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- 430 pages
- English
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Differential Geometry of Curves and Surfaces
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About This Book
Differential Geometry of Curves and Surfaces, Second Edition takes both an analytical/theoretical approach and a visual/intuitive approach to the local and global properties of curves and surfaces. Requiring only multivariable calculus and linear algebra, it develops students' geometric intuition through interactive computer graphics applets suppor
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Yes, you can access Differential Geometry of Curves and Surfaces by Thomas F. Banchoff, Stephen T. Lovett in PDF and/or ePUB format, as well as other popular books in Matematica & Matematica generale. We have over one million books available in our catalogue for you to explore.
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CHAPTER 1
Plane Curves: Local Properties
Just as calculus courses introduce real functions of one variable before tackling multivariable calculus, so it is natural to study curves before addressing surfaces and higher-dimensional objects. This first chapter presents local properties of plane curves, where by local property we mean properties that are defined in a neighborhood of a point on the curve. For the sake of comparison with calculus, the derivative f′(a) of a function f at a point a is a local property of the function since we only need knowledge of f(x) for x in (a − ε, a + ε), where ε is any positive real number, to define f′(a). In contrast, the definite integral of a function over an interval is a global property since we need knowledge of the function over the whole interval to calculate the integral. In contrast to this present chapter, Chapter 2 introduces global properties of plane curves.
1.1 Parametrizations
Borrowing from a physical understanding of motion in the plane, we can think about plane curves by specifying the coordinates x and y as functions of a time variable t, which give the position of a point traveling along the curve. Thus we need two functions x(t) and y(t). Using vector notation to locate a point on the curve, we often write for this pair of coordinate functions and call a vector function into ℝ2. From a mathematical standpoint, t does not have to refer to time and is simply called the parameter of the ...
Table of contents
- Cover
- Half Title
- Title Page
- Copyright Page
- Table of Contents
- Preface
- Acknowledgements
- 1 Plane Curves: Local Properties
- 2 Plane Curves: Global Properties
- 3 Curves in Space: Local Properties
- 4 Curves in Space: Global Properties
- 5 Regular Surfaces
- 6 The First and Second Fundamental Forms
- 7 The Fundamental Equations of Surfaces
- 8 The Gauss-Bonnet Theorem and Geometry of Geodesics
- 9 Curves and Surfaces in n-dimensional Euclidean Space
- A Tensor Notation
- Bibliography
- Index