Chapter 13.4, Problem 55PS

a.

To determine

To find: The given expressions ${\displaystyle \underset{D}{\iint }\left\{\left(f{\nabla }^{2}g\right)+\left(\nabla f\right)\left(\nabla g\right)\right\}dxdy}={\displaystyle {\oint }_{C}f\frac{\partial g}{\partial n}ds}$ .

b.

To determine

To find: The given expressions ${\displaystyle \underset{D}{\iint }\left\{f{\nabla }^{2}g-g{\nabla }^{2}f\right\}dxdy}={\displaystyle {\oint }_C\left(f\frac{\partial g}{\partial n}-g\frac{\partial f}{\partial n}\right)ds}$ .