= Stokes flux through an annular paraboloid
{c}
{title2=$I=3\pi(R^4-r_0^4)/2$}
For an upward-oriented band of the circular <elliptic paraboloid> between radii $r_0$ and $R$, the <vector field> $B=(-y^3,x^3,z^3)$ has vertical <curl> $3(x^2+y^2)$. Its <surface integral> is $3\pi(R^4-r_0^4)/2$. <Stokes theorem> gives the same value from the counterclockwise outer circle minus the counterclockwise inner circle. Equivalently the induced inner boundary direction is clockwise. Forgetting the inner boundary or its reversed orientation changes the answer.
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