A circle C in the plane xplusypluszequals0 has a radius of 5 and center left parenthesis negative 2 comma 1 comma 1 right parenthesis. Evaluate Contour integral Subscript Upper C Superscript Baseline Bold Upper F times d Bold r for Bold Upper Fequalsleft angle 0 comma negative 3 z comma negative 2 y right angle​, where C has counterclockwise orientation when viewed from above. Does the circulation depend on the radius of the​ circle? Does it depend on the location of the center of the​ circle?

To evaluate the contour integral \(\oint_C \mathbf{F} \cdot d\mathbf{r}\) for \(\mathbf{F} = \langle 0, -3z, -2y \rangle\), we will use Stokes' Theorem. Stokes' Theorem relates a surface integral over a surface \(S\) to a line integral over the boundary curve \(C\) of the surface. Specifically, Stokes' Theorem states:

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

where \(\nabla \times \mathbf{F}\) is the curl of \(\mathbf{F}\).

First, we need to compute the curl of \(\mathbf{F}\). The vector field \(\mathbf{F}\) is given by:

\[
\mathbf{F} = \langle 0, -3z, -2y \rangle
\]

The curl of \(\mathbf{F}\) is given by:

\[
\nabla \times \mathbf{F} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
0 & -3z & -2y
\end{vmatrix}
\]

Evaluating this determinant, we get:

\[
\nabla \times \mathbf{F} = \left( \frac{\partial (-2y)}{\partial y} - \frac{\partial (-3z)}{\partial z} \right) \mathbf{i} - \left( \frac{\partial (0)}{\partial z} - \frac{\partial (0)}{\partial x} \right) \mathbf{j} + \left( \frac{\partial (-3z)}{\partial x} - \frac{\partial (0)}{\partial y} \right) \mathbf{k}
\]

Simplifying each term, we find:

\[
\nabla \times \mathbf{F} = (-2 - (-3)) \mathbf{i} + 0 \mathbf{j} + 0 \mathbf{k} = \mathbf{i}
\]

So,

\[
\nabla \times \mathbf{F} = \langle 1, 0, 0 \rangle
\]

Next, we need to compute the surface integral of \(\nabla \times \mathbf{F}\) over a surface \(S\) bounded by the curve \(C\). The surface \(S\) can be taken as the disk in the plane \(x + y + z = 0\).

The unit normal vector to the surface \(S\) (since the plane is \(x + y + z = 0\)) is \(\mathbf{n} = \langle 1, 1, 1 \rangle / \sqrt{3}\).

The differential surface element \(d\mathbf{S}\) is:

\[
d\mathbf{S} = \mathbf{n} \, dS = \frac{1}{\sqrt{3}} \langle 1, 1, 1 \rangle \, dS
\]

Then, the surface integral becomes:

\[
\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \iint_S \langle 1, 0, 0 \rangle \cdot \frac{1}{\sqrt{3}} \langle 1, 1, 1 \rangle \, dS
\]

This simplifies to:

\[
\iint_S \frac{1}{\sqrt{3}} \, dS = \frac{1}{\sqrt{3}} \iint_S dS
\]

The area of the disk \(S\) with radius 5 is \(\pi (5)^2 = 25\pi\). Therefore,

\[
\iint_S dS = 25\pi
\]

Thus,

\[
\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \frac{1}{\sqrt{3}} \times 25\pi = \frac{25\pi}{\sqrt{3}}
\]

Therefore,

\[
\oint_C \mathbf{F} \cdot d\mathbf{r} = \frac{25\pi}{\sqrt{3}}
\]

### Dependence on the Radius and Location of the Center

1. **Dependence on the radius of the circle**: Yes, the circulation depends on the radius of the circle since the integral over the surface \(S\) involves the area of the disk, which is directly proportional to the square of the radius.

2. **Dependence on the location of the center of the circle**: No, the circulation does not depend on the location of the center of the circle as long as the surface lies within the same plane and is bounded by the curve \(C\).

Do you have any further questions or need more details?

### Questions to Explore:
1. What is the divergence theorem, and how is it applied?
2. How do you parameterize a surface for surface integrals?
3. What are some common applications of Stokes' Theorem in physics?
4. How do you calculate the curl of a vector field in spherical coordinates?
5. What is the physical interpretation of the curl of a vector field?

### Tip:
When applying Stokes' Theorem, always ensure that the orientation of the curve and the surface are consistent with the right-hand rule to avoid sign errors.

Learn how to evaluate the contour integral \(\oint_C \mathbf{F} \cdot d\mathbf{r}\) over a circle \(C\) in the plane \(x+y+z=0\) using Stokes' Theorem. Includes step-by-step computation of the curl of \(\mathbf{F}\) and the surface integral over the disk \(S\). Suitable for advanced college-level students and professionals.

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