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191202  eng 
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a 9789811517396

100 
1 

a Kobayashi, Shoshichi

245 
0 
0 
a Differential Geometry of Curves and Surfaces
h Elektronische Ressource
c by Shoshichi Kobayashi

250 


a 1st ed. 2019

260 


a Singapore
b Springer Nature Singapore
c 2019, 2019

300 


a XII, 192 p. 1 illus
b online resource

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0 

a Plane Curves and Space Curves  Local Theory of Surfaces in the Space  Geometry of Surfaces  The GaussBonnet Theorem  Minimal Surfaces.

653 


a Geometry, Differential

653 


a Mathematical analysis

653 


a Analysis

653 


a Manifolds and Cell Complexes

653 


a Manifolds (Mathematics)

653 


a Differential Geometry

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0 
7 
a eng
2 ISO 6392

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b Springer
a Springer eBooks 2005

490 
0 

a Springer Undergraduate Mathematics Series

856 
4 
0 
u https://doi.org/10.1007/9789811517396?nosfx=y
x Verlag
3 Volltext

082 
0 

a 516.36

520 


a This book is a posthumous publication of a classic by Prof. Shoshichi Kobayashi, who taught at U.C. Berkeley for 50 years, recently translated by Eriko Shinozaki Nagumo and Makiko Sumi Tanaka. There are five chapters: 1. Plane Curves and Space Curves; 2. Local Theory of Surfaces in Space; 3. Geometry of Surfaces; 4. Gauss–Bonnet Theorem; and 5. Minimal Surfaces. Chapter 1 discusses local and global properties of planar curves and curves in space. Chapter 2 deals with local properties of surfaces in 3dimensional Euclidean space. Two types of curvatures — the Gaussian curvature K and the mean curvature H —are introduced. The method of the moving frames, a standard technique in differential geometry, is introduced in the context of a surface in 3dimensional Euclidean space. In Chapter 3, the Riemannian metric on a surface is introduced and properties determined only by the first fundamental form are discussed. The concept of a geodesic introduced in Chapter 2 is extensively discussed, and several examples of geodesics are presented with illustrations. Chapter 4 starts with a simple and elegant proof of Stokes’ theorem for a domain. Then the Gauss–Bonnet theorem, the major topic of this book, is discussed at great length. The theorem is a most beautiful and deep result in differential geometry. It yields a relation between the integral of the Gaussian curvature over a given oriented closed surface S and the topology of S in terms of its Euler number χ(S). Here again, many illustrations are provided to facilitate the reader’s understanding. Chapter 5, Minimal Surfaces, requires some elementary knowledge of complex analysis. However, the author retained the introductory nature of this book and focused on detailed explanations of the examples of minimal surfaces given in Chapter 2.
