Differential Geometry and Its Applications

Abstract

Preface. 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 R3. Isometries and Conformal Maps. Geodesics and MAPLE. 6. 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. 7. Minimal Surfaces and Complex Variables. Complex Variables. Isothermal Coordinates. The Weierstrass-Enneper Representations. BjA-rling's Problem. Minimal Surfaces which are not Area Minimizing. Minimal Surfaces and MAPLE. 8. 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. 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.