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Edition: 79

Copyright: 1979

Publisher: Springer-Verlag New York

Published: 1979

International: No

Copyright: 1979

Publisher: Springer-Verlag New York

Published: 1979

International: No

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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.

**Thorpe, J. A. : City University of New York Queens College **

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.

Summary

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.

Author Bio

**Thorpe, J. A. : City University of New York Queens College **

Table of 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.

Publisher Info

Publisher: Springer-Verlag New York

Published: 1979

International: No

Published: 1979

International: No

**Thorpe, J. A. : City University of New York Queens College **

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.