To compute the circulation of the vector field $\mathbf{F} = -y\mathbf{i} + x\mathbf{j}$ around the circle $C$ using Green's Theorem, we need to evaluate the line integral $\oint_C \mathbf{F} \cdot d\mathbf{r}$.

Green's Theorem states:

$\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA$

where $\mathbf{F} = P\mathbf{i} + Q\mathbf{j}$. In our case, $P = -y$ and $Q = x$.

First, compute the partial derivatives:

$\frac{\partial Q}{\partial x} = \frac{\partial x}{\partial x} = 1$ $\frac{\partial P}{\partial y} = \frac{\partial (-y)}{\partial y} = -1$

So, we have:

$\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 1 - (-1) = 2$

Now, we need to integrate this over the region $D$, which is the disk of radius 2 centered at the origin. The area element in polar coordinates is $dA = r \, dr \, d\theta$.

The integral becomes:

$\iint_D 2 \, dA = 2 \iint_D dA$

The area of the disk $D$ with radius 2 is:

$\text{Area}(D) = \pi (2)^2 = 4\pi$

Therefore,

$2 \iint_D dA = 2 \times 4\pi = 8\pi$

So, the circulation of the vector field $\mathbf{F}$ around the circle $C$ is:

$\oint_C \mathbf{F} \cdot d\mathbf{r} = 8\pi$

Would you like more details on any part of this solution or have any other questions?

Here are 5 questions you might want to ask:

1. How is Green's Theorem applied in more complex vector fields?
2. Can Green's Theorem be used to compute area?
3. What are the conditions for using Green's Theorem?
4. How can the circulation be interpreted physically?
5. Can you explain the divergence theorem in relation to Green's Theorem?

Tip: When working with Green's Theorem, always check the orientation of the curve and ensure that the region $D$ is simply connected and the vector field components have continuous partial derivatives.