Say I have a vector field defined on a space such as Actually, it is AdS space im working on but I don't think that matters for what I want to ask. I assume that I can write the coordinates as (time,radial, p-angles)

Now if I take the integral of the divergence of the field

then by the divergence theorem this gives zero if K dies of sufficiently quickly at spatial (and temporal)infinity i.e. (r --->infinity)

Everything works as I want it to if I take it to die of as 1/r or quicker. Unless I have made a mistake, then any other possibility doesn't work like I think it should.

Can any one reassure me about this.