Differential Geometry and Its ApplicationsDifferential geometry has a long, wonderful history and has found relevance in many areas. This book studies the differential geometry of surfaces with the goal of helping students make the transition from the standard university curriculum to a type of mathematics that is a unified whole, by mixing geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. Differential geometry is not just for mathematics majors, but also for students in engineering and the sciences. Into the mix of these ideas comes the opportunity to visualize concepts through the use of computer algebra systems such as Maple. The book emphasizes that this visualization goes hand-in-hand with the understanding of the mathematics behind the computer construction. The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract. |
Contents
Surfaces | 67 |
Curvatures | 107 |
Constant Mean Curvature Surfaces | 161 |
Geodesics Metrics and Isometries | 209 |
Holonomy and the GaussBonnet Theorem | 275 |
The Calculus of Variations and Geometry | 311 |
A Glimpse at Higher Dimensions | 397 |
A List of Examples | 437 |
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Common terms and phrases
algebra angle arclength balloon calculate catenoid circle compute constant coordinates cos(t cos(u cos(v cos² cosh covariant derivative cylinder defined definition denote diff differential equation differential geometry dot product elliptic Enneper's surface Euclidean Euler-Lagrange equation Example Exercise Figure fixed endpoint problem formula function Gauss curvature Gauss map geodesic equations given gives helix Hence Hint holonomy integral isometry k₁ linear Maple mean curvature metric minimal surface normal curvature Note obtain parallel parametrization patch x(u plot procedure Proof radius scaling-constrained shape operator Show simplify simply sin(t sin(u sin(v sinh sn(u solution solve sphere Suppose surface of revolution t₁ tangent plane tangent vector Theorem torus u₁ unduloid unit normal unit speed curve vector field zero ди дх