Undergraduate Texts in Mathematics

By: John A. Thorpe

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Hardcover | 27 October 1994

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276 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
Vector Fields
The Tangent Space
Surfaces
Vector Fields on Surfaces
Orientation
The Gauss Map
Geodesics
Parallel Transport
The Weingarten Map
Curvature of Plane Curves
Arc Length and Line Integrals
Curvature of Surfaces
Convex Surfaces
Parametrized Surfaces
Local Equivalence of Surfaces and Parametrized Surfaces
Focal Points
Surface Area and Volume
Minimal Surfaces
The Exponential Map
Surfaces with Boundary
The Gauss-Bonnet Theorem
Rigid Motions and Congruence
Isometries
Riemannian Metrics
Table of Contents provided by Publisher. All Rights Reserved.

Undergraduate Texts in Mathematics

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