- #1
AriAstronomer
- 48
- 1
Homework Statement
Through what closed, oriented surface in R^3 does the vector field F = <(4x + 2zx^3), -y(x^2 + z^2), -(3x^2z^2 + 4y^2z)> have the greatest flux?
Homework Equations
Flux = double int F.ds
Gauss theorem perhaps (double int F.ds = triple int(DivF)dV)
The Attempt at a Solution
So I figured that since they want max flux, go with the gradient and set it equal to zero (if rate of change is 0, must be at a critical point). So Flux = double int F.ds, and taking the gradient we have grad(Flux) = 0. I thought Gauss might make it easier, so plugging in Gauss:
grad(Flux) = grad(triple int(DivF)dV)= 0.
Take the divergence of F and we get (after simplifying): divF = 4-4y^2 - x^2 -z^2. So now we have have:
grad(triple int(4-4y^2 - x^2 -z^2)dV)= 0
Now I'm stuck though, because I'm not sure how to integrate this since that's the whole point of the question. Is there a way to bring the gradient inside or something?
Thanks in advance