# Electromagnetic field tensor: cyclic derivative relation

Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 7; Problem 7.6.

We’ve used the following relation between the derivatives of the electromagnetic field tensor ${F^{ij}}$ to get several of Maxwell’s equations.

$\displaystyle \partial_{i}F_{jk}+\partial_{k}F_{ij}+\partial_{j}F_{ki}=0 \ \ \ \ \ (1)$

Here we verify that this relation is true when ${F^{ij}}$ is written in terms of the four-potential, that is

$\displaystyle F^{ij}=\partial^{i}A^{j}-\partial^{j}A^{i} \ \ \ \ \ (2)$

We can lower both indices in this equation and the plug it into the first equation:

 $\displaystyle \partial_{i}F_{jk}+\partial_{k}F_{ij}+\partial_{j}F_{ki}$ $\displaystyle =$ $\displaystyle \partial_{i}\left(\partial_{j}A_{k}-\partial_{k}A_{j}\right)+\partial_{k}\left(\partial_{i}A_{j}-\partial_{j}A_{i}\right)+\partial_{j}\left(\partial_{k}A_{i}-\partial_{i}A_{k}\right)\ \ \ \ \ (3)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \partial_{i}\partial_{j}A_{k}-\partial_{j}\partial_{i}A_{k}+\partial_{k}\partial_{i}A_{j}-\partial_{i}\partial_{k}A_{j}+\partial_{j}\partial_{k}A_{i}-\partial_{k}\partial_{j}A_{i}\ \ \ \ \ (4)$ $\displaystyle$ $\displaystyle =$ $\displaystyle 0 \ \ \ \ \ (5)$

The terms in the second line cancel in pairs since the order of the partials doesn’t matter.