This is how the differential Maxwell’s equations look like using vector notation:
It turns out that these equations can be written in a more compact form:
known as the differential geometric formulation.
The main idea to derive this alternative formulation is as follows. First, we define the electromagnetic tensor which contains the components of both and :
This matrix is skew-symmetric, so we can encode it as a 2-form:
where the 2-forms and encode the entries of vectors and in the following way:
Next, one defines an operation known as Hodge dual and checks that
Finally, one has to define the exterior derivative and check that
Using these equations we can rewrite and in terms of and and verify that the resulting equations are equivalent to the original Maxwell’s equations.
For more details see: Differential geometric formulation of Maxwell’s equations.