Unfortunately I don't have a good reference for you, but I remember these two proofs. They both follow a similar procedure using the divergence theorem and Stokes theorem around arbitrary orientable closed surfaces and closed plane curves respectively. Start with the second one, it's a little simpler.
I think this is called Helmholtz's theorem in E&M (Electricity and Magnetism). The div(curl A)=0 in all cases and also curl (grad V)=0 in all cases, but the converse that there exists a field, etc. is Helmholtz's theorem.