Consider a region $V$ bounded by the paraboloid $z=5-4x^2-4y^2$ and the $xy$-plane.
The surface integral for the vector field $$\vec{F}=\bigtriangledown\times \vec{G}=2\vec{i}+2y^2\vec{j}+z\vec{k}$$ over the circle in the $xy$-plane is 15.
What is the value for the surface integral over the paraboloid.
I know that the answer is -15. But I have no idea that how to find out the answer.Would you mind to give some advice for solve this question? Thank you so much.
This kind of problem screams for integration by parts, in particular, the use of one of the theorems that replaces a surface integral with a volume integral. The fact that the flux through the bottom part of the surface is $15$ and through the rest is $-15$ should suggest that this has to do with the divergence of the flow of the vector field, because total flux zero means that its flow doesn't create any net volume in the region. Indeed, your vector field is the curl of another, and $\nabla\cdot\nabla\times = 0$. So ... can you put the pieces together to finish the problem?
