Question

Compute the surface integral over the given oriented surface: F = <0,8,x2>,hemisphere x2+y2+z2=81 , z>=0 , ,outward-pointing normal.

Answer 1

Writing r(u,v) = <9 cos u sin v, 9 sin u sin v, 9 cos v> with v = [0, p/2].
r_u x r_v = <-81 cos u sin^2(v), -81 sin u sin^2(v), -81 sin v cos v>
==> n = <81 cos u sin^2(v), 81 sin u sin^2(v), 81 sin v cos v> so n points out of S.

So, ??s F · dS
= ? v=0 to pi/2? u= -pi to +pi <0, 8,( 9 cos u sin v)2> · <81 cos u sin^2(v), 81 sin u sin^2(v), 81 sin v cos v> dvdu


u could follow this one

Answer 2

?? F · dS
= ??? -?·F dV ....... by the divergence theorem (orientation to the origin)
... ?·F = ?/?x x + ?/?y (-z) + ?/?z y = 1
= -1 ??? dV
= -1/8 (4/3) p (8)³ ....... sphere: r=8 in the first octant, V = 1/8 (4/3) p r³
= -256p/3