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Mathematical Methods in Physics, Engineering, and Chemistry

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Mathematical Methods in Physics, Engineering, and Chemistry

Brett Borden, James Luscombe

ISBN: 978-1-119-57965-6 November 2019 448 Pages

Hardcover
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£94.95
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Description

A concise and up-to-date introduction to mathematical methods for students in the physical sciences 

Mathematical Methods in Physics, Engineering and Chemistry offers an introduction to the most important methods of theoretical physics. Written by two physics professors with years of experience, the text puts the focus on the essential math topics that the majority of physical science students require in the course of their studies. This concise text also contains worked examples that clearly illustrate the mathematical concepts presented and shows how they apply to physical problems.

This targeted text covers a range of topics including linear algebra, partial differential equations, power series, Sturm-Liouville theory, Fourier series, special functions, complex analysis, the Green’s function method, integral equations, and tensor analysis. This important text:

  • Provides a streamlined approach to the subject by putting the focus on the mathematical topics that physical science students really need
  • Offers a text that is different from the often-found definition-theorem-proof scheme
  • Includes more than 150 worked examples that help with an understanding of the problems presented
  • Presents a guide with more than 200 exercises with different degrees of difficulty

Written for advanced undergraduate and graduate students of physics, materials science, and engineering, Mathematical Methods in Physics, Engineering and Chemistry includes the essential methods of theoretical physics. The text is streamlined to provide only the most important mathematical concepts that apply to physical problems.  

Preface vii

1 Vectors and linear operators 1

1.1 The linearity of physical phenomena 1

1.2 Vector spaces 2

1.3 Inner products and orthogonality 8

1.4 Operators and matrices 14

1.5 Eigenvectors and their role in representing operators 30

1.6 Hilbert space: Infinite-dimensional vector space 36

2 Sturm-Liouville theory 43

2.1 Second-order differential equations 44

2.2 Sturm-Liouville systems 49

2.3 The Sturm-Liouville eigenproblem 51

2.4 The Dirac delta function 54

2.5 Completeness 56

2.6 Recap 58

3 Partial differential equations 61

3.1 A survey of partial differential equations 61

3.2 Separation of variables and the Helmholtz equation 65

3.3 The paraxial approximation 72

3.4 The three types of linear PDEs 73

3.5 Outlook 76

4 Fourier analysis 79

4.1 Fourier series 79

4.2 The exponential form of Fourier series 83

4.3 General intervals 85

4.4 Parseval’s theorem 90

4.5 Back to the delta function 91

4.6 Fourier transform 93

4.7 Convolution integral 96

5 Series solutions of ordinary differential equations 105

5.1 The Frobenius method 106

5.2 Wronskian method for obtaining a second solution 119

5.3 Bessel and Neumann functions 120

5.4 Legendre polynomials 124

6 Spherical harmonics 129

6.1 Properties of the Legendre polynomials, Pl(x) 130

6.2 Associated Legendre functions, Pml(x) 137

6.3 Spherical harmonic functions 138

6.4 Addition theorem for Yml(ѳ,ф) 140

6.5 Laplace equation in spherical coordinates 145

7 Bessel functions 151

7.1 Small-argument and asymptotic forms 151

7.2 Properties of the Bessel functions, Jn(x) 153

7.3 Orthogonality 157

7.4 Bessel series 159

7.5 The Fourier-Bessel transform 161

7.6 Spherical Bessel functions 162

7.7 Expansion of plane waves in spherical harmonics 166

8 Complex analysis 171

8.1 Complex functions 171

8.2 Analytic functions: Differentiable in a region 172

8.3 Contour integrals 177

8.4 Integrating analytic functions 180

8.5 Cauchy integral formulas 183

8.6 Taylor and Laurent series 186

8.7 Singularities and residues 189

8.8 Definite integrals 193

8.9 Meromorphic functions 199

8.10 Approximation of integrals 200

8.11 The analytic signal 205

8.12 The Laplace transform 211

9 Inhomogeneous differential equations 219

9.1 The method of Green functions 219

9.2 Poisson equation 227

9.3 Helmholtz equation 232

9.4 Diffusion equation 237

9.5 Wave equation 243

9.6 The Kirchhoff integral theorem 246

10 Integral equations 251

10.1 Introduction 251

10.2 Classification of integral equations 254

10.3 Neumann series 255

10.4 Integral transform methods 256

10.5 Separable kernels 259

10.6 Self-adjoint kernels 261

10.7 Numerical approaches 264

11 Tensor analysis 279

11.1 Once over lightly: A quick intro to tensors 279

11.2 Transformation properties 286

11.3 Contraction and the quotient theorem 297

11.4 The metric tensor 298

11.5 Raising and lowering indices 300

11.6 Geometric properties of covariant vectors 302

11.7 Relative tensors 306

11.8 Tensors as operators 308

11.9 Symmetric and anti-symmetric tensors 310

11.10 The Levi-Civita tensor 311

11.11 Pseudotensors 314

11.12 Covariant differentiation of tensors 317

Appendices 329

A Vector calculus 331

B Power series 353

C The gamma function, Г(x) 355

D Boundary conditions for PDEs 359

Bibliography 367

Index 371