Chapter 6.10, Problem 16P

The following equations are variously known as Green’s first and second identities or formulas or theorems. Derive them, as indicated, from the divergence theorem.

(1) ${\displaystyle {\int }_{\begin{array}{c}\text{valume }\tau \\ \text{ inside o }\end{array}}\left(\varphi {\nabla }^2\psi +\nabla \varphi \cdot \nabla \psi \right)}d\tau ={\displaystyle {\oint }_{\begin{array}{c}\text{clased }\\ \text{ surface }\sigma \end{array}}(\varphi \nabla \psi )}\cdot nd\sigma$.

To prove this, let $V=\varphi \nabla \psi$ in the divergence theorem.

(2) ${\displaystyle {\int }_{\text{volumer }\tau }\left(\varphi {\nabla }^2\psi -\psi {\nabla }^2\varphi \right)}d\tau ={\displaystyle {\oint }_{\text{linitide }}(\varphi \nabla \psi -\psi \nabla \varphi )}\cdot$

n $d\sigma$.

To prove this, copy Theorem 1 above as is and also with $\varphi$ and $\psi$ interchanged; then subtract the two equations.