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
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