I had doubts whether to post this here or in in the physics section but I did here because I'm more interested in a purely mathematical view in this case. I understand a solenoidal vector field implies the existence of another vector field, of wich it is the curl: [tex]S=\nabla X A[/tex] because the divergence of the curl of any vector field is zero. But what if the vector field is conservative instead? I guess in this case it is not necessarly implied the existence of a vector potential. How about in the case of a laplacian vector field, that is both conservative and solenoidal, does it imply the existence of a vector potential?