Mathematical Methods For Mechanical Sciences

by Michael S Howe

Overview

A mathematical model of a physical system provides the engineer with the insight and intuitive understanding required to make efficient system design changes or other modifications. In this context, a simple formula is often worth a thousand numerical simulations, and connections between different control parameters can be immediately revealed that might otherwise take hours or weeks to deduce from a computational analysis. This book supplies the undergraduate engineer with the basic mathematical tools for developing and understanding such models, and is also suitable as a review for engineering graduate students. A firm grasp of the topics covered will also enable the working engineer (educated to bachelor's degree level) to understand, write and otherwise make sensible use of technical reports and papers.

Product Details

ISBN-13:	9781783266647
Publisher:	Imperial College Press
Publication date:	09/22/2015
Pages:	332
Product dimensions:	6.80(w) x 9.80(h) x 0.90(d)

Table of Contents

Preface ix

1 Linear Ordinary Differential Equations 1

1.1 First-Order Equations 1

1.2 Second-Order Equations with Constant Coefficients 3

1.3 Euler's Homogeneous Equation 6

1.4 Method of Reduction of Order 7

1.5 Particular Integrals of Second-Order Equations 9

1.6 Method of Variation of Parameters 13

1.7 Method of Frobenius 15

1.8 Bessel Functions of Integer Order 25

1.9 The Sturm-Liouville Equation 27

1.10 Fourier Series 32

1.11 Generalised Functions and Green's Function 38

2 Vector Calculus 53

2.1 Elementary Operations with Vectors 53

2.2 Scalar and Vector Fields 56

2.3 The Divergence and the Divergence Theorem 59

2.4 Stokes' Theorem and Curl 64

2.5 Green's Identities 68

2.6 Orthogonal Curvilinear Coordinates 70

2.7 Evaluation of Line and Surface Integrals 74

2.8 Suffix Notation 79

3 Complex Variables 83

3.1 Complex Numbers 83

3.2 Functions of a Complex Variable 87

3.3 Integration in the Complex Plane 92

3.4 Cauchy's Theorem 97

3.5 Cauchy's Integral Formula 101

3.6 Taylor's Theorem 103

3.7 Laurent's Expansion 104

3.8 Poles and Essential Singularities 106

3.9 Cauchy's Residue Theorem 107

3.10 Applications of the Residue Theorem to Evaluate Real Integrals 111

3.11 Contour Integration Applied to the Summation of Series 121

3.12 Conformal Representation 123

3.13 Laplace's Equation in Two Dimensions 128

3.14 Applications to Hydrodynamics 131

4 Partial Differential Equations 139

4.1 Classification of Second-Order Equations 139

4.2 Boundary Conditions for Well-Posed Problems 144

4.3 Method of Separation of Variables 147

4.4 Problems with Cylindrical Boundaries 157

4.5 Application of Green's Second Identity: Green's Function 163

4.6 The Dirac Delta Function in Three Dimensions 166

4.7 The Method of Images 168

4.8 Green's Function for the Wave Equation 172

4.9 Fourier Transforms 179

4.10 Application of Fourier Transforms to the Solution of Partial Differential Equations 187

5 Special Functions 205

5.1 The Gamma Function Γ(x) 205

5.2 The Beta Function 210

5.3 Legendre Polynomials 213

5.4 The Error Function erf(x) 224

6 Matrix Algebra and Linear Equations 227

6.1 Definitions 227

6.2 Algebra of Matrices 228

6.3 Linear Equations 230

6.4 Further Discussion of Compatibility 237

6.5 Determinants 239

6.6 Inverse of a Square Matrix 242

6.7 Cramer's Rule 246

6.8 Eigenvalue Problems 253

6.9 Real Symmetric Matrices 258

6.10 The Cayley-Hamilton Equation 267

7 Variational Calculus 271

7.1 Taylor's Theorem for Several Variables 271

7.2 Maxima and Minima 274

7.3 Constrained Maxima and Minima: Lagrange Multipliers 280

7.4 Stationary Definite Integrals 287

7.5 Isoperimetric Problems 294

Useful Formulae 299

Bibliography 307

Index 311