To rewrite the surface integral as a line integral, we will use Stokes' Theorem, which states:

$\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r}$

Given:

• The vector field $\mathbf{F} = (x - y)\mathbf{i} + (y - z)\mathbf{j} + (z - x)\mathbf{k}$.
• The surface $S$ is the disk $x^2 + y^2 = 4$ at $z = 1$, and the curve $C$ is the boundary of this disk.

Step 1: Compute the Curl of $\mathbf{F}$

We already computed the curl: $\nabla \times \mathbf{F} = -\mathbf{i} - \mathbf{j} - \mathbf{k}$

Step 2: Surface Integral Setup

The surface integral is: $\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$

Step 3: Use Stokes' Theorem

Stokes' Theorem allows us to rewrite this surface integral as a line integral around the boundary curve $C$:

$\oint_C \mathbf{F} \cdot d\mathbf{r}$

Step 4: Express the Line Integral

The curve $C$ is the circle $x^2 + y^2 = 4$ in the plane $z = 1$.

In parametric form, the curve can be represented as: $\mathbf{r}(t) = (2\cos t)\mathbf{i} + (2\sin t)\mathbf{j} + \mathbf{k}, \quad 0 \leq t \leq 2\pi$ Here, $t$ is the parameter.

Step 5: Find $d\mathbf{r}$

The differential $d\mathbf{r}$ is: $d\mathbf{r} = \frac{d\mathbf{r}}{dt} dt = (-2\sin t)\mathbf{i} + (2\cos t)\mathbf{j} \, dt$

Step 6: Substitute into the Line Integral

Now substitute $\mathbf{F}$ and $d\mathbf{r}$ into the line integral: $\oint_C \mathbf{F} \cdot d\mathbf{r} = \int_0^{2\pi} \mathbf{F}(\mathbf{r}(t)) \cdot \frac{d\mathbf{r}}{dt} \, dt$ Where $\mathbf{F}(\mathbf{r}(t))$ is $\mathbf{F}$ evaluated at $\mathbf{r}(t) = (2\cos t, 2\sin t, 1)$: $\mathbf{F}(t) = (2\cos t - 2\sin t)\mathbf{i} + (2\sin t - 1)\mathbf{j} + (1 - 2\cos t)\mathbf{k}$

So the dot product $\mathbf{F}(t) \cdot \frac{d\mathbf{r}}{dt}$ becomes: $\left[(2\cos t - 2\sin t)(-2\sin t) + (2\sin t - 1)(2\cos t)\right] dt$

This expression simplifies to: $\oint_C \mathbf{F} \cdot d\mathbf{r} = \int_0^{2\pi} \left[-4\cos t \sin t + 4\sin t \cos t - 2\sin t\right] dt$

This integral can now be computed to evaluate the line integral, completing the conversion from a surface integral to a line integral using Stokes' Theorem.

Would you like any more details or assistance in solving the integral?

8 Related Questions:

1. How do you find the parametric equations for a curve given its geometric description?
2. How does the orientation of the curve affect the outcome of the line integral?
3. What would change if the vector field $\mathbf{F}$ were different?
4. Why is it necessary to compute the curl of the vector field in Stokes' Theorem?
5. Can Stokes' Theorem be applied to any surface with a given boundary?
6. What is the physical interpretation of a line integral in this context?
7. How do you handle cases where the curve $C$ is not simple (e.g., has multiple components)?
8. What would the integral look like if the surface were parameterized differently?

Tip:

When parameterizing curves, ensure the parameterization covers the entire curve exactly once and maintains the correct orientation.