I don't like this jargon because I think its not rigorous. But I've seen respectable people use it, so I'm beginning to wonder: is there a mathematical reason for this being true?
People say Gauss's law for magnetism can be proven because lines are born with charges. Because there are no magnetic charges the integral $$\int\int_{surface} \mathbf{B} \cdot \mathbf{dA}=0$$ is zero and this can be shown by counting the lines that come in and out of the surface. Is this rigorous enough?
Thanks.
Counting flow lines is not going to work halfway rigorously, of course -- there are infinitely many of them.
What does work is to say that if the flow lines that cross surface $A$ are the same ones that cross surface $B$, then the flux through the two surfaces are the same. In that case, whenever we select a subset of all flow lines that we declare "count" there will be as many of them through $A$ as through $B$, also if the count is finite.
In particular, we can choose $A$ and $B$ as the regions of a (possibly closed) surface where the flow points inward and outward, respectively. Then we can conclude that if every flow line that goes "in" through a surface also comes out and vice versa (such that any single flow line has its beginning and end on the same side of the surface), then the total flux is zero.
(This all assumes that the vector field in question has zero divergence in the region of interest, of course).