Let $\vec{F}(x, y, z) = 2x \hat{i} + 3y \hat{j} + z \hat{k}$. The outward flux of $\vec{F}$ across the closed surface of the unit sphere $S: x^2 + y^2 + z^2 = 1$ is:

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### 1. Strategy & Theoretical Principles By Gauss's Divergence Theorem: $$\iint_S \vec{F} \cdot \hat{n} \, dS = \iiint_V (\nabla \cdot \vec{F}) \, dV$$ where $V$ is the volume enclosed by the surface $S$. ### 2. Analytical Formulation & Calculation 1. Compute the divergence of $\vec{F} = 2x \hat{i} + 3y \hat{j} + z \hat{k}$: $$\nabla \cdot \vec{F} = \frac{\partial(2x)}{\partial x} + \frac{\partial(3y)}{\partial y} + \frac{\partial(z)}{\partial z} = 2 + 3 + 1 = 6$$ 2. The divergence is constant, so factor it out of the volume integral: $$\iint_S \vec{F} \cdot \hat{n} \, dS = 6 \iiint_V dV = 6 \times \text{Volume}(V)$$ 3. Volume of the unit sphere ($R = 1$): $$V = \frac{4}{3} \pi R^3 = \frac{4}{3} \pi (1)^3 = \frac{4}{3} \pi$$ 4. Total outward flux: $$\text{Flux} = 6 \times \left(\frac{4}{3}\pi\right) = 8\pi$$ Evaluating numerically: $$\text{Flux} = 8 \times 3.14159265 \approx 25.1327 \approx 25.13$$ ### 3. Distractor Forensics - Forgetting to multiply by 6 gives $4.19$ (volume only). - Forgetting $4/3$ factor gives $18.85$. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Gauss's Divergence Theorem converts a 2D surface integral into a 3D volume integral. - **Standard Reference:** Kreyszig, E., *Advanced Engineering Mathematics*, Chapter 10.

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