evaluate the line iterative integral of F * dr by evaluating the surface integral in Stokes theorem with an appropriate choice of S. Assume that C has a counterclockwise orientation when viewed from above.

F = (4xy sin z, 2x^2 sin z, 2x^2 y cos z)
C is the boundary of the plane z = 15 - 5x - 3y in the first octant

Rewrite the given integral as an area integral over the appropriate region of the xy plane

To evaluate the line integral \(\int_C \mathbf{F} \cdot d\mathbf{r}\) using Stokes' Theorem, we first need to understand the relationship between the line integral and the surface integral provided by Stokes' Theorem. Stokes' Theorem states that:

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

where \(\mathbf{F} = (4xy \sin z, 2x^2 \sin z, 2x^2 y \cos z)\) and \(C\) is the boundary of the plane \(z = 15 - 5x - 3y\) in the first octant.

### Step 1: Compute \(\nabla \times \mathbf{F}\)

First, we find the curl of \(\mathbf{F}\):

\[
\nabla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right)
\]

Given \(\mathbf{F} = (4xy \sin z, 2x^2 \sin z, 2x^2 y \cos z)\):

- \(F_1 = 4xy \sin z\)
- \(F_2 = 2x^2 \sin z\)
- \(F_3 = 2x^2 y \cos z\)

Compute each component of the curl:

1. \(\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}\):

\[
\frac{\partial (2x^2 y \cos z)}{\partial y} = 2x^2 \cos z
\]

\[
\frac{\partial (2x^2 \sin z)}{\partial z} = 2x^2 \cos z
\]

Thus,

\[
\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} = 2x^2 \cos z - 2x^2 \cos z = 0
\]

2. \(\frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}\):

\[
\frac{\partial (4xy \sin z)}{\partial z} = 4xy \cos z
\]

\[
\frac{\partial (2x^2 y \cos z)}{\partial x} = 4xy \cos z
\]

Thus,

\[
\frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} = 4xy \cos z - 4xy \cos z = 0
\]

3. \(\frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}\):

\[
\frac{\partial (2x^2 \sin z)}{\partial x} = 4x \sin z
\]

\[
\frac{\partial (4xy \sin z)}{\partial y} = 4x \sin z
\]

Thus,

\[
\frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} = 4x \sin z - 4x \sin z = 0
\]

So, we have:

\[
\nabla \times \mathbf{F} = (0, 0, 0)
\]

### Step 2: Apply Stokes' Theorem

Since \(\nabla \times \mathbf{F} = \mathbf{0}\), we have:

\[
\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \iint_S \mathbf{0} \cdot d\mathbf{S} = 0
\]

Therefore,

\[
\int_C \mathbf{F} \cdot d\mathbf{r} = 0
\]

### Conclusion

The line integral of \(\mathbf{F}\) over the curve \(C\), which is the boundary of the plane \(z = 15 - 5x - 3y\) in the first octant, evaluates to zero when using Stokes' Theorem.

### Relative Questions

1. What are the conditions for applying Stokes' Theorem?
2. How do you compute the curl of a vector field?
3. What is the physical interpretation of the curl of a vector field?
4. How does the orientation of the curve \(C\) affect the result in Stokes' Theorem?
5. Can Stokes' Theorem be applied to non-planar surfaces? If so, how?
6. What happens if \(\nabla \times \mathbf{F} \neq 0\)?
7. How can Stokes' Theorem be used in electromagnetism?
8. What are some practical applications of Stokes' Theorem in engineering?

### Tip

When applying Stokes' Theorem, always ensure that the surface \(S\) and its boundary curve \(C\) are properly oriented according to the right-hand rule.

Learn how to evaluate the line integral \(\int_C \mathbf{F} \cdot d\mathbf{r}\) using Stokes' Theorem with a detailed step-by-step solution. Understand the relationship between line and surface integrals, and apply it to a vector field \(\mathbf{F}\) in three dimensions. Suitable for college-level mathematics and physics students.

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