Cover -- Half-title -- Title -- Copyright -- Contents -- Preface -- 1 Vector and tensor analysis -- Vectors and scalars -- Direction angles and direction cosines -- Vector algebra -- Equality of vectors -- Vector addition -- Multiplication by a scalar -- The scalar product -- The vector (cross or outer) product -- The triple scalar product A (B × C) -- The triple vector product -- Change of coordinate system -- The linear vector space V -- Vector differentiation -- Space curves -- Motion in a plane -- A vector treatment of classical orbit theory -- Vector differentiation of a scalar field and the gradient -- Conservative vector field -- The vector differential operator… -- Vector differentiation of a vector field -- The divergence of a vector -- The operator…the Laplacian -- The curl of a vector -- Formulas involving… -- Orthogonal curvilinear coordinates -- Special orthogonal coordinate systems -- Cylindrical coordinates (Rho, Phi, z) -- Spherical coordinates (r, Theta, Phi) -- Vector integration and integral theorems -- Gauss' theorem (the divergence theorem) -- Continuity equation -- Stokes' theorem -- Green's theorem -- Green's theorem in the plane -- Helmholtz's theorem -- Some useful integral relations -- Tensor analysis -- Contravariant and covariant vectors -- Tensors of second rank -- Basic operations with tensors -- Quotient law -- The line element and metric tensor -- Associated tensors -- Geodesics in a Riemannian space -- Covariant differentiation -- Problems -- 2 Ordinary differential equations -- First-order differential equations -- Separable variables -- Exact equations -- Integrating factors -- Bernoulli's equation -- Second-order equations with constant coeffcients -- Nature of the solution of linear equations -- General solutions of the second-order equations -- Finding the complementary function.

Finding the particular integral -- Particular integral and the operator D (=d/dx) -- Rules for D operators -- The Euler linear equation -- Solutions in power series -- Ordinary and singular points of a differential equation -- Frobenius and Fuchs theorem -- Simultaneous equations -- The gamma and beta functions -- Problems -- 3 Matrix algebra -- Definition of a matrix -- Four basic algebra operations for matrices -- Equality of matrices -- Addition of matrices -- Multiplication of a matrix by a number -- Matrix multiplication -- The commutator -- Powers of a matrix -- Functions of matrices -- Transpose of a matrix -- Symmetric and skew-symmetric matrices -- The matrix representation of a vector product -- The inverse of a matrix -- A method for finding… -- Systems of linear equations and the inverse of a matrix -- Complex conjugate of a matrix -- Hermitian conjugation -- Hermitian/anti-hermitian matrix -- Orthogonal matrix (real) -- Unitary matrix -- Rotation matrices -- Trace of a matrix -- Orthogonal and unitary transformations -- Similarity transformation -- The matrix eigenvalue problem -- Determination of eigenvalues and eigenvectors -- Eigenvalues and eigenvectors of hermitian matrices -- Diagonalization of a matrix -- Eigenvectors of commuting matrices -- Cayley-Hamilton theorem -- Moment of inertia matrix -- Normal modes of vibrations -- Direct product of matrices -- Problems -- 4 Fourier series and integrals -- Periodic functions -- Fourier series -- Euler-Fourier formulas -- Gibb's phenomena -- Convergence of Fourier series and Dirichlet conditions -- Half-range Fourier series -- Change of interval -- Parseval's identity -- Alternative forms of Fourier series -- Integration and differentiation of a Fourier series -- Vibrating strings -- The equation of motion of transverse vibration -- Solution of the wave equation -- RLC circuit.

Orthogonal functions -- Multiple Fourier series -- Fourier integrals and Fourier transforms -- Fourier sine and cosine transforms -- Heisenberg's uncertainty principle -- Wave packets and group velocity -- Heat conduction -- Head conduction equation -- Fourier transforms for functions of several variables -- The Fourier integral and the delta function -- Parseval's identity for Fourier integrals -- The convolution theorem for Fourier transforms -- Calculations of Fourier transforms -- The delta function and the Green's function method -- Problems -- 5 Linear vector spaces -- Euclidean n-space E -- General linear vector spaces -- Subspaces -- Linear combination -- Linear independence, bases, and dimensionality -- Inner product spaces (unitary spaces) -- The Gram-Schmidt orthogonalization process -- The Cauchy-Schwarz inequality -- Dual vectors and dual spaces -- Linear operators -- Matrix representation of operators -- The algebra of linear operators -- Eigenvalues and eigenvectors of an operator -- Some special operators -- The inverse of an operator -- The adjoint operators -- Hermitian operators -- Unitary operators -- The projection operators -- Change of basis -- Commuting operators -- Function spaces -- Problems -- 6 Functions of a complex variable -- Complex numbers -- Basic operations with complex numbers -- Polar form of complex numbers -- De Moivre's theorem and roots of complex numbers -- Functions of a complex variable -- Mapping -- Branch lines and Riemann surfaces -- The differential calculus of functions of a complex variable -- Limits and continuity -- Derivatives and analytic functions -- The Cauchy-Riemann conditions -- Harmonic functions -- Singular points -- Elementary functions of z -- The exponential function…(or exp(z)) -- Trigonometric and hyperbolic functions -- The logarithmic function w = ln z -- Hyperbolic functions.

Complex integration -- Line integrals in the complex plane -- Cauchy's integral theorem -- Cauchy's integral formulas -- Cauchy's integral formula for higher derivatives -- Series representations of analytic functions -- Complex sequences -- Complex series -- Ratio test -- Uniform convergence and the Weierstrass M-test -- Power series and Taylor series -- Taylor series of elementary functions -- Laurent series -- Integration by the method of residues -- Residues -- The residue theorem -- Evaluation of real definite integrals -- Improper integrals of the rational function… -- Integrals of the rational functions of… -- Fourier integrals of the form… -- Other types of real improper integrals -- Problems -- 7 Special functions of mathematical physics -- Legendre's equation -- Rodrigues' formula for… -- The generating function for… -- Orthogonality of Legendre polynomials -- The associated Legendre functions -- Orthogonality of associated Legendre functions -- Hermite's equation -- Rodrigues' formula for Hermite polynomials… -- Recurrence relations for Hermite polynomials -- Generating function for the… -- The orthogonal Hermite functions -- Laguerre's equation -- The generating function for the Laguerre polynomials… -- Rodrigues' formula for the Laguerre polynomials… -- The orthogonal Laguerre functions -- The associated Laguerre polynomials… -- Generating function for the associated Laguerre polynomials -- Associated Laguerre function of integral order -- Bessel's equation -- Bessel functions of the second kind… -- Hanging flexible chain -- Generating function for… -- Bessel's integral representation -- Recurrence formulas for… -- Approximations to the Bessel functions -- Orthogonality of Bessel functions -- Spherical Bessel functions -- Sturm-Liouville systems -- Property 1 -- Property 2 -- Problems -- 8 The calculus of variations.

The Euler-Lagrange equation -- Variational problems with constraints -- Hamilton's principle and Lagrange's equation of motion -- Rayleigh-Ritz method -- Hamilton's principle and canonical equations of motion -- The modified Hamilton's principle and the Hamilton-Jacobi equation -- Variational problems with several independent variables -- Problems -- 9 The Laplace transformation -- Definition of the Lapace transform -- Existence of Laplace transforms -- Laplace transforms of some elementary functions -- Shifting (or translation) theorems -- The first shifting theorem -- The second shifting theorem -- The unit step function -- Laplace transform of a periodic function -- Laplace transforms of derivatives -- Laplace transforms of functions defined by integrals -- A note on integral transformations -- Problems -- 10 Partial differential equations -- Linear second-order partial differential equations -- Solutions of Laplace's equation: separation of variables -- Solutions of the wave equation: separation of variables -- Solution of Poisson's equation. Green's functions -- Laplace transform solutions of boundary-value problems -- Problems -- 11 Simple linear integral equations -- Classification of linear integral equations -- Some methods of solution -- Separable kernel -- Neumann series solutions -- Transformation of an integral equation into a differential equation -- Laplace transform solution -- Fourier transform solution -- The Schmidt-Hilbert method of solution -- Relation between differential and integral equations -- Use of integral equations -- Abel's integral equation -- Classical simple harmonic oscillator -- Quantum simple harmonic oscillator -- Problems -- 12 Elements of group theory -- Definition of a group (group axioms) -- Cyclic groups -- Group multiplication table -- Isomorphic groups -- Group of permutations and Cayley's theorem.

Subgroups and cosets.