Can we always use the right hand side of Stokes's theorem? $$ \oint \vec{F} \cdot \vec{dl} = \int\int \nabla \times \vec{F}\cdot \vec{n}\ dS $$
For example,
$$\vec{F} = z^2\hat{i} - 3xy \hat{j} + x^3y^3\hat{k}$$
$$z= 5-x^2 -y^2, z=1$$
Using the left hand side was easy, but I'm wondering if this is possible to compute this integral using the right hand side. Or, maybe sometimes only the right hand side works?