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see also gauge_theory and Gauge Symmetry
\begin{align} &\frac {\partial F^{\mu\nu}_d}{\partial A_a^{\mu}}=-gf_{dac}A_c^{\nu}\\ &\frac {\partial F^d_{\mu\nu}}{\partial A_a^{\mu}}=-gf_{dac}A_c^{\nu}g_{\mu\mu}g_{\nu\nu}\\ &\frac {\partial \mathcal{L}}{\partial A_a^{\mu}}=\frac 12 gf_{abc}A_b^{\nu}F^c_{\mu\nu}\\ &\frac {\partial F^{\mu\nu}_d}{\partial \left(\partial^{\nu}A_a^{\mu}\right)}=-\delta_{ad}\\ &\frac {\partial F^d_{\mu\nu}}{\partial \left(\partial^{\nu}A_a^{\mu}\right)}=\delta_{ad}(g_{\mu\nu}g_{\nu\mu}-g_{\mu\mu}g_{\nu\nu})\\ &\frac {\partial \mathcal{L}}{\partial \left(\partial^{\nu}A_a^{\mu}\right)}=\frac 12 F_{\mu\nu}^a \end{align}
The Yang-Mills equations can be expressed with the Hodge star operator as \begin{equation} 0=d_A F=d_a *F \quad F=dA+A\wedge A \end{equation} where $d_A$ is the gauge-covariant extension of the exterior derivative. The gauge field $A$ is a one-form \begin{equation} A(x)=A_{\mu}^a(x)t^adx^{\mu} \end{equation} with the values on the Lie algebra of a compact simple Lie group $G$. The curvature is a two-form \begin{equation} \begin{gathered} F=dA+A\wedge A \\ F=F_{\mu\nu}^at^adx^{\mu}\wedge dx^{\nu}\\ F=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+f^{abc}A_{\mu}^bA_{\nu}^c \end{gathered} \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