Evaluate Surface Integral f.n ds for Sphere x^2+y^2+z^2=a^2

[anand]

Problem : Evaluate [double integral]f.n ds where f=xi+yj-2zk and S is the surface of the sphere x^2+y^2+z^2=a^2 above x-y plane.

My effort:: I know that the sphere's orthogonal projection has to be taken on the x-y plane,but I'm having trouble with the integration.Please help!

 

[Galileo]

Calculating flux integrals can be a bit tedious. Although some are very easy when you invoke the right theorem. Like this one.

 

[mjsd]

Problem : Evaluate [double integral]f.n ds where f=xi+yj-2zk and S is the surface of the sphere x^2+y^2+z^2=a^2 above x-y plane.

My effort:: I know that the sphere's orthogonal projection has to be taken on the x-y plane,but I'm having trouble with the integration.Please help!

you want to find f.n where n is obviously the normal to the surface .. find that first... the easiest way to do this is probably change to spherical coordinates...given the symmetry of the problem

 

[HallsofIvy]

Didn't we just have this question? Or was it also posted on a different board?

The vector function, f(x,y,z)= xi+ yj- 2zk, is obviously "anti-symmetric" about the origin: f(-x,-y,-z)= -(f(x,y,z)), while the region of integration, a sphere centered at the origin, is symmetric. What does that tell you?

Or you can use the "Divergence theorem" and integrate $\nabla \cdot f$ over the interior of the sphere, as Galileo suggested. Here $\nabla \cdot f$ is particularly simple.