I am currently learning about tensors as they come up in the mathematics behind continuum mechanics.
I was fairly disappointed with my initial foray into tensors, as presented in the book Classical and Computational Solid Mechanics, by Fung and Tong. That experience caused me to write up the following question: I feel that (physics) notation for tensor calculus is awful. Are there any alternative notations worth looking into?
As you can see, I got an interesting answer, recommending that I read through the following paper: Tensor Decompositions and Applications. However, that text is still beyond me at the moment, as it assumes a fair bit of background knowledge on tensors.
After doing some researching here, I found the following answers:
- Suggestion for good notation - abstract index notation
- Harmful heuristics one should avoid - "A tensor is a multidimensional array of numbers that transforms in the following way under a change of coordinates..."
The above heuristic is exactly what is presented in Fung and Tong's book. I would really like to understand tensors properly. I want to avoid harmful heuristics, and unnecessarily tedious notation. I would like to be able to understand for instance, the following sentence (touted as being the "real definition of tensors" according to one of the comments):
Let M be a smooth manifold; let FM be the space of smooth functions on M, let VM be the space of smooth vector fields on M, and let V*M be the space of smooth covector fields on M. A (k,l) tensor is a multilinear map from (VM)^k x (V*M)^l to FM.
What resources should I pick up in order to begin?