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The post was also a poor explanation for someone doing modern physics. [edit: not true actually I should have read the rest of the post - it’s a good post]

Wald's approach in General Relativity is much better - he treats Tensors as a multilinear map from vectors and dual vectors to scalars.

He then derives the underlying coordinate transformaton rules, for the vector spaces used in differential geometry. But

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That’s the approach I used as well in the second half of the article - I just mentioned the transformation law in the beginning since that’s what most physics students encounter first.

Most of the article tries to provide some intuition behind why multilinear maps, which sound like a fairly abstract concept, might be relevant in physics. The key link being the importance of coordinate invariance.

I didn’t go into deriving the coordinate transforms from the multilinear map definition as I didn’t feel that it’d provide much better intuition, but I did mention the equivalence near the end.

Yeah sorry you’re right - I should have read the rest of your post, which is excellent and describes precisely why the coordinates/transformations focused definition is bad for one’s intuition.

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