Maxwell's equations in differential form notation appeared as a motivating example in a mathematical physics book I'm reading. However, being a mathematical physics book it doesn't delve much into the physical aspects of the problem. It deduces the equations by setting dF equal to zero and d(*F) equal to J, but it doesn't explain why it is doing so. This article I found online does exactly the same thing. I'm left with the impression that this condition is just set in order to obtain the equations. Is that so? If not, is there any physical interpretation here to the exterior derivative? And what about the exterior derivative of the Hodge operator applied to the tensor being equal to the current density?