Evaluate the surface integral: I = $\int \int_S \ F \cdot n \ dS$ over the surface of the sphere described by $x^2 + y^2 + z^2 = 16$ if $F = xy^2i + yz^2j+ zx^2k$

Attempt at problem: I use the divergence theorem: $\int \int_S \ F \cdot n \ dS = \int \int \int_V \nabla \cdot F \ dV$ I found $\nabla \cdot F$ to be $x^2 + y^2 + z^2 + 2xz + 2xy + 2yz$. All of this implies $\int \int \int_V x^2 + y^2 + z^2 + 2xz + 2xy + 2yz \ dV $ Since there is a sphere, the integral becomes (converting rectangular to spherical coordinates) $\int_0^{\pi} \int_0^{2\pi} \int_0^4 [r^2(r^2 + 2r^2(\cos{\theta}\sin{\phi}\cos{\phi} + \cos{\theta}\sin{\theta}\sin^2{\phi} + \sin{\theta}\sin{\phi}\cos{\phi})]$

  • $\begingroup$ You have miscalculated the divergence of $F$. $\endgroup$ – Paul Sinclair Oct 19 '15 at 1:13
  • $\begingroup$ How so? I believe I also should have got a numerical value. $\endgroup$ – WAS Oct 19 '15 at 1:22
  • $\begingroup$ $\nabla \cdot F = x^2 + y^2 + z^2$ I have no idea where you got the other terms from. $\endgroup$ – Paul Sinclair Oct 19 '15 at 1:26

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.