Book chapter
Covariant Differentiation
Introduction to Tensor Analysis and the Calculus of Moving Surfaces, pp 105-132
10 Aug 2013
Abstract
Chapter 6 demonstrated that the tensor property is the key to invariance. However, a partial derivative \documentclass[12pt]{minimal}
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$$\partial /\partial {Z}^{i}$$
\end{document} of a tensor is itself not a tensor. This is a major obstacle in the way of developing differential geometry using the coordinate approach. For example, the expression \documentclass[12pt]{minimal}
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$$\partial {T}^{i}/\partial {Z}^{i}$$
\end{document} cannot be used as a definition of divergence since it evaluates to different values in different coordinates. Similarly, \documentclass[12pt]{minimal}
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$${Z}^{ij}{\partial }^{2}T/\partial {Z}^{i}\partial {Z}^{j}$$
\end{document} is not invariant and is therefore not a legitimate definition of the Laplacian.
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Details
- Title
- Covariant Differentiation
- Creators
- Pavel Grinfeld
- Publication Details
- Introduction to Tensor Analysis and the Calculus of Moving Surfaces, pp 105-132
- Publisher
- Springer New York; New York, NY
- Resource Type
- Book chapter
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991019312432804721