To find the flux of the vector field $\mathbf{F} = 6z\mathbf{i} - 5y\mathbf{j} + 6x\mathbf{k}$ out of a sphere of radius $R = 3$ centered at the origin, we can use the Divergence Theorem.

Step 1: Divergence Theorem Formula

The Divergence Theorem states: $\iint_S \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_V (\nabla \cdot \mathbf{F}) \, dV$ Here:

• $S$ is the surface (sphere in this case),
• $V$ is the volume enclosed by $S$,
• $\nabla \cdot \mathbf{F}$ is the divergence of $\mathbf{F}$.

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

The divergence of $\mathbf{F}$ is: $\nabla \cdot \mathbf{F} = \frac{\partial (6z)}{\partial z} + \frac{\partial (-5y)}{\partial y} + \frac{\partial (6x)}{\partial x}$ $\nabla \cdot \mathbf{F} = 6 - 5 + 6 = 7$

Step 3: Compute the Volume Integral

The sphere is given by $x^2 + y^2 + z^2 = R^2$ with $R = 3$. The volume of the sphere is: $V = \frac{4}{3} \pi R^3 = \frac{4}{3} \pi (3)^3 = 36\pi$

The volume integral is: $\iiint_V (\nabla \cdot \mathbf{F}) \, dV = \iiint_V 7 \, dV = 7 \cdot \text{Volume of sphere} = 7 \cdot 36\pi = 252\pi$

Step 4: Flux through the Surface

By the Divergence Theorem: $\iint_S \mathbf{F} \cdot \mathbf{n} \, dS = 252\pi$

Final Answer:

The flux of $\mathbf{F}$ out of the sphere is: $\boxed{252\pi}$

Would you like a detailed explanation of any of these steps or further examples?

Related Questions:

1. How does the Divergence Theorem simplify flux calculations for symmetric shapes?
2. What is the physical interpretation of flux in this context?
3. How would the calculation change if the sphere were not centered at the origin?
4. Can we compute the flux directly using the surface integral without the Divergence Theorem?
5. How does the divergence of a vector field relate to its sources and sinks?

Tip:

For symmetric surfaces like spheres, always check if the Divergence Theorem can simplify the problem—it often reduces surface integrals to straightforward volume integrals!