Given $u(x,y)$, show that there exists a $v(x,y)$ satisfying \begin{equation} \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\quad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}\end{equation} only if $u$ satisfies the Laplacian equation $u_{xx} + u_{yy}=0$.
I am sorry to bother you with this. I don't know what my professor is asking here. Could someone give me some direction on this?
If you're trying to prove the two equations imply it solves Laplace's equation, then: $$u_{xx}=(u_x)_x=v_{yx}, u_{yy}=(u_y)_y=(-v_x)_y=-v_{xy}$$ And so summing these up, and on the lenient condition that $v_{xy}=v_{yx}$: $$u_{xx}+u_{yy}=v_{yx}-v_{xy}=0$$