Could someone try and explain with the differential form means? I've only taken p to calculus 2 so I'm not really sure what divergence in the sense of this equation means. Also what is the difference in the two. I mean the integral form looks at an electric field and charge over a region, so what does the differential form represent?
Always in physics we use to write equation in differential form because they explain how it is the behavior of a system locally... the div operator acting on a vector field suggest us if the field is divergenceless (solenoidal) or not. Physically this mean that it counts the source or sink of the field. the abjective soleinoidal come from the magnetic field B, in fact we know that divB=0!! To obtain the integral equation we just use the divergence theorem, and we obtain the relation with the flux of the field trough a surface... i think you can find many things on wiki or somewhere else that explain things better than me. regards marco
The differential forms of Maxwell's equations, like Gauss's Law help tell us how a field behaves at a point, which the integral forms cannot tell us about. Other than this one difference, they describe the same physical phenomena. Whether you use one form or another depends on how useful that form is to the problem your working on. The divergence is just a vector derivative: [tex]\nabla\cdot\vec{v}= \frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z}[/tex] In fact, just as there are two ways to multiply vectors (dot and cross products), there are two ways to differentiate them. You can take a vector's divergence, or it's curl. Both are derivatives, but they tell you different things. The divergence tells you how much the vector field diverges from a point, i.e. the electric field from a positive point charge had a high divergence. (It always points away from the source of the field. In other words, it diverges from that point.) On the other hand, the magnetic field of an infinite current-carrying wire, which loops around the wire has zero divergence (the field lines are always the same distance from the wire), but they have a high curl (the field lines "curl" back on themselves.)
The differential form and the integral form are the same thing. Once you learn the divergence theorem it will be plain as daylight. Edit: Of course the differential form is much more pretty, and describes what is happening as a point, as others already mentioned.
You find the differential form of Maxwell's equatons prettier? I find the integral forms easier to understand and more enlightening.
think of the differential form as exactly the same as the integral form, but with the milli-smidgen of volume of the little cube being arbitrarily small. it's the same thing.