Page 6 - Basic Structured Grid Generation

Contents

Preface ix

1. Mathematical preliminaries – vector and tensor analysis 1
1.1 Introduction 1
1.2 Curvilinear co-ordinate systems and base vectors in E 3 1
1.3 Metric tensors 4
1.4 Line, area, and volume elements 8
1.5 Generalized vectors and tensors 8
1.6 Christoffel symbols and covariant differentiation 14
1.7 Div, grad, and curl 19
1.8 Summary of formulas in two dimensions 23
1.9 The Riemann-Christoffel tensor 26
1.10 Orthogonal curvilinear co-ordinates 27
1.11 Tangential and normal derivatives – an introduction 28

2. Classical differential geometry of space-curves 30
2.1 Vector approach 30
2.2 The Serret-Frenet equations 32
2.3 Generalized co-ordinate approach 35
2.4 Metric tensor of a space-curve 38
3. Differential geometry of surfaces in E 3 42
3.1 Equations of surfaces 42
3.2 Intrinsic geometry of surfaces 46
3.3 Surface covariant differentiation 51
3.4 Geodesic curves 54
3.5 Surface Frenet equations and geodesic curvature 57
3.6 The second fundamental form 60
3.7 Principal curvatures and lines of curvature 63
3.8 Weingarten, Gauss, and Gauss-Codazzi equations 67
3.9 Div, grad, and the Beltrami operator on surfaces 70