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Books > Science & Mathematics > Mathematics > Geometry > Analytic geometry
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Differential Geometry of Curves and Surfaces (Paperback, 1st ed. 2019)
Loot Price: R1,051
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Differential Geometry of Curves and Surfaces (Paperback, 1st ed. 2019)
Series: Springer Undergraduate Mathematics Series
Expected to ship within 12 - 17 working days
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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 3-dimensional
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 3-dimensional 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 (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.
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