I’ve written before about how the word “tensor” is applied to several different things, and that the connection between them isn’t immediately obvious. I thought about writing more about that some day, but I recently became aware of a paper that does this more thoroughly than I ever could.
The paper looks at three notions of a tensor
- a multi-indexed object that satisfies certain transformation rules
- a multilinear map
- an element of a tensor product of vector spaces
and explores (for 210 pages!) how they are related. The length comes primarily from the number of examples. The author is pulling on a thread that runs throughout a lot of mathematics.
Here’s the paper:
Lek-Heng Lim. Tensors in computations. Acta Numerica (2021), pp. 1–210. doi:10.1017/S0962492921000076. Available here.