Is there any sort of intuitive relationship between vector fields with the same curl? For example, indefinite integrals, (functions that have the same derivatives), are all related such that they are all off by a constant.
But there doesn't seem a clear cut relationship between vector fields with the same curl. For instance, in two dimensions the vector fields $\langle -y,0\rangle$ and $\langle 0,x\rangle$ both have the same two dimensional curl of $1$. But I cannot see a relationship between these two fields. Does one exist?