# Maxwell’s Equations¶

This is only a summary of the Maxwell’s equations under the language of Differential Geometry. There is no intention to introduce in detail — you are expected to cover them in your Electrodynamics courses.

First, we have the electromagnetic tensor \(F_{\mu\nu}\) satisfying

The Maxwell’s equations can be expressed using \(F_{\mu\nu}\) as

You can verify that the above equations indeed imply the ordinary form of Maxwell’s equation. The second equality should imply that the tensor \(F_{\mu\nu}\) can be expressed as

where \(A_\mu\) is a “vector” (dual vector or 1-form to be precise). It is easily identified that \(A_\mu\) represents the electromagnetic potential. With the above construction, the second equation of the Maxwell’s equations automatically holds, and hence the Maxwell’s equations is reduced to

It is easily verified that the Lagrangian of the electromagnetic field is

If you find any of the above content unfamiliar, please find and review a textbook about Electrodynamics, and make sure you have fully understood all the contents before proceeding.