Elementary Topics in Differential Geometry

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Springer Science & Business Media, Oct 27, 1994 - Mathematics - 256 pages
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In the past decade there has been a significant change in the freshman/ sophomore mathematics curriculum as taught at many, if not most, of our colleges. This has been brought about by the introduction of linear algebra into the curriculum at the sophomore level. The advantages of using linear algebra both in the teaching of differential equations and in the teaching of multivariate calculus are by now widely recognized. Several textbooks adopting this point of view are now available and have been widely adopted. Students completing the sophomore year now have a fair preliminary under standing of spaces of many dimensions. It should be apparent that courses on the junior level should draw upon and reinforce the concepts and skills learned during the previous year. Unfortunately, in differential geometry at least, this is usually not the case. Textbooks directed to students at this level generally restrict attention to 2-dimensional surfaces in 3-space rather than to surfaces of arbitrary dimension. Although most of the recent books do use linear algebra, it is only the algebra of ~3. The student's preliminary understanding of higher dimensions is not cultivated.
 

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Contents

Graphs and Level Sets
1
Vector Fields
6
The Tangent Space
13
Surfaces
16
Vector Fields on Surfaces Orientation
23
The Gauss Map
31
Geodesics
38
Parallel Transport
45
Local Equivalence of Surfaces and Parametrized Surfaces
121
Focal Points
132
Surface Area and Volume
139
Minimal Surfaces
156
The Exponential Map
163
Surfaces with Boundary
177
The GaussBonnet Theorem
190
Rigid Motions and Congruence
210

The Weingarten Map
53
Curvature of Plane Curves
62
Arc Length and Line Integrals
68
Curvature of Surfaces
82
Convex Surfaces
95
Parametrized Surfaces
108
Isometries
220
Riemannian Metrics
231
Bibliography
245
Notational Index
247
Subject Index
249
Copyright

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Page 246 - Hurewicz, W. (1958). Lectures on Ordinary Differential Equations, Cambridge, Mass., MIT Press.

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