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Good Introduction August 17, 2011
After trying several others, I found this textbook the best for individual study. The worked out solutions to all exercises gives you a good way to check your understanding. The writing is clear and the textbook has adequate illustrations to help you see what's going on. I also liked presentation in Banchoff's book, Differential Geometry of Curves and Surfaces, but it has no answers or solutions to the exercises.
Elementary Differential Geometry: stars based on 1 user reviews.
The Springer Undergraduate Mathematics Series (SUMS) is designed for undergraduates in the mathematical sciences. From core foundational material to final year topics, SUMS books take a fresh and modern approach and are ideal for self-study or for a one- or two-semester course. Each book includes numerous examples, problems and fully-worked solutions.
Table of Contents
|Curves in the plane and in space|
|What is a curve?||p. 1|
|Closed curves||p. 19|
|Level curves versus parametrized curves||p. 23|
|How much does a curve curve?|
|Plane curves||p. 34|
|Space curves||p. 46|
|Global properties of curves|
|Simple closed curves||p. 55|
|The isoperimetric inequality||p. 58|
|The four vertex theorem||p. 62|
|Surfaces in three dimensions|
|What is a surface?||p. 67|
|Smooth surfaces||p. 76|
|Smooth maps||p. 82|
|Tangents and derivatives||p. 85|
|Normals and orientability||p. 89|
|Examples of surfaces|
|Level surfaces||p. 95|
|Quadric surfaces||p. 97|
|Ruled surfaces and surfaces of revolution||p. 104|
|Compact surfaces||p. 109|
|Triply orthogonal systems||p. 111|
|Applications of the inverse function theorem||p. 116|
|The first fundamental form|
|Lengths of curves on surfaces||p. 121|
|Isometries of surfaces||p. 126|
|Conformal mappings of surfaces||p. 133|
|Equiareal maps mid a theorem of Archimedes||p. 139|
|Spherical geometry||p. 148|
|Curvature of surfaces|
|The second fundamental form||p. 159|
|The Gauss and Weingarten maps||p. 162|
|Normal and geodesic curvatures||p. 165|
|Parallel transport and covariant derivative||p. 170|
|Gaussian, mean and principal curvatures|
|Gaussian and mean curvatures||p. 179|
|Principal curvatures of a surface||p. 187|
|Surfaces of constant Gaussian curvature||p. 196|
|Flat surfaces||p. 201|
|Surfaces of constant mean curvature||p. 206|
|Gaussian curvature of compact surfaces||p. 212|
|Definition and basic properties||p. 215|
|Geodesic equations||p. 220|
|Geodesics on surfaces of revolution||p. 227|
|Geodesics as shortest paths||p. 235|
|Geodesic coordinates||p. 242|
|Gauss' Theorema Egregium|
|The Gauss and Codazzi-Mainardi equations||p. 247|
|Gauss' remarkable theorem||p. 252|
|Surfaces of constant Gaussian curvature||p. 257|
|Geodesic mappings||p. 263|
|Upper half-plane model||p. 270|
|Isometries of H||p. 277|
|Poincaré disc model||p. 283|
|Hyperbolic parallels||p. 290|
|Beltrami-Klein model||p. 295|
|Plateau's problem||p. 305|
|Examples of minimal surfaces||p. 312|
|Gauss map of a minimal surface||p. 320|
|Conformal parametrization of minimal surfaces||p. 322|
|Minimal surfaces and holomorphic functions||p. 325|
|The Gauss-Bonnet theorem|
|Gauss-Bonnet for simple closed curves||p. 335|
|Gauss-Bonnet for curvilinear polygons||p. 342|
|Integration on compact surfaces||p. 346|
|Gauss-Bonnet for compact surfaces||p. 349|
|Map colouring||p. 357|
|Holonomy and Gaussian curvature||p. 362|
|Singularities of vector fields||p. 365|
|Critical points||p. 372|
|Inner product spaces and self-adjoint linear maps|
|Isometries of Euclidean spaces|
|Hints to selected exercises|
|Table of Contents provided by Ingram. All Rights Reserved.|