Using Gauss's Divergence Theorem, evaluate the outward flux $\iint_S \vec{F} \cdot \hat{n} \, dS$ for the vector field $\vec{F} = 2x\hat{i} + 3y\hat{j} + 4z\hat{k}$ over the closed sphere $S: x^2 + y^2 + z^2 = 1$ ________ (round off to 1 decimal place, take $\pi = 3.1416$).
Answer Key:
37.7
Step-by-Step Solution & Explanation
By Gauss's Divergence Theorem:
$\iint_S \vec{F} \cdot \hat{n} \, dS = \iiint_V (\nabla \cdot \vec{F}) \, dV$
Calculate the divergence of $\vec{F}$:
$\nabla \cdot \vec{F} = \frac{\partial}{\partial x}(2x) + \frac{\partial}{\partial y}(3y) + \frac{\partial}{\partial z}(4z) = 2 + 3 + 4 = 9$
Therefore:
$\iint_S \vec{F} \cdot \hat{n} \, dS = 9 \iiint_V dV = 9 \cdot \text{Volume of sphere}$
The radius of the sphere is $R = 1$, so $V = \frac{4}{3}\pi (1)^3 = \frac{4}{3}\pi$.
$\text{Flux} = 9 \times \frac{4}{3}\pi = 12\pi \approx 12 \times 3.14159 = 37.7$