Where is the Curl operator in the PDE physics?

Hello,

I tried to modify the PDE from the rotating machinery physics a little and wanted to define it in the PDE physic.
The problem I have is that I can't see any curl operators in the coeficient or the general form. The original equations are:

sigma* dA/dt + curl(H) - sigma v x B = J
and
B= curl(A)

I'm sure there is a way to define both equations. Maybe I could use the diveregence operator with a matrix or something else...

Thanks in advance