Give a necessary condition for a given vector field $\mathbf{J}$ to be the curl of another vector field $\mathbf{B}$. Is the vector field $\mathbf{B}$ unique? If not, explain why not.

State Stokes' theorem and use it to evaluate the area integral

$$\int_{S}\left(y^{2}, z^{2}, x^{2}\right) \cdot \mathbf{d} \mathbf{A}$$

where $S$ is the half of the ellipsoid

$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$$

that lies in $z \geqslant 0$, and the area element dA points out of the ellipsoid.