Consider in $\mathbb{S}^2 \setminus \{N\}$ (where $N$ is the north pole) the $1$-forms $\omega, \eta$ which with respect to the stereographical projection from the north pole have the following local expressions: $$\omega = \mathrm{d} x, \eta = \mathrm{d} x \wedge \mathrm{d} y$$
Do they have smooth extensions to the north pole?
As I understand it, once we identify in the obvious manner the tangent spaces of $\mathbb{S}^2$ with affine subspaces of $\mathbb{R}^3$ (that is, for $p \in \mathbb{S}^2$, $T_{p} \mathbb{S}^2 = p^{\perp}$) , the $1$-forms $\mathrm{d} x$ and $\mathrm{d} y$ with respect to the stereographical projection from the north pole are just the restrictions of the differential forms $\mathrm{d} r^1$ and $\mathrm{d} r^2$ to the tangent spaces of the sphere, where I'm defining $\mathrm{d} r^i(p)(e_j) = \delta_{ij} $ (the standard basis differential forms on $\mathbb{R}^3$). So of course they have smooth extensions to the north pole.
But this feels too easy so I'm thinking maybe I'm not understanding the exercise correctly. What did I do wrong here, how should I proceed?