# Extended divergence theorem

1. Sep 18, 2013

### Gauss M.D.

"Extended" divergence theorem

...which enables us to calculate the outward flux of a singular vector field through a surface S by enclosing it in some other arbitrary surface and looking at the inward flux instead.

Is there any other application of this apart from the special case when div(F)=0 you can, informally speaking, "ignore" the singularity?

2. Sep 19, 2013

### lavinia

Taking integrals over arbitrary surfaces is not easy. If you surround the singularity with a second surface that is easy, e.g. a small sphere then the region bounded by the two surfaces contains no singularity and is divergence free. Stokes theorem now says that the flux across the first surface is negative the flux over the second. This reduces a hard integral to an easy one.