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

Question:

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 $\operatorname{curl}_{z}\left(\mathbf{F}^{*}ight)=\Delta \varphi$.

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