The Divergence Theorem states:

$\iint_S \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_V (\nabla \cdot \mathbf{F}) \, dV$

Where:

• $S$ is the closed surface,
• $\mathbf{F}$ is the vector field,
• $\nabla \cdot \mathbf{F}$ is the divergence of $\mathbf{F}$,
• $V$ is the volume enclosed by $S$.

We are given: $\mathbf{F} = \langle 3x, 4y, -2z \rangle$ and $S$ is the sphere $x^2 + y^2 + z^2 = 6$.

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

The divergence of a vector field $\mathbf{F} = \langle F_x, F_y, F_z \rangle$ is: $\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$ Here:

• $F_x = 3x$, so $\frac{\partial F_x}{\partial x} = 3$,
• $F_y = 4y$, so $\frac{\partial F_y}{\partial y} = 4$,
• $F_z = -2z$, so $\frac{\partial F_z}{\partial z} = -2$.

Thus: $\nabla \cdot \mathbf{F} = 3 + 4 - 2 = 5$

Step 2: Use the Divergence Theorem

The volume $V$ is the interior of the sphere $x^2 + y^2 + z^2 = 6$, which has radius $r = \sqrt{6}$. The volume of a sphere is: $\text{Volume of sphere} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (\sqrt{6})^3 = \frac{4}{3} \pi (6\sqrt{6})$

The net outward flux is: $\iint_S \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_V (\nabla \cdot \mathbf{F}) \, dV = \int_V 5 \, dV = 5 \cdot \text{Volume of sphere}$

Substitute the volume: $\iint_S \mathbf{F} \cdot \mathbf{n} \, dS = 5 \cdot \frac{4}{3} \pi (6\sqrt{6}) = \frac{20}{3} \pi (6\sqrt{6}) = 40\sqrt{6} \pi$

Final Answer:

$\text{Net outward flux} = 40\sqrt{6} \pi$

Let me know if you'd like further clarification or detailed steps! 😊

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Here are 5 related questions to explore:

1. How is the Divergence Theorem related to Gauss's Law in physics?
2. What happens if the vector field $\mathbf{F}$ is solenoidal ($\nabla \cdot \mathbf{F} = 0$)?
3. How does the flux calculation change for non-spherical surfaces?
4. Can the Divergence Theorem be applied to surfaces that are not closed?
5. How is the volume element $dV$ computed in cylindrical or spherical coordinates?

Tip: Always verify the divergence computation carefully—it’s the foundation of applying the Divergence Theorem correctly.