Introduction to Differential Geometry for Engineers

by Brian F. Doolin, Clyde F. Martin

Overview

This outstanding guide supplies important mathematical tools for diverse engineering applications, offering engineers the basic concepts and terminology of modern global differential geometry. Suitable for independent study as well as a supplementary text for advanced undergraduate and graduate courses, this volume also constitutes a valuable reference for control, systems, aeronautical, electrical, and mechanical engineers.
The treatment's ideas are applied mainly as an introduction to the Lie theory of differential equations and to examine the role of Grassmannians in control systems analysis. Additional topics include the fundamental notions of manifolds, tangent spaces, vector fields, exterior algebra, and Lie algebras. An appendix reviews concepts related to vector calculus, including open and closed sets, compactness, continuity, and derivative.

Product Details

ISBN-13:	9780486488165
Publisher:	Dover Publications
Publication date:	07/17/2012
Series:	Dover Civil and Mechanical Engineering Series
Pages:	176
Product dimensions:	5.30(w) x 8.40(h) x 0.60(d)

About the Author

Brian F. Doolin (1925–2009) was an engineer with NASA as well as Lockheed. At NASA he was a leader of the Control Theory and Applications Group.
Clyde F. Martin is Paul Whitfield Horn Professor of Mathematics at Texas Tech University.

Table of Contents

Preface v

List of Figures xi

1 Introduction 1

2 Manifolds and their Maps 5

2.1 Differentiable Manifolds 5

2.2 Examples 13

2.3 Manifold Maps 18

3 Tangent Spaces 23

3.1 The Tangent Space of Sphere 24

3.2 Equivalence Classes of Curves 30

3.3 The Tangent Space in General 33

3.4 Tangent Space Maps 36

4 Vector Fields 45

4.1 Vector Fields 46

4.2 Derivations 49

4.3 A Digression on Notation 53

4.4 The Isomorphism 54

4.5 The Algebra 60

4.6 An Example of a Lie Algebra 63

5 Exterior Algebra 69

5.1 Addition of Forms 70

5.2 The Wedge Product 71

5.3 Contraction of Forms; Vectors 75

5.4 Equation of a Plane 79

5.5 Use of Determinants 81

5.6 Solution of Linear Equations 84

5.7 Linear Transformations 86

6 Lie Groups and Actions 91

6.1 Lie Groups 91

6.2 Group Action 93

6.3 One-Parameter Subgroups 95

6.4 The Symplectic Group 101

7 Homogeneous Spaces 107

8 Grassmannian Techniques 115

8.1 Linear Optimal Control 116

8.2 The Grassmannian 120

8.3 An Application 129

9 Concluding Remarks 141

10 Appendix: Vector Calculus 143

10.1 Real Euclidean Space 143

10.2 Topological Spaces 144

10.3 Compactness 147

10.4 Continuity 148

10.5 Derivative 149

10.6 Inverse Function Theorems 151

Bibliography 155

Index 159

Editorial Reviews

Treats "the basic concepts and terminology of modern global differential geometry" for the benefit of engineers--particularly control engineers. And unlike many books (written by authors who seem anxious to vanish as soon as possible over the nearest abstract horizon) which pretend to be "useful", this one shows every sign of having been actually and extensively field-tested. The authors adhere to the absolute essentials of their topic, and have been at pains to write with exquisite clarity. Index and brief bibliography, but no exercises. (NW) Annotation c. Book News, Inc., Portland, OR (booknews.com)

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