Question

Evaluate the surface integral

S

F · dS

for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation.

F(x, y, z) = x i − z j + y k

S is the part of the sphere

x² + y² + z² = 36

in the first octant, with orientation toward the origin

Answer 1

Given that F = xi-zj+yX divF = *** () + (-2) + 2()=1+0+0=1 Given surface Sis the sphere r* +*+z= 6in the first octant By Gauss divergene theorem, we get [F. Nds = [divF av = [lav S S (: First octant) 00 =xVolume of the sphere x* +y?+2 = 36 (where r = 6 = 36 00