Find the flux of $F(x, y, z) = \langle-x, -y, z^3\rangle$ through the surface $S$ when $S$ is the part of the cone
$$z =\sqrt{x^2 + y^2}$$
that lies between the planes $z = 1$ and $z = 3$, oriented upwards.
- If I use the usual method, Im getting $1712\pi/15$
- If I use divergence theorem, Im getting $1916\pi/15$
Is the divergence theorem not valid in here? thanks
Sure, the divergence theorem is valid. But you're applying it to a region whose boundary consists of not only that surface $S$, but also two disks: $\Sigma_1 = \{x^2+y^2\le 1, z=1\}$ and $\Sigma_2 = \{x^2+y^2\le 3, z=3\}$. In addition, be careful with the orientations on those disks.