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

\[E_i = F_{0i}, \ \ \ \ \ B_i = -\frac{1}{2}\varepsilon_{ijk}F^{jk}\]

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

\[\begin{split}\partial^\mu &F_{\mu\nu} = 0\\ \partial_{[\mu} &F_{\rho\sigma]} = 0\end{split}\]

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

\[F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu\]

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

\[\partial^\mu F_{\mu\nu} = 0\]

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

\[\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\]

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.