Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 305 / 3 / b / i / Solution

For $U(x)\in SU(3)$ and $A_{\mu }=A_{\mu }^a{T}^a$, choose $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$. The Yang-Mills gauge transformation iswhich ensures $D_{\mu }^{\prime }(U\Phi )=U{D}_{\mu }\Phi$. Sinceconjugating this commutator immediately givesThus the gauge field strength is gauge covariant.