Discussion
Evaluation of
\[ \begin{aligned} &\iint_{S} \left(e^{x}\hat{i} + 3y\hat{j} - ze^{x}\hat{k}\right) \\ &\quad \cdot \hat{n}\, dA \end{aligned} \]
over a surface \(S: x^{2} + y^{2} + z^{2} = 1\), using Gauss divergence theorem, gives
Answer: B
The surface integral equals 4π.
By the divergence theorem the flux through the closed sphere equals the volume integral of the divergence:
\[ \begin{aligned} \nabla \cdot \vec{F} &= \frac{\partial e^{x}}{\partial x} + \frac{\partial (3y)}{\partial y} + \frac{\partial (-ze^{x})}{\partial z} \\ &= e^{x} + 3 - e^{x} = 3 \\ \iint_{S} \vec{F} \cdot \hat{n}\,dA &= 3 \times \tfrac{4}{3}\pi(1)^{3} = 4\pi \end{aligned} \]
4π/3 is the volume of the unit sphere without the factor 3, and 12π multiplies by 3 twice or uses the surface area 4π. The answer is not 0, because the divergence is a non-zero constant.
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