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Question from someone just starting to study tensors (sorry if it's silly):

So I understand (maybe?) that tensors are basically about coordinate transformations (and things that are invariant under said transformations), and that in writing out a tensor, we use notation that represents functions that perform the coordinate transformations.

But when I see a tensor of, say, rank 3 or 4 written out, it looks like we're jumping from one coordinate system to another to another, before we arrive from the original coordinate system to the final one. If you're only really starting at one coordinate system, and ending at another, why can't you treat each tensor as just a single coordinate transformation, i.e. a function that takes you directly from the original coordinate system to the final one?

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    I'm not sure what you mean. You don't need to think about tensors in terms of changing coordinates at all.2012-08-13
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    One place where higher order tensors occur in analysis is Stokes' theorem. Generally, high rank tensors are just general multilinear maps, so they are quite natural...2012-08-13
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    (just expanding on Qiaochu Yuan's comment) $A = A_mdx^m$ or $A = \bar{A}_j d\bar{x}^j$ the one-form $A$ could either be written in the $(x^{\mu})$ or in the $({\bar{x}}^{j})$ coordinate charts. The object $A$ is itself coordinate-free. The coordinate transformations of the differentials then force particular transformations on the components of $A$ in barred or unbarred coordinate charts. It's not that $A$ is a coordinate change, rather, it's components change. $A$ in contrast is what it is no matter how you observe it.2012-08-13
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    In representation theory, new representations can be created from old ones by forming tensor powers of the original representation, and often such powers higher than the second are important.2012-08-13

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