This problem involves verifying Stokes' Theorem for the given surface $S$ and its boundary $\partial S$, with a vector field $\mathbf{F}$.

Stokes' Theorem:

$\iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{dS} = \oint_{\partial S} \mathbf{F} \cdot \mathbf{ds}$

Given:

• Surface $S$: $z = 1 - x^2 - y^2, \, z \geq 0$, which is a paraboloid capped at $z = 0$.
• Boundary $\partial S$: $x^2 + y^2 = 1$, which is the circle at $z = 0$.
• Vector field $\mathbf{F}$: $\mathbf{F} = z \mathbf{i} + x \mathbf{j} + (6zx + 2xy) \mathbf{k}.$

Steps to Verify:

1. Compute the Curl $\nabla \times \mathbf{F}$: The curl of $\mathbf{F}$ is calculated as:

   \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ z & x & 6zx + 2xy \end{vmatrix}.$$ Expanding the determinant gives: $$\nabla \times \mathbf{F} = (6z + 2x) \mathbf{i} + (-6z) \mathbf{j} + (1 - 1) \mathbf{k} = (6z + 2x) \mathbf{i} - 6z \mathbf{j}.$$

2. Surface Integral $\iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{dS}$:

   • The normal vector to the surface $S$ is upward-pointing.
   • Convert to polar coordinates $x = r \cos\theta, y = r \sin\theta, z = 1 - r^2$ on $S$, with $0 \leq r \leq 1, 0 \leq \theta \leq 2\pi$.
   • Substitute $\nabla \times \mathbf{F}$ and integrate over $S$. The final result simplifies to $\pi$.

3. Line Integral $\oint_{\partial S} \mathbf{F} \cdot \mathbf{ds}$:

   • The boundary $\partial S$ lies in the $z = 0$ plane, with $x^2 + y^2 = 1$.
   • Parameterize $\partial S$: $x = \cos\theta, y = \sin\theta, z = 0, 0 \leq \theta \leq 2\pi$.
   • Evaluate $\mathbf{F} \cdot \mathbf{ds}$ along the curve. The result simplifies to $\pi$.

Conclusion:

Both the surface integral and the line integral equal $\pi$, confirming Stokes' Theorem for this setup.

Let me know if you'd like detailed computations for any of the steps!

Expansion Questions:

1. How is the curl of a vector field computed and what does it represent geometrically?
2. Why does Stokes' theorem require the boundary $\partial S$ to be oriented counterclockwise when viewed from above?
3. What role does the parameterization of surfaces and curves play in simplifying integrals in vector calculus?
4. Can Stokes' theorem be extended to surfaces with more complex boundaries? How?
5. What are some practical applications of Stokes' theorem in physics and engineering?

Tip: Always check the orientation of the surface and its boundary when applying Stokes' theorem—it ensures consistency in the signs of the integrals.