This question is related to the answer in this post: https://physics.stackexchange.com/questions/776112/showing-that-the-power-is-p-ui-using-the-poynting-vector
There it is said that:
The first term is 0 because the curl of a vector has no divergence and the surface integral of a divergence-free field is 0.
I understand that this follows from the divergence theorem, because the volume integral of the divergence is zero and this equals the surface integral. The problem here is that the function is not defined inside the entire surface. So does this hold in some other way than using the divergence theorem? Because it is divergence free at the surface, but the field is not defined some places inside the surface?, so we can't take the volume integral?