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Embedded Curves

In: Introduction to Tensor Analysis and the Calculus of Moving Surfaces

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  • Pavel Grinfeld

    (Drexel University, Department of Mathematics)

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 B αβ. 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.

Suggested Citation

  • Pavel Grinfeld, 2013. "Embedded Curves," Springer Books, in: Introduction to Tensor Analysis and the Calculus of Moving Surfaces, edition 127, chapter 0, pages 215-233, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-7867-6_13
    DOI: 10.1007/978-1-4614-7867-6_13
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