What are the prerequisites to learn tensor calculus?

Answer 1

Linear algebra and mathematical maturity.

I think the most confusing thing for me when I was learning this subject was the number of seemingly different objects which people called a "tensor". There are a lot of related phenomena, so its really useful at the beginning to understand what a tensor is, and why its a useful object. I'll try to explain it as best i can. If my explanation makes sense, it may help you gauge if you are ready.

Suppose you live in a world where cartesian product (ordered pairs) dont exist. and you wanted to study multivariable functions. This would be a giant pain. every function comes with a different number of inputs, and composing them would be so complicated. But we do have ordered pairs, and so the theory becomes very simple: a multivariable function of n inputs is just a single variable function which takes an ordered pair of n arguments in as a single object. The cartesian product of sets reduces the study of multivariable functions to the study of single variable functions: functions from A and B to C correspond to functions from AxB to C.

Now, imagine you are coming out of your linear algebra course, and you start stumbling across "multilinear transformations". These are functions which take multiple vectors as input, and output a single vector. If you only vary one input at a time, it behaves linearly. For example, a multilinear function of 2 variables would satisfy T(u₁ + u₂, v) = T(u₁, v) + T(u₂, v), T(u, v₁ + v₂) = T(u, v₁) + T(u, v₂), and T(au, v) = aT(u,v) = T(u, av).

Once again, we have the same situation. Are you going to have to take a whole new class for multivariable linear algebra? Luckily, you dont! There is an analogue to taking the cartesian product of two sets, which we call the tensor product. You take two vector spaces, U and V, and you make a new vector space U⊗V. Multilinear functions from U and V to W correspond exactly to linear functions from U⊗V to W. Confusingly, we call the vectors in the space U⊗V "tensors".

So that's what they are for, but how do you build them? Well, the best way to think about it is to start with what you want, then build the biggest most general vector space you possibly can, then add in equations which "force" this to work. First, we make a vector space which i will call F(UxV). We take every pair of vectors (one from U and one from V), and give the pair its own basis vector in F(UxV). This is a huge space. It has a direction for (u, v), (3*u, 3*v), (pi*u, 100*v), and so on. Every pair of vectors gets its own direction.

Now, we smush this thing by "gluing" vectors together. In the same way that 1/2 and 2/4 are glued into the same rational number, we can identify certain vectors in F(UxV) together by choosing certain equations. Which equations do we use? Exactly the ones we needed for multilinearity. (u₁ + u₂, v) = (u₁, v) + (u₂, v), (u, v₁ + v₂) = (u, v₁) + (u, v₂), and (au, v) = a(u, v) = (u, av). We made the largest space we possibly could, then smushed it together by identifying only what we need for this setup to work. The pieces which get identified together are "tensors", in the tensor product space of U and V. we write (u, v) in this case as u⊗v. As one final piece of silly terminology, we also call u⊗v the "tensor product of vectors u and v".

There are lots of things to check with what i just said. You have to make sure vector addition plays nice with the equations i just said. You probably want to check the dimension of this space. Its the dimension of U times the dimension of V. There is also lots of business about covariance and contravariance, and you can realize these abstract tensors as something more concrete. But fundamentally thats why they are there. They smush the theory of multilinear things into the theory of linear things.