| Description |
1 online resource (xvi, 829 pages) : illustrations |
| Contents |
880-01 Cover -- Half-title -- Title -- Copyright -- Contents -- Preface -- Review of background topics -- 1 Matrices and vector spaces -- 1.1 Vector spaces -- 1.1.1 Basis vectors -- 1.1.2 The inner product -- 1.1.3 Some useful inequalities -- 1.2 Linear operators -- 1.3 Matrices -- 1.4 Basic matrix algebra -- 1.4.1 Matrix addition and multiplication by a scalar -- 1.4.2 Multiplication of matrices -- 1.4.3 The null and identity matrices -- 1.5 Functions of matrices -- 1.6 The transpose of a matrix -- 1.7 The complex and Hermitian conjugates of a matrix -- 1.8 The trace of a matrix -- 1.9 The determinant of a matrix -- 1.9.1 Properties of determinants -- 1.9.2 Evaluation of determinants -- 1.10 The inverse of a matrix -- 1.11 The rank of a matrix -- 1.12 Simultaneous linear equations -- 1.12.1 The number of solutions -- No solution -- A unique solution -- Infinitely many solutions -- 1.12.2 N simultaneous linear equations in N unknowns -- Gaussian elimination -- Direct inversion -- LU decomposition -- Cramer's rule -- 1.12.3 A geometrical interpretation -- 1.13 Special types of square matrix -- 1.13.1 Diagonal matrices -- 1.13.2 Lower and upper triangular matrices -- 1.13.3 Symmetric and antisymmetric matrices -- 1.13.4 Orthogonal matrices -- 1.13.5 Hermitian and anti-Hermitian matrices -- 1.13.6 Unitary matrices -- 1.13.7 Normal matrices -- 1.14 Eigenvectors and eigenvalues -- 1.14.1 Eigenvectors and eigenvalues of Hermitian and unitary matrices -- 1.14.2 Eigenvectors and eigenvalues of a general square matrix -- 1.14.3 Simultaneous eigenvectors -- 1.15 Determination of eigenvalues and eigenvectors -- 1.15.1 Degenerate eigenvalues -- 1.16 Change of basis and similarity transformations -- 1.17 Diagonalization of matrices -- 1.18 Quadratic and Hermitian forms -- 1.18.1 The stationary properties of the eigenvectors -- 1.18.2 Quadratic surfaces |
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880-01/(S Machine generated contents note: 1. Matrices and vector spaces -- 1.1. Vector spaces -- 1.2. Linear operators -- 1.3. Matrices -- 1.4. Basic matrix algebra -- 1.5. Functions of matrices -- 1.6. transpose of a matrix -- 1.7. complex and Hermitian conjugates of a matrix -- 1.8. trace of a matrix -- 1.9. determinant of a matrix -- 1.10. inverse of a matrix -- 1.11. rank of a matrix -- 1.12. Simultaneous linear equations -- 1.13. Special types of square matrix -- 1.14. Eigenvectors and eigenvalues -- 1.15. Determination of eigenvalues and eigenvectors -- 1.16. Change of basis and similarity transformations -- 1.17. Diagonalization of matrices -- 1.18. Quadratic and Hermitian forms -- 1.19. Normal modes -- 1.20. summation convention -- Summary -- Problems -- Hints and answers -- 2. Vector calculus -- 2.1. Differentiation of vectors -- 2.2. Integration of vectors -- 2.3. Vector functions of several arguments -- 2.4. Surfaces -- 2.5. Scalar and vector fields -- 2.6. Vector operators -- 2.7. Vector operator formulae -- 2.8. Cylindrical and spherical polar coordinates -- 2.9. General curvilinear coordinates -- Summary -- Problems -- Hints and answers -- 3. Line, surface and volume integrals -- 3.1. Line integrals -- 3.2. Connectivity of regions -- 3.3. Green's theorem in a plane -- 3.4. Conservative fields and potentials -- 3.5. Surface integrals -- 3.6. Volume integrals -- 3.7. Integral forms for grad, div and curl -- 3.8. Divergence theorem and related theorems -- 3.9. Stokes' theorem and related theorems -- Summary -- Problems -- Hints and answers -- 4. Fourier series -- 4.1. Dirichlet conditions -- 4.2. Fourier coefficients -- 4.3. Symmetry considerations -- 4.4. Discontinuous functions -- 4.5. Non-periodic functions -- 4.6. Integration and differentiation -- 4.7. Complex Fourier series -- 4.8. Parseval's theorem -- Summary -- Problems -- Hints and answers -- 5. Integral transforms -- 5.1. Fourier transforms -- 5.2. Dirac δ-function -- 5.3. Properties of Fourier transforms -- 5.4. Laplace transforms -- 5.5. Concluding remarks -- Summary -- Problems -- Hints and answers -- 6. Higher-order ordinary differential equations -- 6.1. General considerations -- 6.2. Linear equations with constant coefficients -- 6.3. Linear recurrence relations -- 6.4. Laplace transform method -- 6.5. Linear equations with variable coefficients -- 6.6. General ordinary differential equations -- Summary -- Problems -- Hints and answers -- 7. Series solutions of ordinary differential equations -- 7.1. Second-order linear ordinary differential equations -- 7.2. Ordinary and singular points of an ODE -- 7.3. Series solutions about an ordinary point -- 7.4. Series solutions about a regular singular point -- 7.5. Obtaining a second solution -- 7.6. Polynomial solutions -- Summary -- Problems -- Hints and answers -- 8. Eigenfunction methods for differential equations -- 8.1. Sets of functions -- 8.2. Adjoint, self-adjoint and Hermitian operators -- 8.3. Properties of Hermitian operators -- 8.4. Sturm-Liouville equations -- 8.5. Superposition of eigenfunctions: Green's functions -- Summary -- Problems -- Hints and answers -- 9. Special functions -- 9.1. Legendre functions -- 9.2. Associated Legendre functions -- 9.3. Spherical harmonics -- 9.4. Chebyshev functions -- 9.5. Bessel functions -- 9.6. Spherical Bessel functions -- 9.7. Laguerre functions -- 9.8. Associated Laguerre functions -- 9.9. Hermite functions -- 9.10. gamma function and related functions -- Summary -- Problems -- Hints and answers -- 10. Partial differential equations -- 10.1. Important partial differential equations -- 10.2. General form of solution -- 10.3. General and particular solutions -- 10.4. wave equation -- 10.5. diffusion equation -- 10.6. Boundary conditions and the uniqueness of solutions -- Summary -- Problems -- Hints and answers -- 11. Solution methods for PDEs -- 11.1. Separation of variables: the general method -- 11.2. Superposition of separated solutions -- 11.3. Separation of variables in polar coordinates -- 11.4. Integral transform methods -- 11.5. Inhomogeneous problems -- Green's functions -- Summary -- Problems -- Hints and answers -- 12. Calculus of variations -- 12.1. Euler-Lagrange equation -- 12.2. Special cases -- 12.3. Some extensions -- 12.4. Constrained variation -- 12.5. Physical variational principles -- 12.6. General eigenvalue problems -- 12.7. Estimation of eigenvalues and eigenfunctions -- 12.8. Adjustment of parameters -- Summary -- Problems -- Hints and answers -- 13. Integral equations -- 13.1. Obtaining an integral equation from a differential equation -- 13.2. Types of integral equation -- 13.3. Operator notation and the existence of solutions -- 13.4. Closed-form solutions -- 13.5. Neumann series -- 13.6. Fredholm theory -- 13.7. Schmidt-Hilbert theory -- Summary -- Problems -- Hints and answers -- 14. Complex variables -- 14.1. Functions of a complex variable -- 14.2. Cauchy-Riemann relations -- 14.3. Power series in a complex variable -- 14.4. Some elementary functions -- 14.5. Multivalued functions and branch cuts -- 14.6. Singularities and zeros of complex functions -- 14.7. Conformal transformations -- 14.8. Complex integrals -- 14.9. Cauchy's theorem -- 14.10. Cauchy's integral formula -- 14.11. Taylor and Laurent series -- 14.12. Residue theorem -- Summary -- Problems -- Hints and answers -- 15. Applications of complex variables -- 15.1. Complex potentials -- 15.2. Applications of conformal transformations -- 15.3. Definite integrals using contour integration -- 15.4. Summation of series -- 15.5. Inverse Laplace transform -- 15.6. Some more advanced applications -- Summary -- Problems -- Hints and answers -- 16. Probability -- 16.1. Venn diagrams -- 16.2. Probability -- 16.3. Permutations and combinations -- 16.4. Random variables and distributions -- 16.5. Properties of distributions -- 16.6. Functions of random variables -- 16.7. Generating functions -- 16.8. Important discrete distributions -- 16.9. Important continuous distributions -- 16.10. central limit theorem -- 16.11. Joint distributions -- 16.12. Properties of joint distributions -- Summary -- Problems -- Hints and answers -- 17. Statistics -- 17.1. Experiments, samples and populations -- 17.2. Sample statistics -- 17.3. Estimators and sampling distributions -- 17.4. Some basic estimators -- 17.5. Data modeling -- 17.6. Hypothesis testing -- Summary -- Problems -- Hints and answers -- A. Review of background topics -- A.1. Arithmetic and geometry -- A.2. Preliminary algebra -- A.3. Differential calculus -- A.4. Integral calculus -- A.5. Complex numbers and hyperbolic functions -- A.6. Series and limits -- A.7. Partial differentiation -- A.8. Multiple integrals -- A.9. Vector algebra -- A.10. First-order ordinary differential equations -- B. Inner products -- C. Inequalities in linear vector spaces -- D. Summation convention -- E. Kronecker delta and Levi-Civita symbols -- F. Gram-Schmidt orthogonalization -- G. Linear least squares -- H. Footnote answers |
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1.19 Normal modes -- 1.19.1 Typical oscillatory systems -- 1.19.2 Rayleigh -- Ritz method -- 1.20 The summation convention -- Summary -- Problems -- Hints and answers -- 2 Vector calculus -- 2.1 Differentiation of vectors -- 2.1.1 Differentiation of composite vector expressions -- 2.1.2 Differential of a vector -- 2.2 Integration of vectors -- 2.3 Vector functions of several arguments -- 2.4 Surfaces -- 2.5 Scalar and vector fields -- 2.6 Vector operators -- 2.6.1 Gradient of a scalar field -- 2.6.2 Divergence of a vector field -- 2.6.3 Curl of a vector field -- 2.7 Vector operator formulae -- 2.7.1 Vector operators acting on sums and products -- 2.7.2 Combinations of grad, div and curl -- 2.8 Cylindrical and spherical polar coordinates -- 2.8.1 Cylindrical polar coordinates -- 2.8.2 Spherical polar coordinates -- 2.9 General curvilinear coordinates -- 2.9.1 Gradient -- 2.9.2 Divergence -- 2.9.3 Laplacian -- 2.9.4 Curl -- Summary -- Problems -- Hints and answers -- 3 Line, surface and volume integrals -- 3.1 Line integrals -- 3.1.1 Evaluating line integrals -- 3.1.2 Physical examples of line integrals -- 3.1.3 Line integrals with respect to a scalar -- 3.2 Connectivity of regions -- 3.3 Green's theorem in a plane -- 3.4 Conservative fields and potentials -- 3.5 Surface integrals -- 3.5.1 Evaluating surface integrals -- 3.5.2 Vector areas of surfaces -- 3.5.3 Physical examples of surface integrals -- 3.6 Volume integrals -- 3.7 Integral forms for grad, div and curl -- 3.8 Divergence theorem and related theorems -- 3.8.1 Green's theorems -- 3.8.2 Other related integral theorems -- 3.8.3 Physical applications of the divergence theorem -- 3.9 Stokes' theorem and related theorems -- 3.9.1 Related integral theorems -- 3.9.2 Physical applications of Stokes' theorem -- Summary -- Problems -- Hints and answers -- 4 Fourier series -- 4.1 The Dirichlet conditions |
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4.2 The Fourier coefficients -- 4.3 Symmetry considerations -- 4.4 Discontinuous functions -- 4.5 Non-periodic functions -- 4.6 Integration and differentiation -- 4.7 Complex Fourier series -- 4.8 Parseval's theorem -- Summary -- Problems -- Hints and answers -- 5 Integral transforms -- 5.1 Fourier transforms -- 5.1.1 The uncertainty principle -- 5.1.2 Fraunhofer diffraction -- 5.2 The Dirac deltaf-unction -- 5.2.1 Relation of the deltaf-function to Fourier transforms -- 5.3 Properties of Fourier transforms -- 5.3.1 Odd and even functions -- 5.3.2 Convolution and deconvolution -- 5.3.3 Parseval's theorem -- 5.3.4 Fourier transforms in higher dimensions -- 5.4 Laplace transforms -- 5.4.1 Laplace transforms of derivatives and integrals -- 5.4.2 Other properties of Laplace transforms -- 5.5 Concluding remarks -- Summary -- Problems -- Hints and answers -- 6 Higher-order ordinary differential equations -- 6.1 General considerations -- 6.1.1 General form of solution -- 6.1.2 Linear equations -- 6.2 Linear equations with constant coefficients -- 6.2.1 Finding the complementary function yc(x) -- 6.2.2 Finding the particular integral yp(x) -- 6.2.3 Constructing the general solution yc(x)+yp(x) -- 6.3 Linear recurrence relations -- 6.3.1 First-order recurrence relations -- 6.3.2 Second-order recurrence relations -- 6.3.3 Higher-order recurrence relations -- 6.4 Laplace transform method -- 6.5 Linear equations with variable coefficients -- 6.5.1 The Legendre and Euler linear equations -- 6.5.2 Exact equations -- 6.5.3 Partially known complementary function -- 6.5.4 Variation of parameters -- 6.5.5 Green's functions -- 6.6 General ordinary differential equations -- 6.6.1 Dependent variable absent -- 6.6.2 Independent variable absent -- 6.6.3 Equations homogeneous in x or y alone -- Summary -- Problems -- Hints and answers |
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7 Series solutions of ordinary differential equations -- 7.1 Second-order linear ordinary differential equations -- 7.2 Ordinary and singular points of an ODE -- 7.3 Series solutions about an ordinary point -- 7.4 Series solutions about a regular singular point -- 7.4.1 Distinct roots not differing by an integer -- 7.4.2 Repeated root of the indicial equation -- 7.4.3 Distinct roots differing by an integer -- 7.5 Obtaining a second solution -- 7.5.1 The Wronskian method -- 7.5.2 The derivative method -- 7.5.3 Series form of the second solution -- 7.6 Polynomial solutions -- Summary -- Problems -- Hints and answers -- 8 Eigenfunction methods for differential equations -- 8.1 Sets of functions -- 8.2 Adjoint, self-adjoint and Hermitian operators -- 8.3 Properties of Hermitian operators -- 8.3.1 Reality of the eigenvalues -- 8.3.2 Orthogonality and normalization of the eigenfunctions -- 8.3.3 Completeness of the eigenfunctions -- 8.3.4 Construction of real eigenfunctions -- 8.4 Sturm -- Liouville equations -- 8.4.1 Hermitian nature of the Sturm -- Liouville operator -- 8.4.2 Transforming an equation into Sturm -- Liouville form -- 8.5 Superposition of eigenfunctions: Green's functions -- Summary -- Problems -- Hints and answers -- 9 Special functions -- 9.1 Legendre functions -- 9.1.1 Legendre functions for integer l -- 9.1.2 Properties of Legendre polynomials -- 9.2 Associated Legendre functions -- 9.2.1 Associated Legendre functions for integer l -- 9.2.2 Properties of associated Legendre functions ... -- Mutual orthogonality -- Generating function -- Recurrence relations -- 9.3 Spherical harmonics -- 9.4 Chebyshev functions -- 9.4.1 Properties of Chebyshev polynomials -- Generating functions -- Recurrence relations -- 9.5 Bessel functions -- 9.5.1 Bessel functions for non-integer nu -- 9.5.2 Bessel functions for integer nu |
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9.5.3 Properties of Bessel functions J, (x) -- Mutual orthogonality -- Recurrence relations -- Generating function -- Integral representations -- 9.6 Spherical Bessel functions -- 9.7 Laguerre functions -- 9.7.1 Properties of Laguerre polynomials -- Rodrigues' formula -- Mutual orthogonality -- Generating function -- Recurrence relations -- 9.8 Associated Laguerre functions -- 9.8.1 Properties of associated Laguerre polynomials -- Rodrigues' formula -- Mutual orthogonality -- Generating function -- Recurrence relations -- 9.9 Hermite functions -- 9.9.1 Properties of Hermite polynomials -- Rodrigues' formula -- Mutual orthogonality -- Generating function -- 9.10 The gamma function and related functions -- 9.10.1 The gamma function -- 9.10.2 The incomplete gamma function -- 9.10.3 The error function -- Summary -- Problems -- Hints and answers -- 10 Partial differential equations -- 10.1 Important partial differential equations -- 10.1.1 The wave equation -- 10.1.2 The diffusion equation -- 10.1.3 Laplace's equation -- 10.1.4 Poisson's equation -- 10.1.5 Schrodinger's equation -- 10.2 General form of solution -- 10.3 General and particular solutions -- 10.3.1 First-order equations -- 10.3.2 Inhomogeneous equations and problems -- 10.3.3 Second-order equations -- 10.4 The wave equation -- 10.5 The diffusion equation -- 10.6 Boundary conditions and the uniqueness of solutions -- 10.6.1 Uniqueness of solutions -- Summary -- Problems -- Hints and answers -- 11 Solution methods for PDEs -- 11.1 Separation of variables: the general method -- 11.2 Superposition of separated solutions -- 11.3 Separation of variables in polar coordinates -- 11.3.1 Laplace's equation in polar coordinates -- Laplace's equation in plane polars -- Laplace's equation in spherical polars -- 11.3.2 Other equations in polar coordinates -- Helmholtz's equation in plane polars |
| Summary |
"The mathematical methods that physical scientists need for solving substantial problems in their fields of study are set out clearly and simply in this tutorial-style textbook. Students will develop problem-solving skills through hundreds of worked examples, self-test questions and homework problems. Each chapter concludes with a summary of the main procedures and results and all assumed prior knowledge is summarized in one of the appendices. Over 300 worked examples show how to use the techniques and around 100 self-test questions in the footnotes act as checkpoints to build student confidence. Nearly 400 end-of-chapter problems combine ideas from the chapter to reinforce the concepts. Hints and outline answers to the odd-numbered problems are given at the end of each chapter, with fully-worked solutions to these problems given in the accompanying Student Solutions Manual. Fully-worked solutions to all problems, password-protected for instructors, are available at www.cambridge.org/essential"-- Provided by publisher |
| Notes |
Print version record |
| Subject |
Mathematics -- Textbooks
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SCIENCE -- Mathematical Physics.
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Mathematics
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| Genre/Form |
Textbooks
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| Form |
Electronic book
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| Author |
Hobson, M. P. (Michael Paul), 1967-
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| ISBN |
9780511918964 |
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0511918968 |
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9780511915192 |
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0511915195 |
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9780511778506 |
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0511778503 |
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9780521141024 |
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0521141028 |
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0511917007 |
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9780511917004 |
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9780511913402 |
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0511913400 |
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