If $\sigma$ is an arbitrary closed differential 2-form on the unit ball $\mathbb{B}^3=\{(x, y, z)\in \mathbb{R}^3|x^2+y^2+z^2 \leq 1\}$, prove that there exists a point $p$ on the unit sphere $\mathbb{S}^2=\{(x, y, z) \in \mathbb{R}^3|x^2+y^2+z^2 =1\}$ where $\sigma (p)=0$ ($0$, the multilinear zero-mapping).
I'm trying to solve this using Stokes' theorem, because if $\sigma$ is closed, $d\sigma=0$, so $\iiint_{\mathbb{B}^3}d\sigma=\iint_{\mathbb{S}^2}\sigma=0$, and then if $\sigma = Pdy \wedge dz+Qdz\wedge dx+Rdx\wedge dy$, maybe use the continuity of $P, Q, R$ to prove that they have to be $0$ somewhere. The problem I'm facing is, I'm not sure if there has to exist a point where they're all simultaneously $0$.
On the unit sphere the form restricts to a scalar function $\phi$ times $$\omega=x\,dy\wedge dz+y\,dz\wedge dx+z\,dx\wedge dy.$$ This corresponds to the "outward normal". Integrating $\sigma=\phi\omega$ over $S^2$ just corresponds to integrating the scalar $\phi$ with respect to surface area on the sphere. If this integral is zero, then $\phi$ must be zero somewhere.