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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.

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    $\begingroup$ 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. $\endgroup$ – Ninad Munshi Jul 22 at 2:22
  • $\begingroup$ @NinadMunshi Yes actually I've been working over problems in the segment of divergence theorem for now $\endgroup$ – Jamie Jul 22 at 2:47
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    $\begingroup$ To apply the Divergence Theorem, you need a closed surface. $\endgroup$ – Ted Shifrin Jul 22 at 4:02
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    $\begingroup$ 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. $\endgroup$ – Math Lover Jul 22 at 6:54
  • $\begingroup$ Thank you @MathLover I understand the concept now and also the solution $\endgroup$ – Jamie Jul 22 at 7:54

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