What is the Difference Between Differential and Integral Form of Maxwell's Equations?

Both the integral and differential forms of Maxwell's equations are equivalent. As an example, consider the following Maxwell equation in differential form:

The same Maxwell equation in integral form looks like this:

As you can see: In the integral form 2 we integrate over a closed surface \( A \). The differential representation form, on the other hand, does not contain an integral, but a differential operator, the nabla operator \(\nabla\), with which the change of the electric field \( \boldsymbol{E} \) at a single space and time point can be studied.

The advantage of the integral form is that it can be used to solve problems with high symmetry, as for example in the calculation of the electric field of a charged hollow sphere, where we have a spherical symmetry.

The advantage of the differential form is that it can be written compactly and is useful in certain derivations, such as the derivation of electromagnetic waves. Also, with the differential form, it is easier to show the covariance of electrodynamics (that is, compatibility with special relativity).