Question about differential forms spherical - Mathematics Stack Exchange

Question

Consider the $2$-form on $mathbb{R}^3$ given by $$omega = x,dywedge dz+y,dzwedge dx+z,dxwedge dy.$$ We can restrict this to the sphere. If I use spherical coordinates with $phi$ the polar angle from the $z$-axis and $ heta$ the azimuthal angle from the $x$-axis, the resulting form I get from computing pullbacks is $sinphi,dphiwedge d heta$.

This can only be computed where spherical coordinates work, which turns out to be everywhere on $S^2$ except for the north and south pole.

I know for a fact that the form is nonzero everywhere on $S^2$. But it seems like $sinphi,dphiwedge d heta$ gets very "small" near the north pole: the scaling by $sinphi$ approaches zero. So it seems for the form to be continuous, we would need it to be zero at the north pole, but I know this is not the case.

How am I to reconcile these facts, geometrically? It seems like $dphi$ and $d heta$ don't "shrink" near the poles either--I am looking for some geometric intuition. Thanks!

Answer 1

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