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