Elementary topics in differential geometry

John A. Thorpe

This introductory text develops the geometry of n-dimensional oriented surfaces in Rn+1. By viewing such surfaces as level sets of smooth functions, the author is able to introduce global ideas early without the need for preliminary chapters developing sophisticated machinery. the calculus of vector fields is used as the primary tool in developing the theory. Coordinate patches are introduced only after preliminary discussions of geodesics, parallel transport, curvature, and convexity. Differential forms are introduced only as needed for use in integration. The text, which draws significantly on students' prior knowledge of linear algebra, multivariate calculus, and differential equations, is designed for a one-semester course at the junior/senior level.

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  • I Graphs and Level Sets.- 2 Vector Fields.- 3 The Tangent Space.- 4 Surfaces.- 5 Vector Fields on Surfaces
  • Orientation.- 6 The Gauss Map.- 7 Geodesics.- 8 Parallel Transport.- 9 The Weingarten Map.- 10 Curvature of Plane Curves.- 11 Arc Length and Line Integrals.- 12 Curvature of Surfaces.- 13 Convex Surfaces.- 14 Parametrized Surfaces.- 15 Local Equivalence of Surfaces and Parametrized Surfaces.- 16 Focal Points.- 17 Surface Area and Volume.- 18 Minimal Surfaces.- 19 The Exponential Map.- 20 Surfaces with Boundary.- 21 The Gauss-Bonnet Theorem.- 22 Rigid Motions and Congruence.- 23 Isometries.- 24 Riemannian Metrics.- Notational Index.

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書名 Elementary topics in differential geometry
著作者等 Thorpe, John A
Thorpe John A.
シリーズ名 Undergraduate texts in mathematics
出版元 Springer-Verlag
刊行年月 c1979
版表示 1st ed. 1979. Corr. 4th printing 1994
ページ数 xiii, 253 p.
大きさ 24 cm
ISBN 3540903577
NCID BA01305313
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言語 英語
出版国 アメリカ合衆国