
Differential Geometry and Its Applications, 1/e
John Oprea, Cleveland State University Published October, 1996 by Prentice Hall Engineering/Science/Mathematics
 
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Provides brief reviews of relevant ideas from linear algebra and complex variables. Guides readers through over 80 worked examples which provide students with fundamental models and illustrations of important concepts. Offers over 400 exercises which are integrated into the text so that their context is immediately clear.
1. The Geometry of Curves. Introduction. Arclength Parametrization. Frenet Formulas. Nonunit Speed Curves. Some Implications of Curvature and Torsion. The Geometry of Curves and MAPLE. 2. Surfaces. Introduction. The Geometry of Surfaces. The Linear Algebra of Surfaces. Normal Curvature. Plotting Surfaces in MAPLE. 3. 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. 4. Constant Mean Curvature Surfaces. Introduction. First Notions in Minimal Surfaces. Area Minimization. Constant Mean Curvature. Harmonic Functions. 5. Geodesics, Metrics and Isometries. Introduction. The Geodesic Equations and the Clairaut Relation. A Brief Digression on Completeness. Surfaces not in R 6. Holonomy and the GaussBonnet Theorem. Introduction. The Covariant Derivative Revisited. Parallel Vector Fields and Holonomy. Foucault's Pendulum. The Angle Excess Theorem. The GaussBonnet Theorem. Geodesic Polar Coordinates. 7. Minimal Surfaces and Complex Variables. Complex Variables. Isothermal Coordinates. The WeierstrassEnneper Representations. BjÖrling's Problem. Minimal Surfaces which are not Area Minimizing. Minimal Surfaces and MAPLE. 8. The Calculus of Variations and Geometry. The EulerLagrange Equations. The Basic Examples. The Weierstrass EFunction. Problems with Constraints. Further Applications to Geometry and Mechanics. The Pontryagin Maximum Principle. The Calculus of Variations and MAPLE. 9. 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.
