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stabu

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I'm trying to close in on a more intuitive way of understanding tensors. For some reason, they've always held an aura of mystique for me, may be also their similarity to the word "tense" has meant that I've never really warmed to the many defintions and explanations available. So, in many ways, I already have a mental block over them.

Recently, I (re-) watched Dan Fleisch's video at

and the tone is very much what I was looking for, but I now feel he spends too much time on vectors (which I have no problems with), and glosses over the rank 1 tensor (a vector) to rank 2 tensor (tensor "proper" say) move, which actually is the part I find hardest.

I have had conceptual problems like this in the past, and found afterwards that I've made too much of a big deal about them. In other words, that in many of the big, supposedly difficult concepts, there is a smaller, much simpler idea trying to get through. For example, complex numbers. The word "complex" can form quite a barrier, but in fact the underlying idea is actually pretty simple if you give it time.

So I was revisiting tensors in that light. My only way of explaining (say, rank 2) to myself is to imagine a vector represented by velocity (perhaps), suddenly thrust from free space into a field of some sort, so that for every basis vector of velocity, a full set of basis vectors for the field are required. So in 3D, this means we get nine components. However, I'm not entirely happy with this. It's what I call "situational", i.e. depends on a specific situation.

Another way is just to examine how a scalar becomes a vector, and then decide to generalize up to tensor in terms of what I may call imprecisely "directional agents". A scalar has no such thing, it's only a quantity, so it's rank 0. If we are say in 3D, moving up a rank, means we add a "directional agent" which by necessity must have three components. That's rank 1. Now if we add a second "directional agent" it must correspond the vector in the three components available in the current context (3D space), and that correspondance must be three times for each of the vectors components, so that means nine (i.e. not 3 or 6).

Any help or insights welcome, thanks for reading! Cheers.