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Tensors
So I'm attempting to write a basic (ie 1st year undergraduate level) introduction to tensors at the moment, where the idea is that I avoid doing the "a tensor is a set of quantities which satisfies the following transformation rule" thing, because that confused the hell out of me when I learnt tensors. I'm going for the "a tensor is a multilinear map" route instead.
I thought that I could do it all without going through dual spaces (just focusing on Cartesian tensors) but I've now hit a brick wall - I simply can't find a good way of describing contraction from the linear maps viewpoint. I'd rather not talk about covectors and dual spaces if I can avoid it, but I'd rather do that than have to resort to describing contraction in terms of indices.
Any ideas?
I thought that I could do it all without going through dual spaces (just focusing on Cartesian tensors) but I've now hit a brick wall - I simply can't find a good way of describing contraction from the linear maps viewpoint. I'd rather not talk about covectors and dual spaces if I can avoid it, but I'd rather do that than have to resort to describing contraction in terms of indices.
Any ideas?
For cartesian tensors, they're really just generalised multidimensional matrices. I really mean multi-rank matrices but multidimensional seems more obvious when you think of matrices as 2d. Quickly, you need to introduce the concept of rank and keep dimension for it's usual meaning otherwise things are going to get too confusing.
Contraction, when done on a single tensor, is just the trace of the matrix. When the rank is 3 or more, you're doing a trace operation for each 2d "slice" of the tensor, so you get a tensor of rank two less than before (trace of a matrix is just a number or rank 0 tensor).
Cross product gives you a big tensor with every possible product in it, often not terribly useful, but contraction across an index from each component tensor in the product gives you a generalisation of matrix multiplication.
I've often wondered about the whole column vector = vector and row vector = covector. The dot product then works, and you can disallow transforming one to the other by straight transposition but forcing the use of the metric. I think this breaks down pretty quickly though as when you write down matrices as being one up and one down index tensors, you can't really have a consistent notation for writing down tensors of rank 2 with all up or all down indices.
At this point index notation, and repeated index summation convention become very necessary but I think that contraction across outer products being like matrix multiplication is a useful understanding point.
Contraction, when done on a single tensor, is just the trace of the matrix. When the rank is 3 or more, you're doing a trace operation for each 2d "slice" of the tensor, so you get a tensor of rank two less than before (trace of a matrix is just a number or rank 0 tensor).
Cross product gives you a big tensor with every possible product in it, often not terribly useful, but contraction across an index from each component tensor in the product gives you a generalisation of matrix multiplication.
I've often wondered about the whole column vector = vector and row vector = covector. The dot product then works, and you can disallow transforming one to the other by straight transposition but forcing the use of the metric. I think this breaks down pretty quickly though as when you write down matrices as being one up and one down index tensors, you can't really have a consistent notation for writing down tensors of rank 2 with all up or all down indices.
At this point index notation, and repeated index summation convention become very necessary but I think that contraction across outer products being like matrix multiplication is a useful understanding point.
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