 Solution

The derivatives needed for varying $A_{\nu }$ areTherefore the electromagnetic Euler-Lagrange equation isAntisymmetry of $F$ and commutation of derivatives imply ${\partial }_{\nu }{\partial }_{\mu }{F}^{\mu \nu }=0$, hence ${\partial }_{\nu }{J}^{\nu }=0$. In ordinary derivatives, $J^{\mu }=j^{\mu }-2q^2{A}^{\mu }|\varphi {|}^2$; the interaction contribution must not be omitted.