Suppose F x33 y33 z33 Find the net outward flux s F ndS whe

Suppose F = (x^3/3, y^3/3, z^3/3 Find the net outward flux: s (F. n)dS, where n is the unit outward normal. The solid D is the solid unit sphere x^2 + y^2 + z^2 lessthanorequalto 1.

Solution

given F=<x³_/3,y³_/3.z³_/3>

divF=x²+y²+z²

in spherical coordinates

x=sincos,y=sinsin,z=cos

x²+y²+z²=²

x²+y²+z²<=1

==>0<=<=1

dV=²sin d d d

0<=<=2,0<=<=,0<=<=1

divF=²

by divergence throrem net outward flux _S(F.n) dS= (divF)dV

= _{[0 to 2] [0 to ] [0 to 1]} ^{2 2}sin d d d

= _{[0 to 2] [0 to ] [0 to 1]} ⁴ sin d d d

= _{[0 to 2] [0 to ][0 to 1]} (1_/5)⁵ sin d d

= _{[0 to 2] [0 to ]}  (1_/5)(1⁵ -0⁵)sin d d

= _{[0 to 2] [0 to ]}  (1_/5)sin d d

= _{[0 to 2][0 to ]}  (1_/5)(-cos)d

= _{[0 to 2]}   (1_/5)(-cos+cos0)d

= _{[0 to 2]}   (1_/5)(1+1)

=(1_/5)(1+1)(2-0)

=4_/5

_S(F.n) dS= 4_/5