Use the Divergence Theorem to calculate the surface integral S F · dS; that is, calculate the flux of F across S. F(x, y, z) = x2 sin(y)i + x cos(y)j − xz sin(y)k, S is the "fat sphere" x8 + y8 + z8 = 8.
Accepted Solution
A:
[tex]\vec F(x,y,z)=x^2\sin y\,\vec\imath+x\cos y\,\vec\jmath-xz\sin y\,\vec k[/tex]has divergence[tex]\mathrm{div}\vec F(x,y,z)=2x\sin y-x\sin y-x\sin y=0[/tex]so that the flux of [tex]\vec F[/tex] across [tex]S[/tex] is 0 by the divergence theorem.