Covariant formulation of classical electrodynamics

This course aims to give a concise, complete and mathematically intuitive description of the fundamental laws of electromagnetism, namely: Maxwell's equations, Lorentz force, electromagnetic energy momentum tensor, etc.

The following concepts are used extensively: tensors, Minkowski metric, lagrangian mechanics, which were introduced by the instructor in the course "Mathematical intuition behind Special and General Relativity".

The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is clearly invariant under Lorentz transformations, in the formalism of special relativity (therefore using inertial coordinate systems). These expressions simply prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, as well as provide a way to treat the fields and forces in different reference frames. However, this is not as general as Maxwell's equations in curved spacetime (i.e. non-rectilinear coordinate systems). Maxwell's equations can also be extended to curved spacetime without great effort.

We will derive Maxwell's equations in vacuum, where they can be written as two tensor equations (instead of 4 vector equations).

We will also see how to derive the electromagnetic tensor, starting from an intuitive Lagrangian approach, and also calculate the energy-momentum-tensor related to electromagnetic fields, by recalling some expressions derived in the course on General Relativity ("Mathematical Intuition behind Special and General Relativity").

Note (September 2021): I have improved the speaking fluency of the entire course, which should now be easier to follow. When I created this course, I focused more on the concepts rather than on the appearance of the course or speech fluency. However, the way concepts are delivered is also very important, so I deem this improvement to be relevant.