Among the fundamental facts of physics that were first recognized by Newton, are the observations that certain entities are conserved; they do not change. Moreover, they do not appear somewhere and disappear somewhere else hazardly. They change only by moving continuously or "flowing" from one place to another. Quantities that are conserved in this way include: matter (in most circumstances), energy, electric charge, momentum, angular momentum, and others as well.
Let C be a conserved quantity. (Think of electric charge)
Given a tiny region of space, of volume dV we define the amount of C in it to be rdV; we define jdV to be the amount of C in it times its average velocity in this volume. Thus j = v, where v is the average velocity of the C in dV.
It is given by the flow integral of j over C, and thus by over S.
The statement of local conservation of C has the form:
the change in the quantity of C in V = -outflow from V + nflow to V or,
which we write as
The divergence theorem tells us
Putting the divergence theorem and the conservation law together we get for any volume V:
Physicists add the final leap to the differential form of the conservation law: they say that if this is true for any volume at all it is true everywhere, so they conclude with:
This equation applies no matter what conserved entity C represents. For our purposes below it will be electric charge.
Application to Electric and Magnetic Phenomena
Exactly the same chain of reasoning can be used to convert the four empirical laws: Gausses Theorem of electrostatics, Faraday's Law of Induction, Ampere's Law for steady currents, and the absence of sources and sinks of magnetic fields, into differential equations.