Playing with vectors!
If two vectors and have the same divergence and curl at every point in volume V and have the same normal component at every point on the surface S enclosing V .
Then , and are equal everywhere .
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1 solution
The divergence and curl are partial derivatives, so there could be a constant term present in one vector field and not the other, that would cancel out of the curl and divergence. So equivalence of the divergence and curl is not enough to ensure that the vector fields are everywhere equal. However, such a constant term would presumably make the surface normal components unequal (given equal curl and divergence). Since the curl, divergence, and surface normal components are all equal, the fields should be the same. This is not so much a proof, as it is an educated guess.