Second order differential forms vanishing?

Question

I understand $dx dx = 0$ but why can’t we say the same for $dx dy$ as they’re both infinitesimal? I came across this when looking up an alternative derivation of change of variables in several dimensions.

Answer 1

There are certainly other cases where a differential element squared is formally zero and the intuitive reason is that the square of something small is much smaller, but I don't think that's the most useful way to think about it here. in In the sense of forms, the reason $dxdx=0,$ isn't because it is infinitesimal. We think of $dx\wedge dy$ intuitively as the area of a parallelogram with sides $dx$ and $dy.$ $dx\wedge dx$ is zero not cause of the infinitesimal-ness but because two parallel vectors make a parallelogram with zero area. In fact, it's generally the case that $v\wedge v=0$ even for non-infinitesimal $v.$ (Except in more exotic cases where the components of $v$ are non-commuting objects rather than numbers.)

(Image provided by user Steffen Plunder.)