Schaum's Outline of Advanced Mathematics for Engineers & Scientists

Schaum's Outline of Advanced Mathematics for Engineers & Scientists

by Murray R. Spiegel


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Product Details

ISBN-13: 9780070602168
Publisher: McGraw-Hill Companies, The
Publication date: 06/01/1971
Series: Schaum's Outline Series
Pages: 416
Product dimensions: 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 1Review of Fundamental Concepts1
Real numbers
Rules of algebra
Special types of functions
Differentiation formulas
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 2Ordinary Differential Equations38
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
Electric circuits
Orthogonal trajectories
Deflection of beams
Miscellaneous problems
Numerical methods for solving differential equations
Chapter 3Linear Differential Equations71
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
Chapter 4Laplace Transforms98
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 5Vector Analysis121
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
Gradient, divergence, curl and Laplacian in orthogonal curvilinear
Special curvilinear coordinates
Chapter 6Multiple, Line and Surface Integrals and Integral Theorems147
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 7Fourier Series182
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 8Fourier Integrals201
The Fourier integral
Equivalent forms of Fourier's integral theorem
Fourier transforms
Parseval's identities for Fourier integrals
The convolution theorem
Chapter 9Gamma, Beta and Other Special Functions210
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 10Bessel Functions224
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 11Legendre Functions and Other Orthogonal Functions242
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 12Partial Differential Equations258
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 13Complex Variables and Conformal Mapping286
Limits and continuity
Cauchy-Riemann equations
Cauchy's theorem
Cauchy's integral formulas
Taylor's series
Singular points
Laurent's series
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 14Complex Inversion Formula for Laplace Transforms324
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 15Matrices342
Definition of a matrix
Some special definitions and operations involving matrices
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 16Calculus of Variations375
Maximum or minimum of an integral
Euler's equation
The variational notation
Hamilton's principle
Lagrange's equations
Sturm-Liouville systems and Rayleigh-Ritz methods
Operator interpretation of matrices

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