Differential Geometry and Its Applications, 1/e
John Oprea, Cleveland State University
Published October, 1996 by Prentice Hall Engineering/Science/Mathematics
Copyright 1997, 387 pp.
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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.
Organizes coverage in a spiral fashion so that the same
topics (e.g. geodesics, least area surfaces of revolution, surfaces
of Delaunay) are encountered again and again from different viewpoints,
allowing students to see a unity in mathematics and its applications.
Discusses concepts with a style that is honestly mathematical,
but still easily understandable to science majors.
Offers an interdisciplinary approach to geometry, depicting
the interconnections among various kinds of mathematics and sciences.
Illustrates concepts using the symbolic computational software
Devotes an entire chapter to an exposition of the calculus
of variations from first principles.
- How geodesics may be plotted on surfaces using MAPLE
and an illustration of the sometimes confusing Clairaut relation.
- How particles move under the influence of gravity, but
constrained to a surface.
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.
- Exercises range from the easy to the very difficult.
- Many are computational, while others require proofs or
characterizations of geometric phenomena.
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.
Introduction. The Geometry of Surfaces. The Linear Algebra of Surfaces. Normal Curvature. Plotting Surfaces in MAPLE.
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. BjÖrling'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.