Question

Give conditions on the constants a, b and c that ensure that the flux integral integral_S F middot dS, where F = ai + bj + ck and S is the lower half of the unit sphere centered at the origin and oriented downward, is positive. (For each, if there is no restriction on the constant select = and enter anything for the value.)

Answer 1

Parameterize $S$ by

$\vec s(u,v)=\cos u\sin v\,\vec\imath+\sin u\sin v\,\vec\jmath+\cos v\,\vec k$

with $0\le u\le2\pi$ and $\frac\pi2\le v\le\pi$. The downward normal vector to $S$ is

$\dfrac{\partial\vec s}{\partial v}\times\dfrac{\partial\vec s}{\partial u}=\cos u\sin^2v\,\vec\imath+\sin u\sin^2v\,\vec\jmath+\cos v\sin v\,\vec k$

Then the flux of $\vec F$ across $S$ is

$\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\int_{\pi/2}^\pi\int_0^{2\pi}(a\,\vec\imath+b\,\vec\jmath+c\,\vec k)\cdot\left(\dfrac{\partial\vec s}{\partial v}\times\dfrac{\partial\vec s}{\partial u}\right)\,\mathrm du\,\mathrm dv$

$=\displaystyle\int_{\pi/2}^\pi\int_0^{2\pi}a\cos u\sin^2v+b\sin u\sin^2v+c\cos v\sin v\,\mathrm du\,\mathrm dv$

Only the last term of the integrand contributes to the integral; we have

$\displaystyle\int_{\pi/2}^\pi\cos v\sin v=\frac12\int_{\pi/2}^\pi\sin(2v)\,\mathrm dv=-\frac12$

so that the flux is $-\frac c2$. In order for this to be positive, we must have $c<0$. $a$ and $b$ can be any constants.