables 445 6.22 Improper Integrals Versus Infinite Series 447 6.23 Improper Integrals Depending on a Parameter ~ Uniform Convergence 453 6.24 Principal Value of Improper Integrals 455 6.25 Laplace Transformation ~ F-Function and B-Function 457 6.26 Convergence of Improper Multiple Integrals 462
7 Fourier Series and Orthogonal Functions 467 7.1 Trigonometric Series 467 7.2 Fourier Series 469 7.3 Convergence of Fourier Series 471 7.4 Examples ~ Minimizing of Square Error 473 7.5 Generalizations ~ Fourier Cosine Series ~ Fourier Sine Series 479 7.6 Remarks on Applications of Fourier Series 485 7.7 Uniqueness Theorem 486 7.8 Proof of Fundamental Theorem for Continuous, Periodic, and Piecewise Very Smooth Functions 489 7.9 Proof of Fundamental Theorem 490 7.10 Orthogonal Functions 495 7.11 Fourier Series of Orthogonal Functions ~ Completeness 499 7.12 Sufficient Conditions for Completeness 502 7.13 Integration and Differentiation of Fourier Series 504 7.14 Fourier-Legendre Series 508 7.15 Fourier-Bessel Series 512 7.16 Orthogonal Systems of Functions of Several Variables 517 7.17 Complex Form of Fourier Series 518 7.18 Fourier Integral 519 7.19 The Laplace Transform as a Special Case of the Fourier Transform 521 7.20 Generalized Functions 523
8 Fuctions of a Complex Variable 531 8.1 Complex Functions 531 8.2 Complex-Valued Functions of a Real Variable 532 8.3 Complex-Valued Functions of a Complex Variable ~ Limits and Continuity 537 8.4 Derivatives and Differentials 539 8.5 Integrals 541 8.6 Analytic Functions ~Cauchy-Riemann Equations 544 8.7 The Functions log z, az, Za, sin-l z, cos-1 z 549 8.8 Integrals of Analytic Functions ~Cauchy Integral Theorem 553 8.9 Cauchy's Integral Formula 557 8.10 Power Series as Analytic Functions 559 8.11 Power Series Expansion of General Analytic Function 562 8.12 Power Series in Positive and Negative Powers ~ Laurent Expansion 566 8.13 Isolated Singularities of an Analytic Function ~ Zeros and Poles 569 8.14 The Complex Number 572 8.15 Residues 575 8.16 Residue at Infinity 579 8.17 Logarithmic Residues ~ Argument Principle 582 8.18 Partial Fraction Expansion of Rational Functions 584 8.19 Application of Residues to Evaluation of Real Integrals 587 8.20 Definition of Conformal Mapping 591 8.21 Examples of Conformal Mapping 594 8.22 Applications of Conformal Mapping ~ The Dirichlet Problem 603 8.23 Dirichlet Problem for the Half-Plane 604 8.24 Conformal Mapping in Hydrodynamics 612 8.25 Applications of Conformal Mapping in the Theory of Elasticity 614 8.26 Further Applications of Conformal Mapping 616 8.27 General Formulas for One-to-One Mapping ~ Schwarz-Christoffel Transformation 617 9 Ordinary Differential Equations 625 9.1 Differential Equations 625 9.2 Solutions 626 9.3 The Basic Problems ~ Existence Theorem 627 9.4 Linear Differential Equations 629 9.5 Systems of Differential Equations ~ Linear Systems 636 9.6 Linear Systems with Constant Coefficients 640 9.7 A Class of Vibration Problems 644 9.8 Solution of Differential Equations by Means of Taylor Series 646 9.9 The Existence and Uniqueness Theorem 651
10 Partial Differential Equations 659 10.1 Introduction 659 10.2 Review of Equation for Forced Vibrations of a Spring 661 10.3 Case of Two Particles 662 10.4 Case of N Particles 668 10.5 Continuous Medium ~Fundamental Partial Differential Equation 674 10.6 Classification of Partial Differential Equations ~Basic Problems 676 10.7 The Wave Equation in One Dimension ~Harmonic Motion 678 10.8 Properties of Solutions of the Wave Equation 681 10.9 The One-Dimensional Heat Equation ~Exponential Decay 685 10.10 Properties of Solutions of the Heat Equation 687 10.11 Equilibrium and Approach to Equilibrium 688 10.12 Forced Motion 690 10.13 Equations with Variable Coefficients ~Sturm-Liouville Problems 695 10.14 Equations in Two and Three Dimensions ~Separation of Variables 698 10.15 Unbounded Regions ~Continuous Spectrum 700 10.16 Numerical Methods 703 10.17 Variational Methods 705 10.18 Partial Differential Equations and Integral Equations 707 Answers to Problems 713 Index 733
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