The Covariant Surface Derivative

Pavel Grinfeld
Additional contact information
Pavel Grinfeld: Drexel University, Department of Mathematics

Chapter Chapter 11 in Introduction to Tensor Analysis and the Calculus of Moving Surfaces, 2013, pp 185-197 from Springer

Abstract: Abstract In the preceding chapter, we gave an overview of embedded surfaces. We now turn to the important question of covariant differentiation on the surface. We will divide the construction of the covariant derivative into two parts. We will first define this operator for objects with surface indices. The definition will be completely analogous to that of the covariant derivative in the ambient space. While the definition will be identical, some of the important characteristics of the surface covariant derivative will be quite different. In particular, surface covariant derivatives do not commute. Our proof of commutativity for the ambient derivative was based on the existence of affine coordinates in Euclidean spaces. Since affine coordinates may not be possible on a curved surface, that argument is no longer available. We will also discover that the surface covariant derivative is not metrinilic with respect to the covariant basis S α. This will prove fundamental and will give rise to the curvature tensor, which will be further developed in Chap. 12

Keywords: Surface Covariant Derivative; Embedded Surfaces; Affine Coordinates; Basic Coverage; Curvature Tensor (search for similar items in EconPapers)
Date: 2013
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4614-7867-6_11

Ordering information: This item can be ordered from
http://www.springer.com/9781461478676

DOI: 10.1007/978-1-4614-7867-6_11

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().