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.
Auflage
Sprache
Verlagsort
Verlagsgruppe
Zielgruppe
Maße
Höhe: 236 mm
Breite: 158 mm
Dicke: 24 mm
Gewicht
ISBN-13
978-0-13-065246-1 (9780130652461)
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Schweitzer Klassifikation
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. Bjoerling'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.
