What's the flux of the vector field F(x,y,z) = (e^-y) i - (y) j + (x sinz) k across σ with outward orientation where σ is the portion of the elliptic cylinder r(u,v) = (2cos v) i + (sin v) j + (u) k with 0 ≤ u ≤ 5, 0 ≤ v ≤ 2pi?

Question

irishboy123 · Answer

maybe use divergence theorem?

$\nabla \bullet E = x \cos z - 1$


so you have $\int\limits_{z = 0}^{5} \int\limits_{y = 0}^{1} \int\limits_{x = -2 \sqrt{1-y^2}}^{2 \sqrt{1-y^2}} \quad x \cos z - 1 \; dx \; dy \; dz  $

then subtract the end caps using a normal flux surface integral

$\Phi_{z=0} = \iint \vec E \bullet \hat n \; da = \int x \sin(0) dx \; dy = 0$

$\Phi_{z=5} = \sin(5) \int\limits_{y=-1}^{1} x \; \int\limits_{x = -2 \sqrt{1-y^2}}^{2 \sqrt{1-y^2}} \; dx \; dy = 0$

but haven't actually computed it out

personally reckon cartesian is the way to go