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雷达卡
高清 pdf 文件，898页 包含微积分
by
C F Chan Man Fong
D De Kee
Tulane University, USA
P N Kaloni
University of Windsor, Canada
2002
目录内容：
CONTENTS
Chapter 1 Review of Calculus and Ordinary Differential Equations 1
1.1 Functions of One Real Variable 1
1.2 Derivatives 3
Mean Value Theorem 4
Cauchy Mean Value Theorem 4
L'Hopital's Rule 5
Taylor's Theorem 5
Maximum and Minimum 6
1.3 Integrals 7
Integration by Parts 8
Integration by Substitution 9
Integration of Rational Functions 10
1.4 Functions of Several Variables 13
1.5 Derivatives 13
Total Derivatives 16
1.6 Implicit Functions 19
1.7 Some Theorems 21
Euler's Theorem 21
Taylor's Theorem 22
1.8 Integral of a Function Depending on a Parameter 22
1.9 Ordinary Differential Equations (O.D.E.) - Definitions 25
1.10 First-Order Differential Equations 26
1.11 Separable First-Order Differential Equations 26
1.12 Homogeneous First-Order Differential Equations 27
1.13 Total or Exact First-Order Differential Equations 31
1.14 Linear First-Order Differential Equations 36
1.15 Bernoulli's Equation 39
1.16 Second-Order Linear Differential Equations with Constant Coefficients 41
1.17 S olutions by Laplace Transform 46
Heaviside Step Function and Dirac Delta Function 49
1.18 Solutions Using Green's Functions 57
1.19 Modelling of Physical Systems 65
Problems 74
Chapter 2 Series Solutions and Special Functions 89
2.1 Definitions 89
2.2 Power Series 92
2.3 Ordinary Points 95
2.4 Regular Singular Points and the Method of Frobenius 99
2.5 Method of Variation of Parameters 118
2.6 Sturm Liouville Problem 120
2.7 Special Functions 126
Legendre's Functions 126
Bessel Functions 134
Modified Bessel's Equation 141
2.8 Fourier Series 144
Fourier Integral 158
2.9 Asymptotic Solutions 162
Parameter Expansion 170
Problems 179
Chapter 3 Complex Variables 191
3.1 Introduction 191
3.2 Basic Properties of Complex Numbers 192
3.3 Complex Functions 198
3.4 Elementary Functions 213
3.5 Complex Integration 220
Cauchy's Theorem 227
Cauchy's Integral Formula 233
Integral Formulae for Derivatives 233
Morera's Theorem 237
Maximum Modulus Principle 237
3.6 Series Representations of Analytic Functions 238
Taylor Series 242
Laurent Series 246
3.7 Residue Theory 253
Cauchy's Residue Theorem 257
Some Methods of Evaluating the Residues 258
Triginometric Integrals 263
Improper Integrals of Rational Functions 265
Evaluating Integrals Using Jordan's Lemma 269
3.8 Conformal Mapping 275
Linear Transformation 279
Reciprocal Transformation 282
Bilinear Transformation 284
Schwarz-Christoffel Transformation 287
Joukowski Transformation 289
Problems 293
Chapter 4 Vector and Tensor Analysis 301
4.1 Introduction 301
4.2 Vectors 301
4.3 Line, Surface and Volume Integrals 305
4.4 Relations Between Line, Surface and Volume Integrals 327
Gauss' (Divergence) Theorem 327
Stokes' Theorem 333
4.5 Applications 336
Conservation of Mass 336
Solution of Poisson's Equation 338
Non-Existence of Periodic Solutions 340
Maxwell's Equations 341
4.6 General Curvilinear Coordinate Systems and Higher Order Tensors 342
Cartesian Vectors and Summation Convention 342
General Curvilinear Coordinate Systems 347
Tensors of Arbitrary Order 354
Metric and Permutation Tensors 357
Covariant, Contravariant and Physical Components 361
4.7 Covariant Differentiation 366
Properties of Christoffel Symbols 368
4.8 Integral Transforms 379
4.9 Isotropic, Objective Tensors and Tensor-Valued Functions 382
Problems 391
Chapter 5 Partial Differential Equations I 401
5.1 Introduction 401
5.2 First Order Equations 402
Method of Characteristics 402
Lagrange' s Method 411
Transformation Method 414
5.3 Second Order Linear Equations 420
Classification 420
5.4 Method of Separation of Variables 427
Wave Equation 427
D'Alembert's Solution 431
Diffusion Equation 435
Laplace's Equation 437
5.5 Cylindrical and Spherical Polar Coordinate Systems 439
5.6 Boundary and Initial Conditions 447
5.7 Non-Homogeneous Problems 450
5.8 Laplace Transforms 460
5.9 Fourier Transforms 474
5.10 Hankel and Mellin Transforms 483
5.11 Summary 494
Problems 496
Chapter 6 Partial Differential Equations II 511
6.1 Introduction 511
6.2 Method of Characteristics 511
6.3 Similarity Solutions 523
6.4 Green's Functions 532
Dirichlet Problems 532
Neumann Problems 540
Mixed Problems (Robin's Problems) 545
Conformal Mapping 547
6.5 Green's Functions for General Linear Operators 548
6.6 Quantum Mechanics 551
Limitations of Newtonian Mechanics 551
Schrodinger Equation 552
Problems 562
Chapter 7 Numerical Methods 569
7.1 Introduction 569
7.2 Solutions of Equations in One Variable 570
Bisection Method (Internal Halving Method) 571
Secant Method 572
Newton's Method 574
Fixed Point Iteration Method 576
7.3 Polynomial Equations 580
Newton's Method 580
7.4 Simultaneous Linear Equations 584
Gaussian Elimination Method 585
Iterative Method 595
7.5 Eigenvalue Problems 600
Householder Algorithm 601
The QR Algorithm 608
7.6 Interpolation 611
Lagrange Interpolation 611
Newton's Divided Difference Representation 614
Spline Functions 619
Least Squares Approximation 623
7.7 Numerical Differentiation and Integration 625
Numerical Differentiation 625
Numerical Integration 629
7.8 Numerical Solution of Ordinary Differential Equations, Initial Value Problems 637
First Order Equations 638
Euler's method 639
Taylor's method 640
Heun's method 643
Runge-Kutta methods 643
Adams-Bashforth method 646
Higher Order or Systems of First Order Equations 648
7.9 Boundary Value Problems 659
Shooting Method 659
Finite Difference Method 665
7.10 Stability 671
Problems 673
Chapter 8 Numerical Solution of Partial Differential Equations 681
8.1 Introduction 681
8.2 Finite Differences 681
8.3 Parabolic Equations 683
Explicit Method 684
Crank-Nicolson Implicit Method 686
Derivative Boundary Conditions 690
8.4 Elliptic Equations 696
Dirichlet Problem 697
Neumann Problem 701
Poisson's and Helmholtz's Equations 703
8.5 Hyperbolic Equations 706
Difference Equations 706
Method of Characteristics 710
8.6 Irregular Boundaries and Higher Dimensions 715
8.7 Non-Linear Equations 717
8.8 Finite Elements 722
One-Dimensional Problems 722
Variational method 723
Galerkin method 724
Two-Dimensional Problems 727
Problems 734
Chapter 9 Calculus of Variations 739
9.1 Introduction 739
9.2 Function of One Variable 740
9.3 Function of Several Variables 742
9.4 Constrained Extrema and Lagrange Multipliers 745
9.5 Euler-Lagrange Equations 747
9.6 Special Cases 749
Function f Does Not Depend on y' Explicitly 749
Function f Does Not Depend on y Explicitly 751
Function f Does Not Depend on x Explicitly 752
Function f Is a Linear Function of y' 754
9.7 Extension to Higher Derivatives 756
9.8 Transversality (Moving Boundary) Conditions 757
9.9 Constraints 760
9.10 Several Dependent Variables 765
9.11 Several Independent Variables 772
9.12 Transversality Conditions Where the Functional Depends on Several Functions 786
9.13 Subsidiary Conditions Where the Functional Depends on Several Functions 790
Problems 803
Chapter 10 Special Topics 809
10.1 Introduction 809
10.2 Phase Space 809
10.3 Hamilton's Equations of Motion 814
10.4 Poisson Brackets 817
10.5 Canonical Transformations 819
10.6 Liouville's Theorem 823
10.7 Discrete Probability Theory 826
10.8 Binomial, Poisson and Normal Distribution 830
10.9 Scope of Statistical Mechanics 840
10.10 Basic Assumptions 842
10.11 Statistical Thermodynamics 845
10.12 The Equipartition Theorem 851
10.13 Maxwell Velocity Distribution 852
10.14 Brownian Motion 853
Problems 856
References 861
Appendices 867
Appendix I - The equation of continuity in several coordinate systems 867
Appendix II - The equation of motion in several coordinate systems 868
Appendix III - The equation of energy in terms of the transport properties 871
in several coordinate systems
Appendix IV - The equation of continuity of species A in several coordinate systems 873
Author Index 875
Subject Index 877
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