Chapter 14.8, Problem 48E

Integration by parts (Gauss' Formula) Recall the Product Rule of Theorem 14.11: ▿ · (uF) = ▿u·F + u(▿·F).

a.    Integrate both sides of this identity over a solid region D with a closed boundary S and use the Divergence Theorem to prove an integration by parts rule:

${\displaystyle \underset{D}{\iiint }u\left(\nabla \cdot F\right)dV}={\displaystyle \underset{S}{\iint }uF\cdot ndS}-{\displaystyle \underset{D}{\iiint }\nabla u\cdot FdV.}$

b.    Explain the correspondence between this rule and the integration by parts rule for single-variable functions.

c.    Use integration by parts to evaluate ${\displaystyle {\iiint }_D\left(x^{2}y+y^{2}z+z^{2}x\right)dV}$, where D is the cube in the first octant cut by the planes x = 1, y = 1, and z = 1.