# Elementary Topics in Differential Geometry

Springer Science & Business Media, Oct 27, 1994 - Mathematics - 256 pages
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

### Popular passages

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