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Bibliografická citace

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BK
Third edition
Cambridge (UK) : Cambridge University Press, [2006]
xxvii, 1333 stran : ilustrace ; 25 cm

objednat
ISBN 978-0-521-67971-8 (brožováno)
Obsahuje rejstřík
001636152
Contents // Preface to the third edition xx // Preface to the second edition xxiii // Preface to the first edition xxv // 1 Preliminary algebra 1 // 1.1 Simple functions and equations 1 // Polynomial equations; factorisation; properties of roots // 1.2 Trigonometric identities 10 // Single angle; compound angles; double- and half-angle identities // 1.3 Coordinate geometry 15 // 1.4 Partial fractions 18 // Complications and special cases // 1.5 Binomial expansion 25 // 1.6 Properties of binomial coefficients 27 // 1.7 Some particular methods of proof 30 // Proof by induction; proof by contradiction; necessary and sufficient conditions // 1.8 Exercises 36 // 1.9 Hints and answers 39 // 2 Preliminary calculus 41 // 2.1 Differentiation 41 // Differentiation from first principles; products; the chain rule; quotients; implicit differentiation; logarithmic differentiation; Leibnitz’ theorem; special points of a function; curvature; theorems of differentiation // v // CONTENTS // 2.2 Integration 59 // Integration from first principles; the inverse of differentiation; by inspection; sinusoidal functions; 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 76 // 2.4 Hints and answers 81 // 3 Complex numbers and hyperbolic functions 83 // 3.1 The need for complex numbers 83 // 3.2 Manipulation of complex
numbers 85 // Addition and subtraction; modulus and argument; multiplication; complex conjugate; division // 3.3 Polar representation of complex numbers 92 // Multiplication and division in polar form // 3.4 de Moivre’s theorem 95 // trigonometric identities; finding the nth roots of unity; solving polynomial equations // 3.5 Complex logarithms and complex powers 99 // 3.6 Applications to differentiation and integration 101 // 3.7 Hyperbolic functions 102 // Definitions; hyperbolic-trigonometric analogies; identities of hyperbolic functions; solving hyperbolic equations; inverses of hyperbolic functions; calculus of hyperbolic functions // 3.8 Exercises 109 // 3.9 Hints and answers 113 // 4 Series and limits 115 // 4.1 Series 115 // 4.2 Summation of series 116 // Arithmetic series; geometric series; arithmetico-geometric series; the difference method; series involving natural numbers; transformation of series // 4.3 Convergence of infinite series 124 // Absolute and conditional convergence ; series containing only real positive // terms; alternating series test // 4.4 Operations with series 131 // 4.5 Power series 131 // Convergence of power series; operations with power series // 4.6 Taylor series 136 // Taylor’s theorem; approximation errors; standard Maclaurin series // 4.7 Evaluation of limits 141 // 4.8 Exercises 144 // 4.9 Hints and answers 149 // CONTENTS // 5 Partial differentiation 151 // 5.1 Definition of the partial derivative 151 // 5.2 The total differential and
total derivative 153 // 5.3 Exact and inexact differentials 155 // 5.4 Useful theorems of partial differentiation 157 // 5.5 The chain rule 157 // 5.6 Change of variables 158 // 5.7 Taylor’s theorem for many-variable functions 160 // 5.8 Stationary values of many-variable functions 162 // 5.9 Stationary values under constraints 167 // 5.10 Envelopes 173 // 5.11 Thermodynamic relations 176 // 5.12 Differentiation of integrals 178 // 5.13 Exercises 179 // 5.14 Hints and answers 185 // 6 Multiple integrals 187 // 6.1 Double integrals 187 // 6.2 Triple integrals 190 // 6.3 Applications of multiple integrals 191 // 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 199 // Change of variables in double integrals; evaluation of the integral I = e~x2 dx; change of variables in triple integrals; general properties of Jacobians // 6.5 Exercises 207 // 6.6 Hints and answers 211 // 7 Vector algebra 212 // 7.1 Scalars and vectors 212 // 7.2 Addition and subtraction of vectors 213 // 7.3 Multiplication by a scalar 214 // 7.4 Basis vectors and components 217 // 7.5 Magnitude of a vector 218 // 7.6 Multiplication of vectors 219 // Scalar product; vector product; scalar triple product; vector triple product // vii // CONTENTS // 7.7 Equations of lines, planes and spheres 226 // 7.8 Using vectors to find distances 229 // Point to line; point to plane; line to line; line to
plane // 7.9 Reciprocal vectors 233 // 7.10 Exercises 234 // 7.11 Hints and answers 240 // 8 Matrices and vector spaces 241 // 8.1 Vector spaces 242 // Basis vectors; inner product; some useful inequalities // 8.2 Linear operators 247 // 8.3 Matrices 249 // 8.4 Basic matrix algebra 250 // Matrix addition; multiplication by a scalar; matrix multiplication // 8.5 Functions of matrices 255 // 8.6 The transpose of a matrix 255 // 8.7 The complex and Hermitian conjugates of a matrix 256 // 8.8 The trace of a matrix 258 // 8.9 The determinant of a matrix 259 // Properties of determinants // 8.10 The inverse of a matrix 263 // 8.11 The rank of a matrix 267 // 8.12 Special types of square matrix 268 // Diagonal; triangular; symmetric and antisymmetric; orthogonal; Hermitian // and anti-Hermitian; unitary; normal // 8.13 Eigenvectors and eigenvalues 272 // Of a normal matrix; of Hermitian and anti-Hermitian matrices; of a unitary matrix; of a general square matrix // 8.14 Determination of eigenvalues and eigenvectors 280 // Degenerate eigenvalues // 8.15 Change of basis and similarity transformations 282 // 8.16 Diagonalisation of matrices 285 // 8.17 Quadratic and Hermitian forms 288 // Stationary properties of the eigenvectors ; quadratic surfaces // 8.18 Simultaneous linear equations 292 // Range; null space; N simultaneous linear equations in N unknowns; singular value decomposition // 8.19 Exercises 307 // 8.20 Hints and answers 314 // 9 Normal modes 316 // 9.1 Typical oscillatory
systems 317 // 9.2 Symmetry and normal modes 322 // viii // CONTENTS // 9.3 Rayleigh-Ritz method 327 // 9.4 Exercises 329 // 9.5 Hints and answers 332 // 10 Vector calculus 334 // 10.1 Differentiation of vectors 334 // Composite vector expressions ; differential of a vector // 10.2 Integration of vectors 339 // 10.3 Space curves 340 // 10.4 Vector functions of several arguments 344 // 10.5 Surfaces 345 // 10.6 Scalar and vector fields 347 // 10.7 Vector operators 347 // Gradient of a scalar field; divergence of a vector field; curl of a vector field // 10.8 Vector operator formulae 354 // Vector operators acting on sums and products; combinations of grad, div and curl // 10.9 Cylindrical and spherical polar coordinates 357 // 10.10 General curvilinear coordinates 364 // 10.11 Exercises 369 // 10.12 Hints and answers 375 // 11 Line, surface and volume integrals 377 // 11.1 Line integrals 377 // Evaluating line integrals; physical examples; line integrals with respect to a scalar // 11.2 Connectivity of regions 383 // 11.3 Green’s theorem in a plane 384 // 11.4 Conservative fields and potentials 387 // 11.5 Surface integrals 389 // Evaluating surface integrals; vector areas of surfaces; physical examples // 11.6 Volume integrals 396 // Volumes of three-dimensional regions // 11.7 Integral forms for grad, div and curl 398 // 11.8 Divergence theorem and related theorems 401 // Greens theorems; other related integral theorems; physical applications // 11.9 Stokes’ theorem and related
theorems 406 // Related integral theorems; physical applications // 11.10 Exercises 409 // 11.11 Hints and answers 414 // 12 Fourier series 415 // 12.1 The Dirichlet conditions 415 // 12.2 The Fourier coefficients 417 // CONTENTS // 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 // 419 // 420 422 424 424 // 426 // 427 431 // 13 Integral transforms // 13.1 Fourier transforms // 433 // 433 // The uncertainty principle; Fraunhofer diffraction; the Dirac ô-function; relation of the ö-function to Fourier transforms; properties of Fourier transforms; odd and even functions; convolution and deconvolution; correlation functions and energy spectra; Parsevals theorem; Fourier transforms in higher dimensions // 13.2 Laplace transforms 453 // Laplace transforms of derivatives and integrals; other properties of Laplace transforms // 13.3 Concluding remarks 459 // 13.4 Exercises 460 // 13.5 Hints and answers 466 // 14 First-order ordinary differential equations 468 // 14.1 General form of solution 469 // 14.2 First-degree first-order equations 470 // 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 480 // Equations
soluble for p; for x; for y ; Clairaut’s equation // 14.4 Exercises 484 // 14.5 Hints and answers 488 // 15 Higher-order ordinary differential equations 490 // 15.1 Linear equations with constant coefficients 492 // Finding the complementary function yc(x); finding the particular integral // yp(x); constructing the general solution yc(x) + yp(x); linear recurrence relations; Laplace transform method // 15.2 Linear equations with variable coefficients 503 // The Legendre and Euler linear equations; exact equations; partially known complementary function ; variation of parameters; Green’s functions ; canonical form for second-order equations // CONTENTS // 15.3 General ordinary differential equations 518 // Dependent variable absent; independent variable absent; non-linear exact equations; isobaric or homogeneous equations; equations homogeneous in x // or ? alone; equations having ? = Aex as a solution // 15.4 Exercises 523 // 15.5 Hints and answers 529 // 16 Series solutions of ordinary differential equations 531 // 16.1 Second-order linear ordinary differential equations 531 // Ordinary and singular points // 16.2 Series solutions about an ordinary point 535 // 16.3 Series solutions about a regular singular point 538 // 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 544 // The Wronskian method; the derivative method; series form of the second solution // 16.5 Polynomial
solutions 548 // 16.6 Exercises 550 // 16.7 Hints and answers 553 // 17 Eigenfunction methods for differential equations 554 // 17.1 Sets of functions 556 // Some useful inequalities // 17.2 Adjoint, self-adjoint and Hermitian operators 559 // 17.3 Properties of Hermitian operators 561 // Reality of the eigenvalues ; orthogonality of the eigenfunctions; construction // of real eigenfunctions // 17.4 Sturm-Liouville equations 564 // Valid boundary conditions; putting an equation into Sturm-Liouville form // 17.5 Superposition of eigenfunctions: Green’s functions 569 // 17.6 A useful generalisation 572 // 17.7 Exercises 573 // 17.8 Hints and answers 576 // 18 Special functions 577 // 18.1 Legendre functions 577 // General solution for integer Ł ; properties of Legendre polynomials // 18.2 Associated Legendre functions 587 // 18.3 Spherical harmonics 593 // 18.4 Chebyshev functions 595 // 18.5 Bessel functions 602 // General solution for non-integer v; general solution for integer v; properties // of Bessel functions // 18.6 Spherical Bessel functions 614 // CONTENTS // 18.7 Laguerre functions 616 // 18.8 Associated Laguerre functions 621 // 18.9 Hermite functions 624 // 18.10 Hypergeometric functions 628 // 18.11 Confluent hypergeometric functions 633 // 18.12 The gamma function and related functions 635 // 18.13 Exercises 640 // 18.14 Hints and answers 646 // 19 Quantum operators 648 // 19.1 Operator formalism 648 // Commutators // 19.2 Physical examples of operators 656
Uncertainty principle; angular momentum; creation and annihilation operators // 19.3 Exercises 671 // 19.4 Hints and answers 674 // 20 Partial differential equations: general and particular solutions 675 // 20.1 Important partial differential equations 676 // The wave equation; the diffusion equation; Laplace’s equation; Poisson’s equation; Schrödinger’s equation // 20.2 General form of solution 680 // 20.3 General and particular solutions 681 // First-order equations; inhomogeneous equations and problems; second-order equations // 20.4 The wave equation 693 // 20.5 The diffusion equation 695 // 20.6 Characteristics and the existence of solutions 699 // First-order equations; second-order equations // 20.7 Uniqueness of solutions 705 // 20.8 Exercises 707 // 20.9 Hints and answers 711 // 21 Partial differential equations: separation of variables // and other methods 713 // 21.1 Separation of variables: the general method 713 // 21.2 Superposition of separated solutions 717 // 21.3 Separation of variables in polar coordinates 725 // Laplace’s equation in polar coordinates; spherical harmonics; other equations // in polar coordinates ; solution by expansion; separation of variables for inhomogeneous equations // 21.4 Integral transform methods 747 // xii // CONTENTS // 21.5 Inhomogeneous problems - Green’s functions 751 // Similarities to Green’s functions for ordinary differential equations; general boundary-value problems; Dirichlet problems; Neumann problems // 21.6
Exercises 767 // 21.7 Hints and answers 773 // 22 Calculus of variations 775 // 22.1 The Euler-Lagrange equation 776 // 22.2 Special cases 777 // F does not contain ? explicitly; F does not contain x explicitly // 22.3 Some extensions 781 // Several dependent variables; several independent variables; higher-order derivatives; variable end-points // 22.4 Constrained variation 785 // 22.5 Physical variational principles 787 // Fermat’s principle in optics; Hamilton’s principle in mechanics // 22.6 General eigenvalue problems 790 // 22.7 Estimation of eigenvalues and eigenfunctions 792 // 22.8 Adjustment of parameters 795 // 22.9 Exercises 797 // 22.10 Hints and answers 801 // 23 Integral equations 803 // 23.1 Obtaining an integral equation from a differential equation 803 // 23.2 Types of integral equation 804 // 23.3 Operator notation and the existence of solutions 805 // 23.4 Closed-form solutions 806 // Separable kernels; integral transform methods; differentiation // 23.5 Neumann series 813 // 23.6 Fredholm theory 815 // 23.7 Schmidt-Hilbert theory 816 // 23.8 Exercises 819 // 23.9 Hints and answers 823 // 24 Complex variables 824 // 24.1 Functions of a complex variable 825 // 24.2 The Cauchy-Riemann relations 827 // 24.3 Power series in a complex variable 830 // 24.4 Some elementary functions 832 // 24.5 Multivalued functions and branch cuts 835 // 24.6 Singularities and zeros of complex functions 837 // 24.7 Conformal transformations 839 // 24.8 Complex integrals 845
xiii // CONTENTS // 24.9 Cauchy’s theorem 849 // 24.10 Cauchy’s integral formula 851 // 24.11 Taylor and Laurent series 853 // 24.12 Residue theorem 858 // 24.13 Definite integrals using contour integration 861 // 24.14 Exercises 867 // 24.15 Hints and answers 870 // 25 Applications of complex variables 871 // 25.1 Complex potentials 871 // 25.2 Applications of conformal transformations 876 // 25.3 Location of zeros 879 // 25.4 Summation of series 882 // 25.5 Inverse Laplace transform 884 // 25.6 Stokes’ equation and Airy integrals 888 // 25.7 WKB methods 895 // 25.8 Approximations to integrals 905 // Level lines and saddle points; steepest descents; stationary phase // 25.9 Exercises 920 // 25.10 Hints and answers 925 // 26 Tensors 927 // 26.1 Some notation 928 // 26.2 Change of basis 929 // 26.3 Cartesian tensors 930 // 26.4 First- and zero-order Cartesian tensors 932 // 26.5 Second- and higher-order Cartesian tensors 935 // 26.6 The algebra of tensors 938 // 26.7 The quotient law 939 // 26.8 The tensors <5,7- and €? 941 // 26.9 Isotropic tensors 944 // 26.10 Improper rotations and pseudotensors 946 // 26.11 Dual tensors 949 // 26.12 Physical applications of tensors 950 // 26.13 Integral theorems for tensors 954 // 26.14 Non-Cartesian coordinates 955 // 26.15 The metric tensor 957 // 26.16 General coordinate transformations and tensors 960 // 26.17 Relative tensors 963 // 26.18 Derivatives of basis vectors and Christoffel symbols 965 // 26.19 Covariant differentiation
968 // 26.20 Vector operators in tensor form 971 // XIV // CONTENTS // 26.21 Absolute derivatives along curves 975 // 26.22 Geodesics 976 // 26.23 Exercises 977 // 26.24 Hints and answers 982 // 27 Numerical methods 984 // 27.1 Algebraic and transcendental equations 985 // Rearrangement of the equation; linear interpolation; binary chopping; Newton-Raphson method // 27.2 Convergence of iteration schemes 992 // 27.3 Simultaneous linear equations 994 // Gaussian elimination; Gauss-Seidel iteration; tridiagonal matrices // 27.4 Numerical integration 1000 // Trapezium rule; Simpsons rule; Gaussian integration; Monte Carlo methods // 27.5 Finite differences 1019 // 27.6 Differential equations 1020 // Difference equations; Taylor series solutions; prediction and correction; Runge-Kutta methods; isoclines // 27.7 Higher-order equations 1028 // 27.8 Partial differential equations 1030 // 27.9 Exercises 1033 // 27.10 Hints and answers 1039 // 28 Group theory 1041 // 28.1 Groups 1041 // Definition of a group; examples of groups // 28.2 Finite groups 1049 // 28.3 Non-Abelian groups 1052 // 28.4 Permutation groups 1056 // 28.5 Mappings between groups 1059 // 28.6 Subgroups 1061 // 28.7 Subdividing a group 1063 // Equivalence relations and classes; congruence and cosets; conjugates and classes // 28.8 Exercises 1070 // 28.9 Hints and answers 1074 // 29 Representation theory 1076 // 29.1 Dipole moments of molecules 1077 // 29.2 Choosing an appropriate formalism 1078 // 29.3 Equivalent representations
1084 // 29.4 Reducibility of a representation 1086 // 29.5 The orthogonality theorem for irreducible representations 1090 // CONTENTS // 29.6 Characters 1092 // Orthogonality property of characters // 29.7 Counting irreps using characters 1095 // Summation rules for irreps // 29.8 Construction of a character table 1100 // 29.9 Group nomenclature 1102 // 29.10 Product representations 1103 // 29.11 Physical applications of group theory 1105 // Bonding in molecules; matrix elements in quantum mechanics; degeneracy of normal modes ; breaking of degeneracies // 29.12 Exercises 1113 // 29.13 Hints and answers 1117 // 30 Probability 1119 // 30.1 Venn diagrams 1119 // 30.2 Probability 1124 // Axioms and theorems; conditional probability; Bayes’ theorem // 30.3 Permutations and combinations 1133 // 30.4 Random variables and distributions 1139 // Discrete random variables; continuous random variables // 30.5 Properties of distributions 1143 // Mean; mode and median; variance and standard deviation; moments; central moments // 30.6 Functions of random variables 1150 // 30.7 Generating functions 1157 // Probability generating functions ; moment generating functions; characteristic functions; cumulant generating functions // 30.8 Important discrete distributions 1168 // Binomial; geometric; negative binomial; hypergeometric ; Poisson // 30.9 Important continuous distributions 1179 // Gaussian; log-normal; exponential; gamma; chi-squared; Cauchy; Breit-Wigner; uniform // 30.10 The central
limit theorem 1195 // 30.11 Joint distributions 1196 // Discrete bivariate; continuous bivariate; marginal and conditional distributions // 30.12 Properties of joint distributions 1199 // Means; variances; covariance and correlation // 30.13 Generating functions for joint distributions 1205 // 30.14 Transformation of variables in joint distributions 1206 // 30.15 Important joint distributions 1207 // Multinominal; multivariate Gaussian // 30.16 Exercises 1211 // 30.17 Hints and answers 1219 // xvi // CONTENTS // 31 Statistics 1221 // 31.1 Experiments, samples and populations 1221 // 31.2 Sample statistics 1222 // Averages; variance and standard deviation; moments; covariance and correlation // 31.3 Estimators and sampling distributions 1229 // Consistency, bias and efficiency; Fisher’s inequality; standard errors; confidence limits // 31.4 Some basic estimators 1243 // Mean; variance; standard deviation; moments; covariance and correlation // 31.5 Maximum-likelihood method 1255 // ML estimator; transformation invariance and bias; efficiency; errors and confidence limits; Bayesian interpretation; large-N behaviour; extended // ML method // 31.6 The method of least squares 1271 // Linear least squares; non-linear least squares // 31.7 Hypothesis testing 1277 // Simple and composite hypotheses; statistical tests; Neyman-Pearson; generalised likelihood-ratio ; Student’s t; Fisher’s F ; goodness of fit // 31.8 Exercises 1298 // 31.9 Hints and answers 1303 // Index 1305 // xvii

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