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Differential Geometry and Its Applications

ISBN-10: 0133407381

ISBN-13: 9780133407389

Edition: 1st 1997

Authors: John Oprea

List price: $100.00
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Description:

Appropriate for undergraduate courses in Differential Geometry. Designed not just for the math major but for all students of science, this text provides an introduction to the basics of the calculus of variations and optimal control theory as well as differential geometry. It then applies these essential ideas to understand various phenomena, such as soap film formation and particle motion on surfaces.
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Book details

List price: $100.00
Edition: 1st
Copyright year: 1997
Publisher: Prentice Hall PTR
Publication date: 10/28/1996
Binding: Hardcover
Pages: 387
Size: 6.10" wide x 9.25" long x 0.71" tall
Weight: 1.342
Language: English

Preface
The Geometry of Curves
Introduction
Arclength Parametrization
Frenet Formulas
Nonunit Speed Curves
Some Implications of Curvature and Torsion
The Geometry of Curves and MAPLE
Surfaces
Introduction
The Geometry of Surfaces
The Linear Algebra of Surfaces
Normal Curvature
Plotting Surfaces in MAPLE
Curvature(s)
Introduction
Calculating Curvature
Surfaces of Revolution
A Formula for Gaussian Curvature
Some Effects of Curvature(s)
Surfaces of Delaunay
Calculating Curvature with MAPLE
Constant Mean Curvature Surfaces
Introduction
First Notions in Minimal Surfaces
Area Minimization
Constant Mean Curvature
Harmonic Functions
Geodesics, Metrics and Isometries
Introduction
The Geodesic Equations and the Clairaut Relation
A Brief Digression on Completeness
Surfaces not in R3
Isometries and Conformal Maps
Geodesics and MAPLE
Holonomy and the Gauss-Bonnet Theorem
Introduction
The Covariant Derivative Revisited
Parallel Vector Fields and Holonomy
Foucault's Pendulum
The Angle Excess Theorem
The Gauss-Bonnet Theorem
Geodesic Polar Coordinates
Minimal Surfaces and Complex Variables
Complex Variables
Isothermal Coordinates
The Weierstrass-Enneper Representations
Bjurling's Problem
Minimal Surfaces which are not Area Minimizing
Minimal Surfaces and MAPLE
The Calculus of Variations and Geometry
The Euler-Lagrange Equations
The Basic Examples
The Weierstrass E-Function
Problems with Constraints
Further Applications to Geometry and Mechanics
The Pontryagin Maximum Principle
The Calculus of Variations and MAPLE
A Glimpse at Higher Dimensions
Introduction
Manifolds
The Covariant Derivative
Christoffel Symbols
Curvatures
The Charming Doubleness
List of Examples, Definitions and Remarks
Answers and Hints to Selected Exercises
References
Index