Let $F$ from $R^3$ to $R$ defined by $F(x, y, z) = (x − yz, xz, y)$. Let $S$ be the surface obtained by rotating the graph of $x=2^z+3^z$ with $z ∈ [0, 1]$, around the $z$-axis (with normal vectors oriented outward). Calculate the flux of $∇×F$ across S.
I know how to compute the flux of vector field: either by Stokes Theorem, or directly. But, I have no idea how on this problem it should be dealt with. Any suggestions would be strongly welcomed.
Since it is always true that $\nabla \cdot (\nabla \times F)=0$, Gauss' Law tells us that the total flux of a curl through any closed surface always vanishes.
A closed surface is formed from ...
the curved surface you wish to calculate the flux through, call this $S_1$,
and a disk of radius $2$ parallel to the $xy$ plane at $z=0$ call this $S_2$ ,
and finally a disk of radius $5$ parallel to the $xy$ plane at $z=1$ call this $S_3$
This closed surface is congruent to the boundary of the volume of revolution formed by the graph of $y=2^x + 3^x$ revolved about the x-axis between $x=0$ and $x=1$ ...
the fluxes through the three surfaces are related by ...
$$\phi_1+\phi_2+\phi_3=0$$ we can calculate $\phi_2$ and $\phi_3$ directly using the fact that the $z$ component of $\nabla \times F$ is just $-2z$
so $\phi_2=0$ and $\phi_3 = -2(25 \pi)$
so $\phi_1 = 50 \pi$