 # Schaum's Outline of Advanced Mathematics for Engineers & Scientists

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## Overview

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## Product Details

ISBN-13: 9780070602168 McGraw-Hill Companies, The 06/01/1971 Schaum's Outline Series 416 8.10(w) x 10.80(h) x 0.64(d)

## About the Author

Murray Speigel, Ph.D., was Former Professor and Chairman of the Mathematics Department at Rensselaer Polytechnic Institute, Hartford Graduate Center.

## Table of Contents

 Chapter 1 Review of Fundamental Concepts 1 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 Chapter 2 Ordinary Differential Equations 38 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 Chapter 3 Linear Differential Equations 71 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 Chapter 4 Laplace Transforms 98 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 Chapter 5 Vector Analysis 121 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 Chapter 6 Multiple, Line and Surface Integrals and Integral Theorems 147 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 Chapter 7 Fourier Series 182 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 Chapter 8 Fourier Integrals 201 The Fourier integral Equivalent forms of Fourier's integral theorem Fourier transforms Parseval's identities for Fourier integrals The convolution theorem Chapter 9 Gamma, Beta and Other Special Functions 210 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 Chapter 10 Bessel Functions 224 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 Chapter 11 Legendre Functions and Other Orthogonal Functions 242 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 Chapter 12 Partial Differential Equations 258 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 Chapter 13 Complex Variables and Conformal Mapping 286 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 Chapter 14 Complex Inversion Formula for Laplace Transforms 324 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 Chapter 15 Matrices 342 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 Chapter 16 Calculus of Variations 375 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 399

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