Questions about Tensors and Contractions

Question

I have a few questions about tensors. I'm still new to this whole thing and I've mostly just been reading out of Needham's VDGF with some occasional videos and lecture notes.

1. I know that for a vector v and a 1-form w, that w(v) represents the contraction of w with v. Is there a similar thing for higher rank tensors? I.e. what does it mean to plug in 1-forms and vectors into a tensor? Does a similar thing happen with tensor contractions, if so, how does it it work?

Related: I know that computing a 1-form is finding its contraction with a vector, what about for higher rank, possibly mixed tensors?

2. Can you give simple examples of mixed tensors without using physics if possible please? I'm struggling to see how a 1-form is used in higher rank tensors

3. What does it really even mean if we plug in 1-forms and vectors into a tensor of rank 2 or higher? For example, for a given tensor, what happens if we plug in a 1-form w and a vector v? Are there any restrictions on what 1-forms or vectors we can plug in to a certain tensor aside from dimensionality?

4. I saw that in the definition for tensors it mentions that the tensor is on a point, how would it be used in a given tensor?

5. What is the purpose or motivation behind defining a tensor contraction?

6. Can you suggest any books that aren’t too heavy on formalisms that might help me? Preferably without too much knowledge past very introductory analysis.

If you can help me or give me some examples while using as little physics as possible that would be very much appreciated. I’ve never really seen any examples of rank 2 or higher tensors being computed, so some things that seem to be obvious might not be so obvious to me. Thank you :DDD

I’m not sure how to tag this so please excuse any mistakes

Answer 1

1. One way that helped me to think about these things is with the "co" terminology. That is, thinking of 1-forms as covectors. More precisely, vectors are elements of the tangent space, and 1-forms are elements of the cotangent space, which is the dual vector space to the tangent space. By definition, the dual $V^*$ of a vector space $V$ consists of linear functionals, i.e. linear maps $V\to \mathbb R$ (assuming that $V$ is a vector space over $\mathbb R$, which is usually the case in differential geometry). Tensors consist of several "normal" dimensions and several "dual" dimensions. This can be summarized by the type of the tensor. For instance, type $(0,1)$ is a 1-form, and type $(1,0)$ is a vector.

2. Continuing in this way, one can define a type $(p,q)$ tensor as a multi-linear map $(V^*)^p\times V^q\to \mathbb R$. There are other ways to define it as well.

3. Suppose I have a tensor $T:V^*\times V\to \mathbb R$. A 1-form is an element $\omega\in V^*$, so plugging $\omega$ into $T$ fills one of its slots: $T(\omega,\cdot)$ still has a spot left for an element in $V$. This is the idea of tensor contraction.

4. Yes, $V$ here is always the tangent space to your surface/manifold. Tangent spaces exist at every point, and you can have a tensor that changes from point to point.

5. I'm not sure how well I could motivate this except for the fact that 1-forms themselves are made for "contraction" with vectors. And it just tends to come up a lot.

6. Dirac's book on General Relativity is one way to go if you want the heavily computational look. I know you're not into physics, but the book is really just about math, especially in the beginning. It's small and nice to work with. As for the multi-linear approach to tensors, I'm not sure of a good reference.