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Mathematical Methods in Physics, Engineering, and Chemistry
Brett Borden (Author), James Luscombe (Author)
9781119579656, Wiley
Hardback, published 5 December 2019
448 pages
25.9 x 21.3 x 2.5 cm, 1.202 kg
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: 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 xi 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 10 1.4 Operators and matrices 16 1.5 Eigenvectors and their role in representing operators 36 1.6 Hilbert space: Infinite-dimensional vector space 43 Exercises 47 2 Sturm–Liouville theory 51 2.1 Second-order differential equations 52 2.2 Sturm–Liouville systems 57 2.3 The Sturm–Liouville eigenproblem 60 2.4 The Dirac delta function 64 2.5 Completeness 66 2.6 Recap 68 Summary 68 Exercises 69 3 Partial differential equations 71 3.1 A survey of partial differential equations 71 3.2 Separation of variables and the Helmholtz equation 76 3.3 The paraxial approximation 83 3.4 The three types of linear PDEs 84 3.5 Outlook 88 Summary 88 Exercises 89 4 Fourier analysis 91 4.1 Fourier series 91 4.2 The exponential form of Fourier series 96 4.3 General intervals 98 4.4 Parseval’s theorem 103 4.5 Back to the delta function 105 4.6 Fourier transform 107 4.7 Convolution integral 111 Summary 115 Exercises 116 5 Series solutions of ordinary differential equations 121 5.1 The Frobenius method 122 5.2 Wronskian method for obtaining a second solution 137 5.3 Bessel and Neumann functions 137 5.4 Legendre polynomials 142 Summary 144 Exercises 145 6 Spherical harmonics 147 6.1 Properties of the Legendre polynomials, Pl(x) 148 6.2 Associated Legendre functions, Pm l (x) 157 6.3 Spherical harmonic functions, Yml (θ, φ) 158 6.4 Addition theorem for Ym l (θ, φ) 160 6.5 Laplace equation in spherical coordinates 166 Summary 167 Exercises 168 7 Bessel functions 173 7.1 Small-argument and asymptotic forms 173 7.2 Properties of the Bessel functions, Jn(x) 175 7.3 Orthogonality 180 7.4 Bessel series 182 7.5 The Fourier-Bessel transform 185 7.6 Spherical Bessel functions 186 7.7 Expansion of plane waves in spherical harmonics 190 Summary 192 Exercises 192 8 Complex analysis 195 8.1 Complex functions 195 8.2 Analytic functions: differentiable in a region 197 8.3 Contour integrals 202 8.4 Integrating analytic functions 206 8.5 Cauchy integral formulas 210 8.6 Taylor and Laurent series 213 8.7 Singularities and residues 217 8.8 Definite integrals 221 8.9 Meromorphic functions 228 8.10 Approximation of integrals 230 8.11 The analytic signal 236 8.12 The Laplace transform 242 Summary 245 Exercises 245 9 Inhomogeneous differential equations 251 9.1 The method of Green functions 251 9.2 Poisson equation 260 9.3 Helmholtz equation 266 9.4 Diffusion equation 272 9.5 Wave equation 279 9.6 The Kirchhoff integral theorem 283 Summary 284 Exercises 284 10 Integral equations 287 10.1 Introduction 287 10.2 Classification of integral equations 290 10.3 Neumann series 291 10.4 Integral transform methods 293 10.5 Separable kernels 295 10.6 Self-adjoint kernels 297 10.7 Numerical approaches 302 Summary 314 Exercises 315 11 Tensor analysis 319 11.1 Once over lightly: A quick intro to tensors 319 11.2 Transformation properties 327 11.3 Contraction and the quotient theorem 340 11.4 The metric tensor 342 11.5 Raising and lowering indices 344 11.6 Geometric properties of covariant vectors 347 11.7 Relative tensors 350 11.8 Tensors as operators 353 11.9 Symmetric and antisymmetric tensors 356 11.10 The Levi-Civita tensor 357 11.11 Pseudotensors 360 11.12 Covariant differentiation of tensors 363 Summary 373 Exercises 374 A Vector calculus 377 A.1 Scalar fields 377 A.1.1 The directional derivative 377 A.1.2 The gradient 378 A.2 Vector fields 379 A.2.1 Divergence 379 A.2.2 Curl 380 A.2.3 The Laplacian 380 A.2.4 Vector operator formulae 381 A.3 Integration 382 A.3.1 Line integrals 382 A.3.2 Surface integrals 383 A.4 Important integral theorems in vector calculus 384 A.4.1 Green’s theorem in the plane 384 A.4.2 The divergence theorem 386 A.4.3 Stokes’ theorem 386 A.4.4 Conservative fields 387 A.4.5 The Helmholtz theorem 389 A.5 Coordinate systems 390 A.5.1 Orthogonal curvilinear coordinates 390 A.5.2 Unit vectors 391 A.5.3 Differential displacement 392 A.5.4 Differential surface and volume elements 393 A.5.5 Transformation of vector components 393 A.5.6 Cylindrical coordinates 394 B Power series 401 C The gamma function, Γ(x) 403 Recursion relation 403 Limit formula 404 Reflection formula 405 Digamma function 405 D Boundary conditions for Partial Differential Equations 409 Summary 417 References 419 Index 421
Subject Areas: Physics [PH]
