Let $\mathcal{S}$ be the part of the sphere $x^2+y^2+z^2=4$, $y \geq 0$, oriented in the direction of the positive $y$-axis. Compute $\iint_\mathcal{S} \vec{F} \cdot d\vec{S}$, where $\vec{F}$ is ..
[thinks]
[sighs]
[gives up]
.. some vector field that has nice derivatives. (i forget.)
when i wrote the solutions, i used a parametrization with coordinates $x$, $z$. out of 100+ homeworks, only 1-2 students did the problem the same way that i did it.
on a related note, i have a soft heart. since it's finals week, i'm letting my undergrad graders off and grading the last homework and quiz myself.
no good deed goes unpunished. i thought that this week would be mathematically luxurious: spend all day with research and other matters. instead, even though classes are over, i am still working on teaching things ..!
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