Differential Form of Maxwell's Equations

Maxwell's four equations explain the fundamentals of electromagnetism.

Applying the divergence theorem to Gauss's law and rewriting the enclosed charge in terms of the total charge density gives the differential form of Gauss's law.

Similarly, applying divergence theorem to Gauss's law for the magnetic field gives the differential form of Gauss's law in magnetism.

Furthermore, rearranging Faraday's law and applying Stoke's theorem to it gives Faraday's law in the differential form.

Consider the Ampère-Maxwell equation, in which the enclosed current can be expressed in terms of the integral of current density.

Now, rearranging the terms and applying Stoke's theorem to it gives the differential form of the Ampère-Maxwell equation.

Grouping the terms of electric and magnetic fields on one side and the sources producing these fields on the other side suggests that all electromagnetic fields are produced by charges and currents.

The integral form of Maxwell's equation applies to fields in a region containing charge or current, whereas the differential form applies at a given point with charge and current densities.