Book chapter
Embedded Curves
Introduction to Tensor Analysis and the Calculus of Moving Surfaces, pp 215-233
10 Aug 2013
Abstract
In this chapter, we apply the methods of tensor calculus to embedded curves. In some ways curves are similar to surfaces and in some ways they are different. Of course, our focus is on the differences. When embedded in Euclidean spaces of dimension greater than two, curves are not hypersurfaces and therefore do not have a well-defined normal N and curvature tensor Bαβ. Furthermore, a number of interesting features of curves can be attributed to their one-dimensional nature. For example, curves are intrinsically Euclidean: they admit Cartesian coordinates (arc length) and their Riemann–Christoffel tensor vanishes. Other properties that stem from the curves’ one-dimensional nature are captured by the Frenet formulas.
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Details
- Title
- Embedded Curves
- Creators
- Pavel Grinfeld - Drexel University
- Publication Details
- Introduction to Tensor Analysis and the Calculus of Moving Surfaces, pp 215-233
- Publisher
- Springer New York; New York, NY
- Resource Type
- Book chapter
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991019312335004721