物理学和工程学中的数学方法

Since the publication of the first edition of this book, both through teaching the material it covers and as a result of receiving helpful comments from colleagues, we have become aware of the desirability of changes in a number of areas. The most important of these is that the mathematical preparation of current senior college and university entrants is now less thorough than it used to be. To match this, we decided to include a preliminary chapter covering areas such as polynomial equations, trigonometric identities, coordinate geometry, partial fractions, binomial expansions, necessary and sufficient condition and proof by induction and contradiction.

preface to the second edition preface to the first edition 1 preliminary algebra 1.1 simple functions and equations polynomial equations; factorisation; properties of roots 1.2 trigonometric identities single angle; compound-angles; double- and half-angle identities 1.3 coordinate geometry 1.4 partial fractions complications and special cases 1.5 binomial expansion 1.6 properties of binomial coefficients 1.7 some particular methods of proof proof by induction; proof by contradiction; necessary and sufficient conditions 1.8 exercises 1.9 hints and answers 2 preliminary calculus 2. 1 differentiation differentiation from first principles: products; the chain rule; quotients; implicit differentiation; logarithmic differentiation; leibnitz' theorem; special points of a function: curvature: theorems of differentiation 2.2 integration .integration from first principles; the inverse of differentiation; by inspection; sinusoidal jhnctions; logarithmic integration; using partial fractions;substitution method; integration by parts; reduction formulae; infinite and improper integrals; plane polar coordinates; integral inequalities; applications of integration 2.3 exercises 2.4 hints and answers 3 complex numbers and hyperbolic functions 3.1 the need for complex numbers 3.2 manipulation of complex numbers addition and subtraction; modulus and argument; multiplication; complex conjugate; division 3.3 polar representation of complex numbers multiplication and division in polar form 3.4 de moivre's theorem trigonometric identities;finding the nth roots of unity: solving polynomial equations 3.5 complex logarithms and complex powers 3.6 applications to differentiation and integration 3.7 hyperbolic functions definitions; hyperbolic-trigonometric analogies; identities of hyperbolic functions: solving hyperbolic equations; inverses of hyperbolic functions;calculus of hyperbolic functions 3.8 exercises 3.9 hints and answers 4 series and limits 4.1 series 4.2 summation of series arithmetic series; geometric series; arithmetico-geometric series; the difference method; series involving natural numbers; transformation of series 4.3 convergence of infinite series absolute and conditional convergence; series containing only real positive terms; alternating series test 4.4 operations with series 4.5 power series convergence of power series; operations with power series 4.6 taylor series taylor's theorem; approximation errors; standard maclaurin series 4.7 evaluation of limits 4.8 exercises 4.9 hints and answers 5 partial differentiation 5.1 definition of the partial derivative 5.2 the total differential and total derivative 5.3 exact and inexact differentials 5.4 useful theorems of partial differentiation 5.5 the chain rule 5.6 change of variables 5.7 taylor's theorem for many-variable functions 5.8 stationary values of many-variable functions 5.9 stationary values under constraints 5.10 envelopes 5.11 thermodynamic relations 5.12 differentiation of integrals 5.13 exercises 5.14 hints and answers 6 multiple integrals 6.1 double integrals 6.2 triple integrals 6.3 applications of multiple integrals areas and volumes; masses, centres of mass and centroids; pappus' theorems; moments of inertia; mean values of functions 6.4 change of variables in multiple integrals change of variables in double integrals; evaluation of the integral i =change of variables in triple integrals; general properties of jacobians 6.5 exercises 6.6 hints and answers 7 vector algebra 7.1 scalars and vectors 7.2 addition and subtraction of vectors 7.3 multiplication by a scalar 7.4 basis vectors and components 7.5 magnitude of a vector 7.6 multiplication of vectors scalar product; vector product; scalar triple product; vector triple product 7.7 equations of lines, planes and spheres 7.8 using vectors to find distances point to line; point to plane; line to line; line to plane 7.9 reciprocal vectors 7.10 exercises 7.11 hints and answers 8 matrices and vector spaces 8.1 vector spaces basis vectors; inner product; some useful inequalities 8.2 linear operators 8.3 matrices 8.4 basic matrix algebra matrix addition; multiplication by a scalar; matrix multiplication 8.5 functions of matrices 8,6 the transpose of a matrix 8.7 the complex and hermitian conjugates of a matrix 8.8 the trace of a matrix 8.9 the determinant of a matrix properties of determinants 8.10 the inverse of a matrix 8.11 the rank of a matrix 8.12 special types of square matrix diagonal; triangular; symmetric and antisymmetric ; orthogonal; hermitian and anti-hermitian; unitary; normal 8.13 eigenvectors and eigenvalues ora normal matrix; of hermitian and anti~herrnitian matrices; ora unitary matrix; ora general square matrix 8.14 determination of eigenvalues and eigenvectors degenerate eigenvalues 8.15 change of basis and similarity transformations 8.16 diagonalisation of matrices 8.17 quadratic and hermitian forms stationary properties of the eigenvectors ; quadratic surfaces 8.18 simultaneous linear equations range; null space; n simultaneous linear equations in n unknowns; singular value decomposition 8.19 exercises 8.20 hintsand answers 9 normal modes 9.1 typical oscillatory systems 9.2 symmetry and normal modes 9.3 rayleigh-ritz method 9.4 exercises 9.5 hints and answers 10 vector calculus 10.1 differentiation of vectors composite vector expressions; differential of a vector 10.2 integration of vectors 10.3 space curves 10.4 vector functions of several arguments 10.5 surfaces 10.6 scalar and vector fields 10.7 vector operators gradient of a scalar field: divergence of a vector field: curl of a vector field 10.8 vector operator formulae vector operators acting on sums and products; combinations of grad, div and curl 10.9 cylindrical and spherical polar coordinates 10.10 general curvilinear coordinates 10.11 exercises 10.12 hints and answers 11 line, surface and volume integrals 11.1 line integrals evaluating line integrals; physical examples; line integrals with respect to a scalar 11.2 connectivity of regions 11.3 green's theorem in a plane 11.4 conservative fields and potentials 11.5 surface integrals evaluating surface integrals; vector areas of surfaces; physical examples 11.6 volume integrals volumes of three-dimensional regions 11.7 integral forms for grad, div and curl 11.8 divergence theorem and related theorems green's theorems; other related integral theorems; physical applications 11.9 stokes' theorem and related theorems related integral theorems: physical applications 11.10 exercises 11.11 hints and answers 12 fourier series 12.1 the dirichlet conditions 12.2 the fourier coefficients 12.3 symmetry considerations 12.4 discontinuous functions 12.5 non-periodic functions 12.6 integration and differentiation 12.7 complex fourier series 12.8 parseval's theorem 12.9 exercises 12.10 hints and answers 13 integral transforms 13.1 fourier transforms the uncertainty principle; fraunhofer diffraction: the dirac &-function: relation of the 6-function to fourier transforms; properties of fourier transjorms; odd and even functions; convolution and deconvolution; correlation functions and energy spectra; parseval's theorem; fourier transforms in higher dimensions 13.2 laplace transforms laplace transforms of derivatives and integrals; other properties of laplace transforms 13.3 concluding remarks 13.4 exercises 13.5 hints and answers 14 first-order ordinary differential equations 14.1 general form of solution 14.2 first-degree first-order equations separable-variable equations; exact equations; inexact equations, integrating factors; linear equations; homogeneous equations; isobaric equations: bernoulli's equation; miscellaneous equations 14.3 higher-degree first-order equations equations soluble for p; for x; for y; clairaut's equation 14.4 exercises 14.5 hints and answers 15 higher-order ordinary differential equations 15.1 linear equations with constant coefficients finding the complementary function yc(x): finding the particular integral yp(x); constructing the general solution ye(x)+ yp(x): linear recurrence relations: laplace transform method 15.2 linear equations with variable coefficients the legendre and euler linear equations; exact equations; partially known complementary function; variation of parameters; green's functions; canonical form for second-order equations 15.3 general ordinary differential equations dependent variable absent; independent variable absent; non-linear exact equations; isobaric or homogeneous equations; equations homogeneous in x or y alone; equations having y = aex as a solution 15.4 exercises 15.5 hints and answers 16 series solutions of ordinary differential equations 16.1 second-order linear ordinary differential equations ordinary and singular points 16.2 series solutions about an ordinary point 16.3 series solutions about a regular singular point distinct roots not differing by an integer; repeated root of the indicial equation; distinct roots differing by an integer 16.4 obtaining a second solution the wronskian method; the derivative method; series form of the second solution 16.5 polynomial solutions 16.6 legendre's equation general solution for integer 1 ; properties of legendre polynomials 16.7 bessers equation general solution for non-integer v; general solution for integer v; properties of bessel functions 16.8 general remarks 16.9 exercises 16.10 hints and answers 17 eigenfunction methods for differential equations 17.1 sets of functions some useful inequalities 17.2 adjoint and hermitian operators 17.3 the properties of hermitian operators reality of the eigenvalues; orthogonality of the eigenfunctions; construction of real eigenfunctions 17.4 sturm-liouville equations valid boundary conditions; putting an equation into sturm-liouville form 17.5 examples of sturm-liouville equations legendre's equation; the associated legendre equation; bessel's equation; the simple harmonic equation; hermite's equation; laguerre's equation; chebyshev's equation 17.6 superposition of eigenfunctions: green's functions 17.7 a useful generalisation 17.8 exercises 17.9 hints and answers 18 partial differential equations: general and particular solutions 18.1 important partial differential equations the wave equation; the diffusion equation; laplace's equation; poisson's equation; schrodinger's equation 18.2 general form of solution 18.3 general and particular solutions first-order equations; inhomogeneous equations and problems; second-order equations 18.4 the wave equation 18.5 the diffusion equation 18.6 characteristics and the existence of solutions first-order equations; second-order equations 18.7 uniqueness of solutions 18.8 exercises 18.9 hints and answers 19 partial differential equations: separation of variables and other methods 19.1 separation of variables: the general method 19.2 superposition of separated solutions 19.3 separation of variables in polar coordinates laplace's equation in polar coordinates: spherical harmonics: other equations in polar coordinates; solution by expansion; separation of variables for inhomogeneous equations 19.4 integral transform methods 19.5 inhomogeneous problems-green's functions similarities to green's functions for ordinary differential equations: general boundary-value problems: dirichlet problems; neumann problems 19.6 exercises 19.7 hints and answers 20 complex variables 20.1 functions of a complex variable 20.2 the cauchy-riemann relations 20.3 power series in a complex variable 20.4 some elementary functions 20.5 multivalued functions and branch cuts 20.6 singularities and zeroes of complex functions 20.7 complex potentials 20.8 conformal transformations 20.9 applications ofconformal transformations 20.10 complex integrals 20.11 cauchy's theorem 20.12 cauchy's integral formula 20.13 taylor and laurent series 20.14 residue theorem 20.15 location of zeroes 20.16 integrals of sinusoidal functions 20.17 some infinite integrals 20.18 integrals of multivalued functions 20.19 summation of series 20.20 inverse laplace transform 20.21 exercises 20.22 hints and answers 21 tensors 21.1 some notation 21.2 change of basis 21.3 cartesian tensors 21.4 first- and zero-order cartesian tensors 21.5 second- and higher-order cartesian tensors 21.6 the algebra of tensors 21.7 the quotient law 21.8 the tensors and 21.9 isotropic tensors 21.10 improper rotations and pseudotensors 21.11 dual tensors 21.t2 physical applications of tensors 21.13 integral theorems for tensors 21.14 non-cartesian coordinates 21.15 the metric tensor 21.16 general coordinate transformations and tensors 21.17 relative tensors 21.18 derivatives of basis vectors and christoffel symbols 21.19 covariant differentiation 21.20 vector operators in tensor form 21.21 absolute derivatives along curves 21.22 geodesics 21.23 exercises 21.24 hints and answers 22 calculus of variations 22.1 the euler-lagrange equation 22.2 special cases f does not contain y explicitly; f does not contain x explicitly 22.3 some extensions several dependent variables; several independent variables; higher-order derivatives: variable end-points 22.4 constrained variation 22.5 physical variational principles fermat's principle in optics; hamilton's principle in mechanics 22.6 general eigenvalue problems 22.7 estimation ofeigenvalues and eigenfunctions 22.8 adjustment of parameters 22.9 exercises 22.10 hints and answers 23 integral equations 23.1 obtaining an integral equation from a differential equation 23.2 types of integral equation 23.3 operator notation and the existence of solutions 23.4 closed-form solutions separable kernels; integral transform methods; differentiation 23.5 neumann series 23.6 fredholm theory 23.7 schmidt-hilbert theory 23.8 exercises 23.9 hints and answers 24 group theory 24.1 groups definition of a group; examples of groups 24.2 finite groups 24.3 non-abelian groups 24.4 permutation groups 24.5 mappings between groups 24.6 subgroups 24.7 subdividing a group equivalence relations and classes; congruence and cosets; conjugates and classes 24.8 exercises 24.9 hints and answers 25 representation theory 25.1 dipole moments of molecules 25.2 choosing an appropriate formalism 25.3 equivalent representations 25.4 reducibility of a representation 25.5 the orthogonality theorem for irreducible representations 25.6 characters orthogonality property of characters 25.7 counting irreps using characters summation rules for irreps 25.8 construction of a character table 25.9 group nomenclature 25.10 product representations 25.11 physical applications of group theory bonding in molecules: matrix elements in quantum mechanics: degeneracy of normal modes: breaking of degeneracies 25.12 exercises 25.13 hints and answers 26 probability 26.1 venn diagrams 26.2 probability axioms and theorems; conditional probability; bayes' theorem 26.3 permutations and combinations 26.4 random variables and distributions discrete random variables; continuous random variables 26.5 properties of distributions mean: mode and median: variance and standard deviation: moments: central moments 26.6 functions of random variables 2617 generating functions probability generating functions; moment generating functions; characteristic functions; cumulant generating functions 26.8 important discrete distributions binomial; geometric; negative binomial; hypergeometric ; poisson 26.9 important continuous distributions gaussian : log-normah exponential; gamma; chi-squared; cauchy ; breitwigner : uniform 26.10 the central limit theorem 26.11 joint distributions discrete bivariate ; continuous bivariate ; marginal and conditional distributions 26.12 properties of joint distributions means; variances; covariance and correlation 26.13 generating functions for joint distributions 26.14 transformation of variables in joint distributions 26.15 important joint distributions multinominah multivariate gaussian 26.16 exercises 26.17 hints and answers 27 statistics 27.1 experiments, samples and populations 27.2 sample statistics averages; variance and standard deviation; moments; covariance and correlation 27.3 estimators and sampling distributions consistency, bias and efficiency; fisher's inequality: standard errors; confidence limits 27.4 some basic estimators mean; variance: standard deviation; moments; covariance and correlation 27.5 maximum-likelihood method ml estimator; trans]ormation invariance and bias; efficiency; errors and confidence limits; bayesian interpretation; large-n behaviour; extended ml method 27.6 the method of least squares linear least squares; non-linear least squares 27.7 hypothesis testing simple and composite hypotheses; statistical tests; neyman-pearson; generalised likelihood-ratio: student's t: fisher's f: goodness of fit 27.8 exercises 27.9 hints and answers 28 numerical methods 28.1 algebraic and transcendental equations rearrangement of the equation; linear interpolation; binary chopping; newton-raphson method 28.2 convergence of iteration schemes 28.3 simultaneous linear equations gaussian elimination; gauss-seidel iteration; tridiagonal matrices 28.4 numerical integration trapezium rule; simpson's rule; gaussian integration; monte carlo methods 28.5 finite differences 28.6 differential equations difference equations; taylor series solutions; prediction and correction; runge-kutta methods; isoclines 28.7 higher-order equations 28.8 partial differential equations 28.9 exercises 28.10 hints and answers appendix gamma, beta and error functions a1.1 the gamma function al.2 the beta function al.3 the error function index