feeling extremely insulted by this. sigh i hate the maths : physics feud.
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Replying to @the_aiju
"pseudovector" is the worst concept, this whole area requires some representation theory to sort out properly
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aw sorry i was just trying to beef for fun not beef for real. i get that physicists do things differently and it works for them and it's not that hard to translate back and forth all things considered
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So what is a pseudovector? I had the idea that maybe it's a linear functional, or perhaps a row vector that you think of as representing a linear functional, but I've never been sure.
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okay this is not going to sound like it explains anything but it does. a pseudovector on a f.d. vector space V of dimension n is an element of the next-to-top exterior power wedge^{n-1}(V). this is aaaalmost an element of the dual V*; there’s a natural pairing...
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...V x wedge^{n-1}(V) -> wedge^n(V) given by exterior product, which lets you write down a natural isomorphism from the space of pseudovectors to V* tensor wedge^n(V). concretely this means a pseudovector transforms almost like a linear functional unde change of coordinates...
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...except that it picks up an extra factor of the determinant! if you now pick an inner product on V and identify V with its dual, a pseudovector transforms exactly like a vector wrt rotations but picks up an extra -1 wrt reflections (corresponding to the sign of the determinant)
in dimension 3 this is the space where cross products live which i think is ultimately where (most of?) their appearances in physics are due to
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