The Laplace operator \(\Delta\) is defined by
\[
\Delta \varphi=\frac{\partial^{2} \varphi}{\partial x^{2}}+\frac{\partial^{2} \varphi}{\partial y^{2}}
\]
For any vector field \(\mathbf{F}=\left\langle F_{1}, F_{2}ightangle\), define the conjugate vector field \(\mathbf{F}^{*}=\left\langle-F_{2}, F_{1}ightangle\).

Show that if \(\mathbf{F}=abla \varphi\), then \(\operatorname{curl}_{z}\left(\mathbf{F}^{*}ight)=\Delta \varphi\).