Introduction to Differential Geometry for Engineers

by Doolin, Brian F.; Martin, Clyde F.

• ISBN13: 9780486488165
• ISBN10: 0486488160
• Format: Paperback
• Copyright: 2012-07-17
• Publisher: Dover Publications

Summary

This book acquaints engineers with the basic concepts and terminology of modern global differential geometry. It introduces the Lie theory of differential equations and examines the role of Grassmannians in control systems analysis. Additional topics include the fundamental notions of manifolds, tangent spaces, and vector fields. 1990 edition.

Author Biography

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	p. v
List of Figures	p. xi
Introduction	p. 1
Manifolds and their Maps	p. 5
Differentiable Manifolds	p. 5
Examples	p. 13
Manifold Maps	p. 18
Tangent Spaces	p. 23
The Tangent Space of Sphere	p. 24
Equivalence Classes of Curves	p. 30
The Tangent Space in General	p. 33
Tangent Space Maps	p. 36
Vector Fields	p. 45
Vector Fields	p. 46
Derivations	p. 49
A Digression on Notation	p. 53
The Isomorphism	p. 54
The Algebra	p. 60
An Example of a Lie Algebra	p. 63
Exterior Algebra	p. 69
Addition of Forms	p. 70
The Wedge Product	p. 71
Contraction of Forms; Vectors	p. 75
Equation of a Plane	p. 79
Use of Determinants	p. 81
Solution of Linear Equations	p. 84
Linear Transformations	p. 86
Lie Groups and Actions	p. 91
Lie Groups	p. 91
Group Action	p. 93
One-Parameter Subgroups	p. 95
The Symplectic Group	p. 101
Homogeneous Spaces	p. 107
Grassmannian Techniques	p. 115
Linear Optimal Control	p. 116
The Grassmannian	p. 120
An Application	p. 129
Concluding Remarks	p. 141
Appendix: Vector Calculus	p. 143
Real Euclidean Space	p. 143
Topological Spaces	p. 144
Compactness	p. 147
Continuity	p. 148
Derivative	p. 149
Inverse Function Theorems	p. 151
Bibliography	p. 155
Index	p. 159
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