To evaluate the surface integral $\iint_S \mathbf{F} \cdot d\mathbf{S}$, where $\mathbf{F} = \langle 3x, -z, y \rangle$, and $S$ is the part of the sphere $x^2 + y^2 + z^2 = 9$ in the first octant, oriented toward the origin, we use the Divergence Theorem.

Step 1: Apply the Divergence Theorem

The Divergence Theorem states:

$\iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{F}) \, dV,$

where $V$ is the volume enclosed by $S$, and $\nabla \cdot \mathbf{F}$ is the divergence of $\mathbf{F}$.

Step 2: Compute $\nabla \cdot \mathbf{F}$

The vector field is $\mathbf{F} = \langle 3x, -z, y \rangle$. Its divergence is:

$\nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}(3x) + \frac{\partial}{\partial y}(-z) + \frac{\partial}{\partial z}(y).$

$\nabla \cdot \mathbf{F} = 3 + 0 + 0 = 3.$

Step 3: Set up the volume integral

The volume $V$ is the portion of the sphere $x^2 + y^2 + z^2 = 9$ in the first octant. The volume of the entire sphere is:

$\text{Volume of sphere: } \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (3)^3 = 36\pi.$

Since we are only considering the first octant, the volume of $V$ is:

$\text{Volume of } V = \frac{1}{8} \times 36\pi = \frac{9\pi}{2}.$

Step 4: Evaluate the integral

Using the Divergence Theorem:

$\iiint_V (\nabla \cdot \mathbf{F}) \, dV = \iiint_V 3 \, dV = 3 \times \text{Volume of } V.$

$\iiint_V 3 \, dV = 3 \times \frac{9\pi}{2} = \frac{27\pi}{2}.$

Final Answer:

$\iint_S \mathbf{F} \cdot d\mathbf{S} = \frac{27\pi}{2}.$

Would you like a detailed step-by-step derivation of any part? Let me know!

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Related Questions:

1. What is the physical significance of the divergence theorem in vector calculus?
2. How would the integral change if the sphere's orientation was outward instead of toward the origin?
3. What happens if we extend the region $S$ to include other octants?
4. How does the vector field $\mathbf{F}$ influence the computation of the integral?
5. Can this problem be solved using parametrization instead of the Divergence Theorem?

Tip: Always check the orientation of the surface while applying the Divergence Theorem; it determines the sign of the flux.