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see also gauge_theory and Gauge Symmetry
\begin{equation} \partial^{\mu}F_{\mu\nu}^a-gf_{abc}A^{\mu}_bF_{\mu\nu}^c=0 \end{equation}
In physics, of course, Maxwell's equations of electromagnetism are linear partial differential equations. Their counterparts, the famous Yang-Mills equations, are non-linear equations which are supposed to govern the forces involved in the structure of matter. The equations are non-linear, because the Yang-Mills equations are essentially matrix versions of Maxwell's equations, and the fact that matrices do not commute is what produces the non-linear term in the equations.
The Aleph of Space by Luciano Boi