State Stokes' Theorem for a vector field $\mathbf{B}(\mathbf{x})$ on $\mathbb{R}^{3}$.

Consider the surface $S$ defined by

$$z=x^{2}+y^{2}, \quad \frac{1}{9} \leqslant z \leqslant 1 .$$

Sketch the surface and calculate the area element $d \mathbf{S}$ in terms of suitable coordinates or parameters. For the vector field

$$\mathbf{B}=\left(-y^{3}, x^{3}, z^{3}\right)$$

compute $\nabla \times \mathbf{B}$ and calculate $I=\int_{S}(\nabla \times \mathbf{B}) \cdot d \mathbf{S}$.

Use Stokes' Theorem to express $I$ as an integral over $\partial S$ and verify that this gives the same result.