Differential Geometry of Curves and Surfaces, 2nd Edition

APPENDIX A

Tensor Notation

A.1 Tensor Notation

In Chapter 6, we gave to the first fundamental form the alternate name of metric tensor and delayed the explanation of what a tensor is. Mathematicians and physicists often present tensors and the tensor product in very different ways, sometimes making it difficult for a reader to see that authors in different fields are talking about the same thing. In this section, we introduce tensor notation in what one might call the “physics style,” which emphasizes how components of objects change under a coordinate transformation. Readers of mathematics who are well aquainted with tensor algebras on vector spaces might find this approach unsatisfactory, but physicists should recognize it. (The reader ...