Chapter 15, Problem 30RE

Let G be a solid with the surface $\sigma$ oriented by outward unit normals, suppose that $\varphi$ has continuous first and second partial derivatives in some open set containing G, and let $D_{\text{n}}\varphi$ be the directional derivative of $\varphi$ , where n is an outward unit normal to $\sigma$ . Show that ${\displaystyle \underset{\sigma }{\iint }{D}_{\text{n}}\varphi dS={\displaystyle \underset{G}{\iiint }\left[\frac{{\partial }^2\varphi }{\partial x^2}+\frac{{\partial }^2\varphi }{\partial y^2}+\frac{{\partial }^2\varphi }{\partial z^2}\right]}}dV$