Hey Brilliant Users!
So I am fairly inexperienced in dealing with tensors (aside from the usual scalars, vectors and linear maps) and am trying to understand the nature of these "objects" a little better. My understanding is as follows, and I'm really not sure if I'm interpreting this correctly so please correct me if I'm wrong:
A 0 order tensor is a scalar. A 1st order tensor is a vector. A second order tensor, by my reasoning, must always be representable as a matrix when one chooses to find such a representation, since every element requires two indices to specify. Now, if we continue this pattern, I reason that a 3rd order tensor must be some kind of geometric object. I imagine this object as a "cube" in which each face is a matrix. This is purely speculation, as I have no proof to back that last claim up but it seems to follow.
As many of you have read in my previous post, I am currently reading Spivak's "Calc on Manifolds" and have recently begun chapter 4, in which I have been formally introduced to tensors for the first time in their full generality. I take back everything I said about the book not being hard, I'll be the first to admit I'm having quite a bit of trouble intuitively understanding what the author is establishing concerning tensors. Any feedback whatsoever is, as usual, greatly appreciated.
I also want to take a brief moment to thank the Brilliant.org community. I've actually learned more from the users on here in the relatively brief time I've been a member of the site than I have from any teacher in any formal class. Great stuff guys!