Use Stokes Theorem to Prove Two Integrals on Differentiable Manifiolds are equivalent I have to answer the following problem:
Let $\omega= ydx + xzdy + xdz$.
Let $S_1$ be the portion of the upper hemisphere given by $\phi_1(u,v)=(u,v,\sqrt{4-u^2-v^2})$; where $u^2 + v^2 \leq 2$.
Let $S_2$ be the disc in the plane $z=\sqrt{2}$, given by $\phi_2(u,v)=(u,v,\sqrt{2})$; where $u^2 + v^2 \leq 2$.
Use Stokes theorem to show $\int\int_{s_1}d\omega=\int\int_{s_2}d\omega$
I know that I could technically calculate each integral to prove this, but since the directions are asking for the use of Stokes theorem, I feel like there's some trick that I'm missing. The two parameterizations look like they have equivalent boundaries, but I'm getting stuck on exactly how to show that. Any help would be greatly appreciated

HINT: When $u^2+v^2=2$, we have $\sqrt{4-u^2-v^2}=\sqrt2$. Indeed,
$$\int_{S_1}d\omega=\int_{\partial S_1} \omega=\int_{\partial S_2} \omega=\int_{S_2} d\omega.$$
