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 s

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.