So I asked this question last week:
Consider the vector field $\mathbf{v}=(x^2y-\sqrt[3]{z}, \sqrt[3]{z}-xy^2,z)$ and a surface given by $\frac{z^2}{2}=x^2+y^2$ and $z\in[0,2]$. Calculate the flux of $\mathbf{v}$ through the surface.
I did calculate the divergence of $\mathbf{v}$ to using the divergence theorem, but I don't know how to write the limits of the integral.
I calculated and i get $\frac{4\pi}{3}$, but all possible values are :
- $\frac{-7\pi}{3}$
- $\frac{-12\pi}{5}$
- $-\pi$
- $\frac{-8\pi}{3}$
- $\frac{-9\pi}{4}$
- $\frac{-13\pi}{5}$
My calculations: $$\int^{2\pi}_{0} \int^{\sqrt{2}}_{0} \int^{2}_{\sqrt{2} \rho} \rho \ dz \ d\rho \ d\phi=2 \pi \int^{\sqrt{2}}_{0} (2\rho-\sqrt{2} \rho^2) \ d\rho=4\frac{\pi}{3}$$
Where did I go wrong?