Mathematical Methods in Physics, Engineering, and Chemistry:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Hoboken, NJ
Wiley
2020
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| Schlagworte: | |
| Beschreibung: | Preface xi; 1 Vectors and linear operators 1; 1.1 The linearity of physical phenomena 1; 1.2 Vector spaces 2; 1.2.1 A word on notation 4; 1.2.2 Linear independence, bases, - and dimensionality 5; 1.2.3 Subspaces 7; 1.2.4 Isomorphism of N-dimensional spaces 8; 1.2.5 Dual spaces 8; 1.3 Inner products and orthogonality 10; 1.3.1 Inner products 10; 1.3.2 The Schwarz inequality 11; 1.3.3 Vector norms 12; 1.3.4 Orthonormal bases and the Gram-Schmidt process 12; 1.3.5 Complete sets of orthonormal vectors 15; 1.4 Operators and matrices 16; 1.4.1 Linear operators 17; 1.4.2 Representing operators with matrices 18; 1.4.3 Matrix algebra 20; 1.4.4 Rank and nullity 22; 1.4.5 Bounded operators 23; 1.4.6 Inverses 24; 1.4.7 Change of basis and the similarity transformation 25; 1.4.8 Adjoints and Hermitian operators 27; 1.4.9 Determinants and the matrix inverse 29; 1.4.10 Unitary operators 33; 1.4.11 The trace of a matrix 35; 1.5 Eigenvectors and their role in representing operators 36; 1.5.1 Eigenvectors and eigenvalues 36; 1.5.2 The eigenproblem for Hermitian and unitary operators 39; 1.5.3 Diagonalizing matrices 40; 1.6 Hilbert space: Infinite-dimensional vector space - 43; Exercises 47; 2 Sturm-Liouville theory 51; 2.1 Second-order differential equations 52; 2.1.1 Uniqueness and linear independence 52; 2.1.2 The adjoint operator 55; 2.1.3 Self-adjoint operator 56; 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; 3.1.1 The continuity equation 71; 3.1.2 The diffusion equation 72; 3.1.3 The free-particle Schroedinger equation 73; 3.1.4 The heat equation 73; 3.1.5 The inhomogeneous diffusion equation 74; 3.1.6 Schroedinger equation for a particle in a potential field 74; 3.1.7 The Poisson equation 74; 3.1.8 The Laplace equation 75; 3.1.9 The wave equation 75; 3.1.10 Inhomogeneous wave equation 76; 3.1.11 Summary of PDEs 76; 3.2 Separation of variables and the Helmholtz equation 76; 3.2.1 Rectangular coordinates 78; 3.2.2 Cylindrical coordinates 80; - 3.2.3 Spherical coordinates 82; 3.3 The paraxial approximation 83; 3.4 The three types of linear PDEs 84; 3.4.1 Hyperbolic PDEs 85; 3.4.2 Parabolic PDEs 87; 3.4.3 Elliptic PDEs 87; 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; 5.1.1 Power series 122; 5.1.2 Introductory example 123; 5.1.3 Ordinary points 125; 5.1.4 Regular singular points 130; 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; 6.1.1 Rodrigues formula 148; 6.1.2 Orthogonality 150; 6.1.3 Completeness 151; 6.1.4 Generating function 152; 6.1.5 Recursion relations 155; 6.2 Associated Legendre functions, Pm l (x) 157; 6.3 Spherical harmonic functions, Yml ( , ) 158; 6.4 Addition theorem for Ym l ( , ) 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; 7.1.1 Limiting forms for small argument 173; 7.1.2 Asymptotic forms for large argument 174; 7.1.3 Hankel functions 174; 7.2 Properties of the Bessel functions, - Jn(x) 175; 7.2.1 Series associated with the generating function 175; 7.2.2 Recursion relations 177; 7.2.3 Integral representation 178; 7.3 Orthogonality 180; 7.4 Bessel series 182; 7.5 The Fourier-Bessel transform 185; 7.6 Spherical Bessel functions 186; 7.6.1 Reduction to elementary functions 186; 7.6.2 Small-argument forms 188; 7.6.3 Asymptotic forms 188; 7.6.4 Orthogonality and completeness 189; 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; 8.2.1 Continuity, differentiability, - and analyticity 197; 8.2.2 Cauchy-Riemann conditions 198; 8.2.3 Analytic functions are functions only of z = x + iy 201; 8.2.4 Useful definitions 201; 8.3 Contour integrals 202; 8.4 Integrating analytic functions 206; 8.5 Cauchy integral formulas 210; 8.5.1 Derivatives of analytic functions 211; 8.5.2 Consequences of the Cauchy formulas 212; 8.6 Taylor and Laurent series 213; 8.6.1 Taylor series 213; 8.6.2 The zeros of analytic functions are isolated 215; 8.6.3 Laurent series 215; 8.7 Singularities and residues 217; 8.7.1 Isolated singularities, residue theorem 217; 8.7.2 Multivalued functions, branch points, - and branch cuts 220; 8.8 Definite integrals 221; 8.8.1 Integrands containing cos and sin 222; 8.8.2 Infinite integrals 223; 8.8.3 Poles on the contour of integration 226; 8.9 Meromorphic functions 228; 8.10 Approximation of integrals 230; 8.10.1 The method of steepest descent 233; 8.10.2 The method of stationary phase 235; 8.11 The analytic signal 236; 8.11.1 The Hilbert transform 237; 8.11.2 Paley-Wiener and Titchmarsh theorems 239; 8.11.3 Is the analytic signal, analytic? 241; 8.12 The Laplace transform 242; Summary 245; Exercises 245; 9 Inhomogeneous differential equations 251; 9.1 The method of Green functions 251; 9.1.1 Boundary conditions 252; 9.1.2 Reciprocity relation: G(x, x') = G(x', x) 253; 9.1.3 Matching conditions 254; 9.1.4 Direct construction of G(x, x') 255; 9.1.5 Eigenfunction expansions 257; 9.2 Poisson equation 260; 9.2.1 Boundary conditions and reciprocity relations 261; 9.2.2 So, - what's the Green function? 263; 9.3 Helmholtz equation 266; 9.3.1 Green function for two-dimensional problems 267; 9.3.2 Free-space Green function for three dimensions 270; 9.3.3 Expansion in spherical harmonics 270; 9.4 Diffusion equation 272; 9.4.1 Boundary conditions, causality, - and reciprocity 272; 9.4.2 Solution to the diffusion equation 274; 9.4.3 Free-space Green function 275; 9.5 Wave equation 279; 9.6 The Kirchhoff integral theorem 283; Summary 284; Exercises 284; 10 Integral equations 287; 10.1 Introduction 287; 10.1.1 Equivalence of integral and differential equations 287; 10.1.2 Role of coordinate systems in capturing boundary data 288; 10.2 Classification of integral equations 290; 10.3 Neumann series 291; 10.4 Integral transform methods 293; 10.4.1 Difference kernels 293; 10.4.2 Fourier kernels 294; 10.5 Separable kernels 295; 10.6 Self-adjoint kernels 297; 10.7 Numerical approaches 302; 10.7.1 Matrix form 302; 10.7.2 Measurement space 303; 10.7.3 The generalized inverse 306; Summary 314; Exercises 315; 11 Tensor analysis 319; 11.1 Once over lightly: A quick intro to tensors 319; 11.2 Transformation properties 327; 11.2.1 The two types of vector: Contravariant and covariant 327; 11.2.2 Coordinate transformations 328; 11.2.3 Contravariant vectors - and tensors 332; 11.2.4 Covariant vectors and tensors 336; 11.2.5 Mixed tensors 339; 11.2.6 Covariant equations 339; 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, (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 |
| Beschreibung: | xiii, 428 pages illustrations 259 grams |
| ISBN: | 9781119579656 |
Internformat
MARC
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| 100 | 1 | |a Borden, Brett |d 1954- |e Verfasser |0 (DE-588)140807799 |4 aut | |
| 245 | 1 | 0 | |a Mathematical Methods in Physics, Engineering, and Chemistry |c Brett Borden and James Luscombe (Naval Postgraduate School, Monterey, CA, USA) |
| 264 | 1 | |a Hoboken, NJ |b Wiley |c 2020 | |
| 300 | |a xiii, 428 pages |b illustrations |c 259 grams | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Preface xi; 1 Vectors and linear operators 1; 1.1 The linearity of physical phenomena 1; 1.2 Vector spaces 2; 1.2.1 A word on notation 4; 1.2.2 Linear independence, bases, | ||
| 500 | |a - and dimensionality 5; 1.2.3 Subspaces 7; 1.2.4 Isomorphism of N-dimensional spaces 8; 1.2.5 Dual spaces 8; 1.3 Inner products and orthogonality 10; 1.3.1 Inner products 10; 1.3.2 The Schwarz inequality 11; 1.3.3 Vector norms 12; 1.3.4 Orthonormal bases and the Gram-Schmidt process 12; 1.3.5 Complete sets of orthonormal vectors 15; 1.4 Operators and matrices 16; 1.4.1 Linear operators 17; 1.4.2 Representing operators with matrices 18; 1.4.3 Matrix algebra 20; 1.4.4 Rank and nullity 22; 1.4.5 Bounded operators 23; 1.4.6 Inverses 24; 1.4.7 Change of basis and the similarity transformation 25; 1.4.8 Adjoints and Hermitian operators 27; 1.4.9 Determinants and the matrix inverse 29; 1.4.10 Unitary operators 33; 1.4.11 The trace of a matrix 35; 1.5 Eigenvectors and their role in representing operators 36; 1.5.1 Eigenvectors and eigenvalues 36; 1.5.2 The eigenproblem for Hermitian and unitary operators 39; 1.5.3 Diagonalizing matrices 40; 1.6 Hilbert space: Infinite-dimensional vector space | ||
| 500 | |a - 43; Exercises 47; 2 Sturm-Liouville theory 51; 2.1 Second-order differential equations 52; 2.1.1 Uniqueness and linear independence 52; 2.1.2 The adjoint operator 55; 2.1.3 Self-adjoint operator 56; 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; 3.1.1 The continuity equation 71; 3.1.2 The diffusion equation 72; 3.1.3 The free-particle Schroedinger equation 73; 3.1.4 The heat equation 73; 3.1.5 The inhomogeneous diffusion equation 74; 3.1.6 Schroedinger equation for a particle in a potential field 74; 3.1.7 The Poisson equation 74; 3.1.8 The Laplace equation 75; 3.1.9 The wave equation 75; 3.1.10 Inhomogeneous wave equation 76; 3.1.11 Summary of PDEs 76; 3.2 Separation of variables and the Helmholtz equation 76; 3.2.1 Rectangular coordinates 78; 3.2.2 Cylindrical coordinates 80; | ||
| 500 | |a - 3.2.3 Spherical coordinates 82; 3.3 The paraxial approximation 83; 3.4 The three types of linear PDEs 84; 3.4.1 Hyperbolic PDEs 85; 3.4.2 Parabolic PDEs 87; 3.4.3 Elliptic PDEs 87; 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; 5.1.1 Power series 122; 5.1.2 Introductory example 123; 5.1.3 Ordinary points 125; 5.1.4 Regular singular points 130; 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, | ||
| 500 | |a - Pl(x) 148; 6.1.1 Rodrigues formula 148; 6.1.2 Orthogonality 150; 6.1.3 Completeness 151; 6.1.4 Generating function 152; 6.1.5 Recursion relations 155; 6.2 Associated Legendre functions, Pm l (x) 157; 6.3 Spherical harmonic functions, Yml ( , ) 158; 6.4 Addition theorem for Ym l ( , ) 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; 7.1.1 Limiting forms for small argument 173; 7.1.2 Asymptotic forms for large argument 174; 7.1.3 Hankel functions 174; 7.2 Properties of the Bessel functions, | ||
| 500 | |a - Jn(x) 175; 7.2.1 Series associated with the generating function 175; 7.2.2 Recursion relations 177; 7.2.3 Integral representation 178; 7.3 Orthogonality 180; 7.4 Bessel series 182; 7.5 The Fourier-Bessel transform 185; 7.6 Spherical Bessel functions 186; 7.6.1 Reduction to elementary functions 186; 7.6.2 Small-argument forms 188; 7.6.3 Asymptotic forms 188; 7.6.4 Orthogonality and completeness 189; 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; 8.2.1 Continuity, differentiability, | ||
| 500 | |a - and analyticity 197; 8.2.2 Cauchy-Riemann conditions 198; 8.2.3 Analytic functions are functions only of z = x + iy 201; 8.2.4 Useful definitions 201; 8.3 Contour integrals 202; 8.4 Integrating analytic functions 206; 8.5 Cauchy integral formulas 210; 8.5.1 Derivatives of analytic functions 211; 8.5.2 Consequences of the Cauchy formulas 212; 8.6 Taylor and Laurent series 213; 8.6.1 Taylor series 213; 8.6.2 The zeros of analytic functions are isolated 215; 8.6.3 Laurent series 215; 8.7 Singularities and residues 217; 8.7.1 Isolated singularities, residue theorem 217; 8.7.2 Multivalued functions, branch points, | ||
| 500 | |a - and branch cuts 220; 8.8 Definite integrals 221; 8.8.1 Integrands containing cos and sin 222; 8.8.2 Infinite integrals 223; 8.8.3 Poles on the contour of integration 226; 8.9 Meromorphic functions 228; 8.10 Approximation of integrals 230; 8.10.1 The method of steepest descent 233; 8.10.2 The method of stationary phase 235; 8.11 The analytic signal 236; 8.11.1 The Hilbert transform 237; 8.11.2 Paley-Wiener and Titchmarsh theorems 239; 8.11.3 Is the analytic signal, analytic? 241; 8.12 The Laplace transform 242; Summary 245; Exercises 245; 9 Inhomogeneous differential equations 251; 9.1 The method of Green functions 251; 9.1.1 Boundary conditions 252; 9.1.2 Reciprocity relation: G(x, x') = G(x', x) 253; 9.1.3 Matching conditions 254; 9.1.4 Direct construction of G(x, x') 255; 9.1.5 Eigenfunction expansions 257; 9.2 Poisson equation 260; 9.2.1 Boundary conditions and reciprocity relations 261; 9.2.2 So, | ||
| 500 | |a - what's the Green function? 263; 9.3 Helmholtz equation 266; 9.3.1 Green function for two-dimensional problems 267; 9.3.2 Free-space Green function for three dimensions 270; 9.3.3 Expansion in spherical harmonics 270; 9.4 Diffusion equation 272; 9.4.1 Boundary conditions, causality, | ||
| 500 | |a - and reciprocity 272; 9.4.2 Solution to the diffusion equation 274; 9.4.3 Free-space Green function 275; 9.5 Wave equation 279; 9.6 The Kirchhoff integral theorem 283; Summary 284; Exercises 284; 10 Integral equations 287; 10.1 Introduction 287; 10.1.1 Equivalence of integral and differential equations 287; 10.1.2 Role of coordinate systems in capturing boundary data 288; 10.2 Classification of integral equations 290; 10.3 Neumann series 291; 10.4 Integral transform methods 293; 10.4.1 Difference kernels 293; 10.4.2 Fourier kernels 294; 10.5 Separable kernels 295; 10.6 Self-adjoint kernels 297; 10.7 Numerical approaches 302; 10.7.1 Matrix form 302; 10.7.2 Measurement space 303; 10.7.3 The generalized inverse 306; Summary 314; Exercises 315; 11 Tensor analysis 319; 11.1 Once over lightly: A quick intro to tensors 319; 11.2 Transformation properties 327; 11.2.1 The two types of vector: Contravariant and covariant 327; 11.2.2 Coordinate transformations 328; 11.2.3 Contravariant vectors | ||
| 500 | |a - and tensors 332; 11.2.4 Covariant vectors and tensors 336; 11.2.5 Mixed tensors 339; 11.2.6 Covariant equations 339; 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 | ||
| 500 | |a - 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, (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 | ||
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| author_GND | (DE-588)140807799 (DE-588)1142246302 |
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1.5.1 Eigenvectors and eigenvalues 36; 1.5.2 The eigenproblem for Hermitian and unitary operators 39; 1.5.3 Diagonalizing matrices 40; 1.6 Hilbert space: Infinite-dimensional vector space </subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a"> - 43; Exercises 47; 2 Sturm-Liouville theory 51; 2.1 Second-order differential equations 52; 2.1.1 Uniqueness and linear independence 52; 2.1.2 The adjoint operator 55; 2.1.3 Self-adjoint operator 56; 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; 3.1.1 The continuity equation 71; 3.1.2 The diffusion equation 72; 3.1.3 The free-particle Schroedinger equation 73; 3.1.4 The heat equation 73; 3.1.5 The inhomogeneous diffusion equation 74; 3.1.6 Schroedinger equation for a particle in a potential field 74; 3.1.7 The Poisson equation 74; 3.1.8 The Laplace equation 75; 3.1.9 The wave equation 75; 3.1.10 Inhomogeneous wave equation 76; 3.1.11 Summary of PDEs 76; 3.2 Separation of variables and the Helmholtz equation 76; 3.2.1 Rectangular coordinates 78; 3.2.2 Cylindrical coordinates 80; </subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a"> - 3.2.3 Spherical coordinates 82; 3.3 The paraxial approximation 83; 3.4 The three types of linear PDEs 84; 3.4.1 Hyperbolic PDEs 85; 3.4.2 Parabolic PDEs 87; 3.4.3 Elliptic PDEs 87; 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; 5.1.1 Power series 122; 5.1.2 Introductory example 123; 5.1.3 Ordinary points 125; 5.1.4 Regular singular points 130; 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, </subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a"> - Pl(x) 148; 6.1.1 Rodrigues formula 148; 6.1.2 Orthogonality 150; 6.1.3 Completeness 151; 6.1.4 Generating function 152; 6.1.5 Recursion relations 155; 6.2 Associated Legendre functions, Pm l (x) 157; 6.3 Spherical harmonic functions, Yml ( , ) 158; 6.4 Addition theorem for Ym l ( , ) 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; 7.1.1 Limiting forms for small argument 173; 7.1.2 Asymptotic forms for large argument 174; 7.1.3 Hankel functions 174; 7.2 Properties of the Bessel functions, </subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a"> - Jn(x) 175; 7.2.1 Series associated with the generating function 175; 7.2.2 Recursion relations 177; 7.2.3 Integral representation 178; 7.3 Orthogonality 180; 7.4 Bessel series 182; 7.5 The Fourier-Bessel transform 185; 7.6 Spherical Bessel functions 186; 7.6.1 Reduction to elementary functions 186; 7.6.2 Small-argument forms 188; 7.6.3 Asymptotic forms 188; 7.6.4 Orthogonality and completeness 189; 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; 8.2.1 Continuity, differentiability, </subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a"> - and analyticity 197; 8.2.2 Cauchy-Riemann conditions 198; 8.2.3 Analytic functions are functions only of z = x + iy 201; 8.2.4 Useful definitions 201; 8.3 Contour integrals 202; 8.4 Integrating analytic functions 206; 8.5 Cauchy integral formulas 210; 8.5.1 Derivatives of analytic functions 211; 8.5.2 Consequences of the Cauchy formulas 212; 8.6 Taylor and Laurent series 213; 8.6.1 Taylor series 213; 8.6.2 The zeros of analytic functions are isolated 215; 8.6.3 Laurent series 215; 8.7 Singularities and residues 217; 8.7.1 Isolated singularities, residue theorem 217; 8.7.2 Multivalued functions, branch points, </subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a"> - and branch cuts 220; 8.8 Definite integrals 221; 8.8.1 Integrands containing cos and sin 222; 8.8.2 Infinite integrals 223; 8.8.3 Poles on the contour of integration 226; 8.9 Meromorphic functions 228; 8.10 Approximation of integrals 230; 8.10.1 The method of steepest descent 233; 8.10.2 The method of stationary phase 235; 8.11 The analytic signal 236; 8.11.1 The Hilbert transform 237; 8.11.2 Paley-Wiener and Titchmarsh theorems 239; 8.11.3 Is the analytic signal, analytic? 241; 8.12 The Laplace transform 242; Summary 245; Exercises 245; 9 Inhomogeneous differential equations 251; 9.1 The method of Green functions 251; 9.1.1 Boundary conditions 252; 9.1.2 Reciprocity relation: G(x, x') = G(x', x) 253; 9.1.3 Matching conditions 254; 9.1.4 Direct construction of G(x, x') 255; 9.1.5 Eigenfunction expansions 257; 9.2 Poisson equation 260; 9.2.1 Boundary conditions and reciprocity relations 261; 9.2.2 So, </subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a"> - what's the Green function? 263; 9.3 Helmholtz equation 266; 9.3.1 Green function for two-dimensional problems 267; 9.3.2 Free-space Green function for three dimensions 270; 9.3.3 Expansion in spherical harmonics 270; 9.4 Diffusion equation 272; 9.4.1 Boundary conditions, causality, </subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a"> - and reciprocity 272; 9.4.2 Solution to the diffusion equation 274; 9.4.3 Free-space Green function 275; 9.5 Wave equation 279; 9.6 The Kirchhoff integral theorem 283; Summary 284; Exercises 284; 10 Integral equations 287; 10.1 Introduction 287; 10.1.1 Equivalence of integral and differential equations 287; 10.1.2 Role of coordinate systems in capturing boundary data 288; 10.2 Classification of integral equations 290; 10.3 Neumann series 291; 10.4 Integral transform methods 293; 10.4.1 Difference kernels 293; 10.4.2 Fourier kernels 294; 10.5 Separable kernels 295; 10.6 Self-adjoint kernels 297; 10.7 Numerical approaches 302; 10.7.1 Matrix form 302; 10.7.2 Measurement space 303; 10.7.3 The generalized inverse 306; Summary 314; Exercises 315; 11 Tensor analysis 319; 11.1 Once over lightly: A quick intro to tensors 319; 11.2 Transformation properties 327; 11.2.1 The two types of vector: Contravariant and covariant 327; 11.2.2 Coordinate transformations 328; 11.2.3 Contravariant vectors </subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a"> - and tensors 332; 11.2.4 Covariant vectors and tensors 336; 11.2.5 Mixed tensors 339; 11.2.6 Covariant equations 339; 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 </subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a"> - 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, (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</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">bicssc / Electronics & communications engineering</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Mathematische Chemie</subfield><subfield code="0">(DE-588)4246882-6</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Mathematische Physik</subfield><subfield code="0">(DE-588)4037952-8</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Mathematische Physik</subfield><subfield code="0">(DE-588)4037952-8</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2="1"><subfield code="a">Mathematische Chemie</subfield><subfield code="0">(DE-588)4246882-6</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Luscombe, James</subfield><subfield code="d">1954-</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)1142246302</subfield><subfield code="4">aut</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-031611129</subfield></datafield></record></collection> |
| id | DE-604.BV046232666 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T08:39:03Z |
| institution | BVB |
| isbn | 9781119579656 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-031611129 |
| oclc_num | 1136487645 |
| open_access_boolean | |
| owner | DE-29T DE-706 |
| owner_facet | DE-29T DE-706 |
| physical | xiii, 428 pages illustrations 259 grams |
| publishDate | 2020 |
| publishDateSearch | 2020 |
| publishDateSort | 2020 |
| publisher | Wiley |
| record_format | marc |
| spelling | Borden, Brett 1954- Verfasser (DE-588)140807799 aut Mathematical Methods in Physics, Engineering, and Chemistry Brett Borden and James Luscombe (Naval Postgraduate School, Monterey, CA, USA) Hoboken, NJ Wiley 2020 xiii, 428 pages illustrations 259 grams txt rdacontent n rdamedia nc rdacarrier Preface xi; 1 Vectors and linear operators 1; 1.1 The linearity of physical phenomena 1; 1.2 Vector spaces 2; 1.2.1 A word on notation 4; 1.2.2 Linear independence, bases, - and dimensionality 5; 1.2.3 Subspaces 7; 1.2.4 Isomorphism of N-dimensional spaces 8; 1.2.5 Dual spaces 8; 1.3 Inner products and orthogonality 10; 1.3.1 Inner products 10; 1.3.2 The Schwarz inequality 11; 1.3.3 Vector norms 12; 1.3.4 Orthonormal bases and the Gram-Schmidt process 12; 1.3.5 Complete sets of orthonormal vectors 15; 1.4 Operators and matrices 16; 1.4.1 Linear operators 17; 1.4.2 Representing operators with matrices 18; 1.4.3 Matrix algebra 20; 1.4.4 Rank and nullity 22; 1.4.5 Bounded operators 23; 1.4.6 Inverses 24; 1.4.7 Change of basis and the similarity transformation 25; 1.4.8 Adjoints and Hermitian operators 27; 1.4.9 Determinants and the matrix inverse 29; 1.4.10 Unitary operators 33; 1.4.11 The trace of a matrix 35; 1.5 Eigenvectors and their role in representing operators 36; 1.5.1 Eigenvectors and eigenvalues 36; 1.5.2 The eigenproblem for Hermitian and unitary operators 39; 1.5.3 Diagonalizing matrices 40; 1.6 Hilbert space: Infinite-dimensional vector space - 43; Exercises 47; 2 Sturm-Liouville theory 51; 2.1 Second-order differential equations 52; 2.1.1 Uniqueness and linear independence 52; 2.1.2 The adjoint operator 55; 2.1.3 Self-adjoint operator 56; 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; 3.1.1 The continuity equation 71; 3.1.2 The diffusion equation 72; 3.1.3 The free-particle Schroedinger equation 73; 3.1.4 The heat equation 73; 3.1.5 The inhomogeneous diffusion equation 74; 3.1.6 Schroedinger equation for a particle in a potential field 74; 3.1.7 The Poisson equation 74; 3.1.8 The Laplace equation 75; 3.1.9 The wave equation 75; 3.1.10 Inhomogeneous wave equation 76; 3.1.11 Summary of PDEs 76; 3.2 Separation of variables and the Helmholtz equation 76; 3.2.1 Rectangular coordinates 78; 3.2.2 Cylindrical coordinates 80; - 3.2.3 Spherical coordinates 82; 3.3 The paraxial approximation 83; 3.4 The three types of linear PDEs 84; 3.4.1 Hyperbolic PDEs 85; 3.4.2 Parabolic PDEs 87; 3.4.3 Elliptic PDEs 87; 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; 5.1.1 Power series 122; 5.1.2 Introductory example 123; 5.1.3 Ordinary points 125; 5.1.4 Regular singular points 130; 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; 6.1.1 Rodrigues formula 148; 6.1.2 Orthogonality 150; 6.1.3 Completeness 151; 6.1.4 Generating function 152; 6.1.5 Recursion relations 155; 6.2 Associated Legendre functions, Pm l (x) 157; 6.3 Spherical harmonic functions, Yml ( , ) 158; 6.4 Addition theorem for Ym l ( , ) 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; 7.1.1 Limiting forms for small argument 173; 7.1.2 Asymptotic forms for large argument 174; 7.1.3 Hankel functions 174; 7.2 Properties of the Bessel functions, - Jn(x) 175; 7.2.1 Series associated with the generating function 175; 7.2.2 Recursion relations 177; 7.2.3 Integral representation 178; 7.3 Orthogonality 180; 7.4 Bessel series 182; 7.5 The Fourier-Bessel transform 185; 7.6 Spherical Bessel functions 186; 7.6.1 Reduction to elementary functions 186; 7.6.2 Small-argument forms 188; 7.6.3 Asymptotic forms 188; 7.6.4 Orthogonality and completeness 189; 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; 8.2.1 Continuity, differentiability, - and analyticity 197; 8.2.2 Cauchy-Riemann conditions 198; 8.2.3 Analytic functions are functions only of z = x + iy 201; 8.2.4 Useful definitions 201; 8.3 Contour integrals 202; 8.4 Integrating analytic functions 206; 8.5 Cauchy integral formulas 210; 8.5.1 Derivatives of analytic functions 211; 8.5.2 Consequences of the Cauchy formulas 212; 8.6 Taylor and Laurent series 213; 8.6.1 Taylor series 213; 8.6.2 The zeros of analytic functions are isolated 215; 8.6.3 Laurent series 215; 8.7 Singularities and residues 217; 8.7.1 Isolated singularities, residue theorem 217; 8.7.2 Multivalued functions, branch points, - and branch cuts 220; 8.8 Definite integrals 221; 8.8.1 Integrands containing cos and sin 222; 8.8.2 Infinite integrals 223; 8.8.3 Poles on the contour of integration 226; 8.9 Meromorphic functions 228; 8.10 Approximation of integrals 230; 8.10.1 The method of steepest descent 233; 8.10.2 The method of stationary phase 235; 8.11 The analytic signal 236; 8.11.1 The Hilbert transform 237; 8.11.2 Paley-Wiener and Titchmarsh theorems 239; 8.11.3 Is the analytic signal, analytic? 241; 8.12 The Laplace transform 242; Summary 245; Exercises 245; 9 Inhomogeneous differential equations 251; 9.1 The method of Green functions 251; 9.1.1 Boundary conditions 252; 9.1.2 Reciprocity relation: G(x, x') = G(x', x) 253; 9.1.3 Matching conditions 254; 9.1.4 Direct construction of G(x, x') 255; 9.1.5 Eigenfunction expansions 257; 9.2 Poisson equation 260; 9.2.1 Boundary conditions and reciprocity relations 261; 9.2.2 So, - what's the Green function? 263; 9.3 Helmholtz equation 266; 9.3.1 Green function for two-dimensional problems 267; 9.3.2 Free-space Green function for three dimensions 270; 9.3.3 Expansion in spherical harmonics 270; 9.4 Diffusion equation 272; 9.4.1 Boundary conditions, causality, - and reciprocity 272; 9.4.2 Solution to the diffusion equation 274; 9.4.3 Free-space Green function 275; 9.5 Wave equation 279; 9.6 The Kirchhoff integral theorem 283; Summary 284; Exercises 284; 10 Integral equations 287; 10.1 Introduction 287; 10.1.1 Equivalence of integral and differential equations 287; 10.1.2 Role of coordinate systems in capturing boundary data 288; 10.2 Classification of integral equations 290; 10.3 Neumann series 291; 10.4 Integral transform methods 293; 10.4.1 Difference kernels 293; 10.4.2 Fourier kernels 294; 10.5 Separable kernels 295; 10.6 Self-adjoint kernels 297; 10.7 Numerical approaches 302; 10.7.1 Matrix form 302; 10.7.2 Measurement space 303; 10.7.3 The generalized inverse 306; Summary 314; Exercises 315; 11 Tensor analysis 319; 11.1 Once over lightly: A quick intro to tensors 319; 11.2 Transformation properties 327; 11.2.1 The two types of vector: Contravariant and covariant 327; 11.2.2 Coordinate transformations 328; 11.2.3 Contravariant vectors - and tensors 332; 11.2.4 Covariant vectors and tensors 336; 11.2.5 Mixed tensors 339; 11.2.6 Covariant equations 339; 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, (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 bicssc / Electronics & communications engineering Mathematische Chemie (DE-588)4246882-6 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 s Mathematische Chemie (DE-588)4246882-6 s DE-604 Luscombe, James 1954- Verfasser (DE-588)1142246302 aut |
| spellingShingle | Borden, Brett 1954- Luscombe, James 1954- Mathematical Methods in Physics, Engineering, and Chemistry bicssc / Electronics & communications engineering Mathematische Chemie (DE-588)4246882-6 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| subject_GND | (DE-588)4246882-6 (DE-588)4037952-8 |
| title | Mathematical Methods in Physics, Engineering, and Chemistry |
| title_auth | Mathematical Methods in Physics, Engineering, and Chemistry |
| title_exact_search | Mathematical Methods in Physics, Engineering, and Chemistry |
| title_full | Mathematical Methods in Physics, Engineering, and Chemistry Brett Borden and James Luscombe (Naval Postgraduate School, Monterey, CA, USA) |
| title_fullStr | Mathematical Methods in Physics, Engineering, and Chemistry Brett Borden and James Luscombe (Naval Postgraduate School, Monterey, CA, USA) |
| title_full_unstemmed | Mathematical Methods in Physics, Engineering, and Chemistry Brett Borden and James Luscombe (Naval Postgraduate School, Monterey, CA, USA) |
| title_short | Mathematical Methods in Physics, Engineering, and Chemistry |
| title_sort | mathematical methods in physics engineering and chemistry |
| topic | bicssc / Electronics & communications engineering Mathematische Chemie (DE-588)4246882-6 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| topic_facet | bicssc / Electronics & communications engineering Mathematische Chemie Mathematische Physik |
| work_keys_str_mv | AT bordenbrett mathematicalmethodsinphysicsengineeringandchemistry AT luscombejames mathematicalmethodsinphysicsengineeringandchemistry |