Question

Apply Gauss’s Divergence theorem to evaluate ∬(lx^2+my^2+nz^2 )dS taken over the sphere (x-a)^2+(y-b)^2+(z-c)^2=ρ^2;l,m,n being the direction cosines of the external normal to the sphere.

Answer 1

Evaluation of given sphere is 8πe³ (a +b +c).  

Step-by-step explanation:  

• By Gauss divergence theorem\\ \iint_s\bar{F.\bar{n}}\space ds=\iiint_v\space div\space dv∬   F.  n ˉ ˉ   ds=∭  v div dv

• where \bar{F}.\bar{n}=lx^2+my^2+nz^2

• let \bar{n}=li +mj + nk\space then \space \bar{F}=xi^2+y@j+n62\bar{k}  

• n  =li+mj+nk then  
•  div F = 2(x + y+ z).  

• use spherical coordinate

       x-a=sin \theta\space cos\theta\\ y-b=sin \theta\space sin\theta\\z-                

   c=cos\thetax−a=sinθ cosθ  

       y−b=sinθ sinθ

          z−c=cosθ  

• dx\space dy\space dz=sin \theta\space cos \thetadx dy dz=sinθ cosθ

      and 0≤\theta≤\pi,0=\theta≤2\pi,0≤e0≤θ≤π,0=θ≤2π,0≤e

• By doing tipple integration with x, y, z  we get,  (8/3)  bπe³

•       By adding these three integrals we get, 8πe³ (a +b +c).