# Facing issue in understanding some concept. Gauss Divergence Theorem problem.

Question. If $$F = (x^2+y-4)i + 3xyj + (2xz+z^2)k$$

Evaluate $$\iint(\nabla\times F)\cdot n\ dS$$

where $$S$$ is the surface of the sphere $$x^2+y^2+z^2=16$$ above the xy-plane.

If we use Gauss Divergence Theorem, then $$\nabla\cdot(\nabla\times F)$$ will be divergence of curl of F that is $$0$$. So I'm unable to understand how to proceed in this question.

Kindly guide me for the same. Thank you.

• Divergence theorem is not the intended solution - it would be Stokes' theorem, but you are welcome to calculate the curl in order to use divergence theorem instead. – Ninad Munshi Jul 22 at 2:22
• @NinadMunshi Yes actually I've been working over problems in the segment of divergence theorem for now – Jamie Jul 22 at 2:47
• To apply the Divergence Theorem, you need a closed surface. – Ted Shifrin Jul 22 at 4:02
• It is a direct application of Stokes' theorem but if this is a problem in the segment of divergence theorem, please first close the surface with a disk at $z = 0$. Now apply divergence theorem. The net flux is zero but it includes the disk at $z = 0$ and so find flux through the disk at $z = 0$ and subtract from the total flux (which is zero). To find flux through the disk, first find the curl of the vector field. Let us know if you get stuck. – Math Lover Jul 22 at 6:54
• Thank you @MathLover I understand the concept now and also the solution – Jamie Jul 22 at 7:54