Applicable Differential Geometry

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Cambridge University Press, 1986 - Mathematics - 394 pages
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This is an introduction to geometrical topics that are useful in applied mathematics and theoretical physics, including manifolds, metrics, connections, Lie groups, spinors and bundles, preparing readers for the study of modern treatments of mechanics, gauge fields theories, relativity and gravitation. The order of presentation corresponds to that used for the relevant material in theoretical physics: the geometry of affine spaces, which is appropriate to special relativity theory, as well as to Newtonian mechanics, is developed in the first half of the book, and the geometry of manifolds, which is needed for general relativity and gauge field theory, in the second half. Analysis is included not for its own sake, but only where it illuminates geometrical ideas. The style is informal and clear yet rigorous; each chapter ends with a summary of important concepts and results. In addition there are over 650 exercises, making this a book which is valuable as a text for advanced undergraduate and postgraduate students.
 

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Contents

The background vector calculus
1
Affline spaces
8
Curves functions and derivatives
29
Vector fields and flows
53
Volumes and subspaces exterior algebra
85
Calculus of forms
117
Frobeniuss theorem
140
Metrics on affline spaces
164
9 Geometry of surfaces
216
10 Manifolds
236
11 Connections
268
12 Lie groups
298
13 The tangent and cotangent bundles
327
14 Fibre bundles
353
15 Connections revisited
371
Copyright

8 Isometries
188

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About the author (1986)

Crampin is a Professor of Mathematics at the Open University.

Pirani is Emeritus Professor of Rational Mechanics at the University of London.

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