# Surface integral

1. Mar 14, 2007

### 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!

2. Mar 16, 2007

### Galileo

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

3. Mar 16, 2007

### mjsd

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

4. Mar 17, 2007

### HallsofIvy

Staff Emeritus
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.