How are the differential forms for Maxwell's Equations used?

Question

I am currently reading up on Maxwell's Equations (specifically Ampere's Circuital Law- with Maxwell's Addition) for a presentation on differential equations.

I chose the topic ignorant of how the differential form of these equations are used, and I cannot seem to find a digestable use of their differential form anywhere.

My understanding of the differential forms is that their mathematical representations in these forms are easier to grasp than their integral forms (they make more physical sense). However, I am certain they are used for more than just this, but I cannot seem to find any of examples of this.

That being said, my questions are:

1. Are the differential forms for Maxwell's Equations used in practice?
2. How are they used? (If possible, please provide a mathematical model and/or link to its derivation or result)

In particular, I am focused on Ampere's Law, so answers involving just this equation is okay too.

Answer 1

The integral forms of Maxwell's equations are fairly useless unless you have situations with very high degrees of symmetry and/or fields aligned along co-ordinate axes. e.g. The beloved examples of undergraduate physics everywhere of spherical and cylindrical charge and current distributions.

Once you move away from these situations then the integral forms become extremely difficult to use in practice because they do not apply at a point. If you wish to numerically solve the equations then it is far easier to do that starting off with a set of differential equations that are already in the form that are amenable to solving on a "grid".

A second reason to move to the differential forms is to show how electromagnetic waves can exist and can be generated from accelerating charge and current distributions.

The differential forms also allow you to intuitively grasp some aspects of electromagnetism far more easily. e.g. If I ask you whether the field $\vec{B} = x\vec{i}$ is a valid description for a magnetic field, it is far easier to say that it can't be because the divergence is non-zero than to perform closed surface integrals for (potentially) an infinite number of possible closed surfaces.