I am working with the covariant derivative and trying to show that the commutator of this derivative $[D_\mu , D_\nu]$ is proportional to the field $F_{\mu \nu}$. That is, I need the final term to be have $(\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu})$ contained in the answer (proportional to the field $F_{\mu\nu}$. Instead the commutator appears to be zero! When I work it all out I get
$\left[ { D }_{ \mu },{ D }_{ \nu } \right] =\left( { \partial }_{ \mu }-iq{ A }_{ \mu } \right) \left( { \partial }_{ \nu }-iq{ A }_{ \nu } \right) -\left( { \partial }_{ \nu }-iq{ A }_{ \nu } \right) \left( { \partial }_{ \mu }-iq{ A }_{ \mu } \right) ={ \partial }_{ \mu }{ \partial }_{ \nu }-{ q }^{ 2 }{ A }_{ \mu }{ A }_{ \nu }-iq\left( { \partial }_{ \mu }{ A }_{ \mu }+{ \partial }_{ \nu }{ A }_{ \mu } \right) -{ \partial }_{ \nu }{ \partial }_{ \mu }+{ q }^{ 2 }{ A }_{ \nu }{ A }_{ \mu }+iq\left( { \partial }_{ \nu }{ A }_{ \mu }+{ \partial }_{ \mu }{ A }_{ \mu } \right) =0$
I have checked this over and over but I cannot get it. The answer is in the book but I'm starting to think maybe the book is wrong? Other books show it is not, I am definitely missing something. If someone can point me to what I'm missing please?