Mathematical Methods in Physics and EngineeringPDF电子书下载

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CHAPTER 1．Algebraic Preliminaries 1

1．1 Linear Equations．Summation Convention 1

1．2 Matrices 5

1．3 Determinants 8

1．4 Systems of Linear Algebraic Equations．Ranks of a Matrix 15

1．5 Vector Spaces 18

1．6 Scalar Product 24

1．7 Orthonormal Basis．Linear Transformations 30

1．8 Infinite-dimensional Vector Spaces．Function Space 35

1．9 Fourier Series 44

1．10 Quadratic Forms．Hermitian Forms 53

1．11 Systems of Ordinary Differential Equations．Vibrations Problems 62

CHAPTER 2．Calculus of Variations 68

2．1 Maxima and Minima of Functions．Lagrange Multipliers 68

2．2 Maxima and Minima of Functionals．Euler＇s Equation 76

2．3 Hamilton＇s Principle．Lagrange＇s Equations 89

2．4 Theory of Small Vibrations 96

2．5 The Vibrating String 99

2．6 Boundary-value Problems of Mathematical Physics 108

2．7 Eigenvalues and Eigenfunctions 117

2．8 Eigenfunction Expansions 125

2．9 Upper and Lower Bounds for Eigenvalues 129

CHAPTER 3．Boundary-value Problems．Separation of Variables 137

3．1 Orthogonal Coordinate Systems．Separation of Variables 137

3．2 Sturm-Liouville Problems．Fourier Series 149

3．3 Series Solutions of Ordinary Differential Equations 162

3．4 Series Solutions of Boundary-value Problems 176

CHAPTER 4．Boundary-value Problems．Green＇s Functions 184

4．1 Nonhomogeneous Boundary-value Problems 184

4．2 One-dimensional Green＇s Functions 190

4．3 Green＇s Functions in Higher Dimensions 201

4．4 Problems in Unbounded Regions 207

4．5 A Problem in Diffraction Theory 219

CHAPTER 5．Integral Equations． 226

5．1 Integral Equations Formulation of Boundary-value Problems 226

5．2 Hilbert-Schmidt Theory 230

5．3 Fredholm Theory 246

5．4 Integral Equations of the First Kind 260

CHAPTER 6．Integral Transform Methods 266

6．1 Fourier Transforms 266

6．2 Applications of Fourier Transforms．Ordinary Differential Equations 277

6．3 Applications of Fourier Transforms．Partial Differential Equations 283

6．4 Applications of Fourier Transforms．Integral Equations 291

6．5 Laplace Transforms．Applications 297

6．6 Other Transform Techniques 310

Index 319

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