| |
| |
Review of Fundamental Concepts | |
| |
| |
Real numbers | |
| |
| |
Rules of algebra | |
| |
| |
Functions | |
| |
| |
Special types of functions | |
| |
| |
Limits | |
| |
| |
Continuity | |
| |
| |
Derivatives | |
| |
| |
Differentiation formulas | |
| |
| |
Integrals | |
| |
| |
Integration formulas | |
| |
| |
Sequences and series | |
| |
| |
Uniform convergence | |
| |
| |
Taylor series | |
| |
| |
Functions of two or more variables | |
| |
| |
Partial derivatives | |
| |
| |
Taylor series for functions of two or more variables | |
| |
| |
Linear equations and determinants | |
| |
| |
Maxima and minima | |
| |
| |
Method of Lagrange multipliers | |
| |
| |
Leibnitz's rule for differentiating an integral | |
| |
| |
Multiple integrals | |
| |
| |
Complex numbers | |
| |
| |
Ordinary Differential Equations | |
| |
| |
Definition of a differential equation | |
| |
| |
Order of a differential equation | |
| |
| |
Arbitrary constants | |
| |
| |
Solution of a differential equation | |
| |
| |
Differential equation of a family of curves | |
| |
| |
Special first order equations and solutions | |
| |
| |
Equations of higher order | |
| |
| |
Existence and uniqueness of solutions | |
| |
| |
Applications of differential equations | |
| |
| |
Some special applications | |
| |
| |
Mechanics | |
| |
| |
Electric circuits | |
| |
| |
Orthogonal trajectories | |
| |
| |
Deflection of beams | |
| |
| |
Miscellaneous problems | |
| |
| |
Numerical methods for solving differential equations | |
| |
| |
Linear Differential Equations | |
| |
| |
General linear differential equation of order n | |
| |
| |
Existence and uniqueness theorem | |
| |
| |
Operator notation | |
| |
| |
Linear operators | |
| |
| |
Fundamental theorem on linear differential equations | |
| |
| |
Linear dependence and Wronskians | |
| |
| |
Solutions of linear equations with constant coefficients | |
| |
| |
Non-operator techniques | |
| |
| |
The complementary or homogeneous solution | |
| |
| |
The particular solution | |
| |
| |
Method of undetermined coefficients | |
| |
| |
Method of variation of parameters | |
| |
| |
Operator techniques | |
| |
| |
Method of reduction of order | |
| |
| |
Method of inverse operators | |
| |
| |
Linear equations with variable coefficients | |
| |
| |
Simultaneous differential equations | |
| |
| |
Applications | |
| |
| |
Laplace Transforms | |
| |
| |
Definition of a Laplace transform | |
| |
| |
Laplace transforms of some elementary functions | |
| |
| |
Sufficient conditions for existence of Laplace transforms | |
| |
| |
Inverse Laplace transforms | |
| |
| |
Laplace transforms of derivatives | |
| |
| |
The unit step function | |
| |
| |
Some special theorems on Laplace transforms | |
| |
| |
Partial fractions | |
| |
| |
Solutions of differential equations by Laplace transforms | |
| |
| |
Applications to physical problems | |
| |
| |
Laplace inversion formulas | |
| |
| |
Vector Analysis | |
| |
| |
Vectors and scalars | |
| |
| |
Vector algebra | |
| |
| |
Laws of vector algebra | |
| |
| |
Unit vectors | |
| |
| |
Rectangular unit vectors | |
| |
| |
Components of a vector | |
| |
| |
Dot or scalar product | |
| |
| |
Cross or vector product | |
| |
| |
Triple products | |
| |
| |
Vector functions | |
| |
| |
Limits, continuity and derivatives of vector functions | |
| |
| |
Geometric interpretation of a vector derivative | |
| |
| |
Gradient, divergence and curl | |
| |
| |
Formulas involving [down triangle, open] | |
| |
| |
Orthogonal curvilinear coordinates | |
| |
| |
Jacobians | |
| |
| |
Gradient, divergence, curl and Laplacian in orthogonal curvilinear | |
| |
| |
Special curvilinear coordinates | |
| |
| |
Multiple, Line and Surface Integrals and Integral Theorems | |
| |
| |
Double integrals | |
| |
| |
Iterated integrals | |
| |
| |
Triple integrals | |
| |
| |
Transformations of multiple integrals | |
| |
| |
Line integrals | |
| |
| |
Vector notation for line integrals | |
| |
| |
Evaluation of line integrals | |
| |
| |
Properties of line integrals | |
| |
| |
Simple closed curves | |
| |
| |
Simply and multiply-connected regions | |
| |
| |
Green's theorem in the plane | |
| |
| |
Conditions for a line integral to be independent of the path | |
| |
| |
Surface integrals | |
| |
| |
The divergence theorem | |
| |
| |
Stokes' theorem | |
| |
| |
Fourier Series | |
| |
| |
Periodic functions | |
| |
| |
Fourier series | |
| |
| |
Dirichlet conditions | |
| |
| |
Odd and even functions | |
| |
| |
Half range Fourier sine or cosine series | |
| |
| |
Parseval's identity | |
| |
| |
Differentiation and integration of Fourier series | |
| |
| |
Complex notation for Fourier series | |
| |
| |
Complex notation for Fourier series | |
| |
| |
Orthogonal functions | |
| |
| |
Fourier Integrals | |
| |
| |
The Fourier integral | |
| |
| |
Equivalent forms of Fourier's integral theorem | |
| |
| |
Fourier transforms | |
| |
| |
Parseval's identities for Fourier integrals | |
| |
| |
The convolution theorem | |
| |
| |
Gamma, Beta and Other Special Functions | |
| |
| |
The gamma function | |
| |
| |
Table of values and graph of the gamma function | |
| |
| |
Asymptotic formula for [Gamma](n) | |
| |
| |
Miscellaneous results involving the gamma function | |
| |
| |
The beta function | |
| |
| |
Dirichlet integrals | |
| |
| |
Other special functions | |
| |
| |
Error function | |
| |
| |
Exponential integral | |
| |
| |
Sine integral | |
| |
| |
Cosine integral | |
| |
| |
Fresnel sine integral | |
| |
| |
Fresnel cosine integral | |
| |
| |
Asymptotic series or expansions | |
| |
| |
Bessel Functions | |
| |
| |
Bessel's differential equation | |
| |
| |
Bessel functions of the first kind | |
| |
| |
Bessel functions of the second kind | |
| |
| |
Generating function for J[subscript n](x) | |
| |
| |
Recurrence formulas | |
| |
| |
Functions related to Bessel functions | |
| |
| |
Hankel functions of first and second kinds | |
| |
| |
Modified Bessel functions | |
| |
| |
Ber, bei, ker, kei functions | |
| |
| |
Equations transformed into Bessel's equation | |
| |
| |
Asymptotic formulas for Bessel functions | |
| |
| |
Zeros of Bessel functions | |
| |
| |
Orthogonality of Bessel functions | |
| |
| |
Series of Bessel functions | |
| |
| |
Legendre Functions and Other Orthogonal Functions | |
| |
| |
Legendre's differential equation | |
| |
| |
Legendre polynomials | |
| |
| |
Generating function for Legendre polynomials | |
| |
| |
Recurrence formulas | |
| |
| |
Legendre functions of the second kind | |
| |
| |
Orthogonality of Legendre polynomials | |
| |
| |
Series of Legendre polynomials | |
| |
| |
Associated Legendre functions | |
| |
| |
Other special functions | |
| |
| |
Hermite polynomials | |
| |
| |
Laguerre polynomials | |
| |
| |
Sturm-Liouville systems | |
| |
| |
Partial Differential Equations | |
| |
| |
Some definitions involving partial differential equations | |
| |
| |
Linear partial differential equations | |
| |
| |
Some important partial differential equations | |
| |
| |
Heat conduction equation | |
| |
| |
Vibrating string equation | |
| |
| |
Laplace's equation | |
| |
| |
Longitudinal vibrations of a beam | |
| |
| |
Transverse vibrations of a beam | |
| |
| |
Methods of solving boundary-value problems | |
| |
| |
General solutions | |
| |
| |
Separation of variables | |
| |
| |
Laplace transform methods | |
| |
| |
Complex Variables and Conformal Mapping | |
| |
| |
Functions | |
| |
| |
Limits and continuity | |
| |
| |
Derivatives | |
| |
| |
Cauchy-Riemann equations | |
| |
| |
Integrals | |
| |
| |
Cauchy's theorem | |
| |
| |
Cauchy's integral formulas | |
| |
| |
Taylor's series | |
| |
| |
Singular points | |
| |
| |
Poles | |
| |
| |
Laurent's series | |
| |
| |
Residues | |
| |
| |
Residue theorem | |
| |
| |
Evaluation of definite integrals | |
| |
| |
Conformal mapping | |
| |
| |
Riemann's mapping theorem | |
| |
| |
Some general transformations | |
| |
| |
Mapping of a half plane on to a circle | |
| |
| |
The Schwarz-Christoffel transformation | |
| |
| |
Solutions of Laplace's equation by conformal mapping | |
| |
| |
Complex Inversion Formula for Laplace Transforms | |
| |
| |
The complex inversion formula | |
| |
| |
The Bromwich contour | |
| |
| |
Use of residue theorem in finding inverse Laplace transforms | |
| |
| |
A sufficient condition for the integral around [Gamma] to approach zero | |
| |
| |
Modification of Bromwich contour in case of branch points | |
| |
| |
Case of infinitely many singularities | |
| |
| |
Applications to boundary-value problems | |
| |
| |
Matrices | |
| |
| |
Definition of a matrix | |
| |
| |
Some special definitions and operations involving matrices | |
| |
| |
Determinants | |
| |
| |
Theorems on determinants | |
| |
| |
Inverse of a matrix | |
| |
| |
Orthogonal and unitary matrices | |
| |
| |
Orthogonal vectors | |
| |
| |
Systems of linear equations | |
| |
| |
Systems of n equations in n unknowns | |
| |
| |
Cramer's rule | |
| |
| |
Eigenvalues and eigenvectors | |
| |
| |
Theorems on eigenvalues and eigenvectors | |
| |
| |
Calculus of Variations | |
| |
| |
Maximum or minimum of an integral | |
| |
| |
Euler's equation | |
| |
| |
Constraints | |
| |
| |
The variational notation | |
| |
| |
Generalizations | |
| |
| |
Hamilton's principle | |
| |
| |
Lagrange's equations | |
| |
| |
Sturm-Liouville systems and Rayleigh-Ritz methods | |
| |
| |
Operator interpretation of matrices | |
| |
| |
Index | |