 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2017/8/b/solution

\iint_{S} F \cdot n d S = ∭_{V} \nabla \cdot F d V

for a sufficiently smooth field and a piecewise smooth closed boundary with outward unit normal. Choose

F = r /3

. Since

\nabla \cdot r = 3

V = \frac{1}{3} \iint_{S} n \cdot r d S .

The factor

1/3

comes from the three spatial dimensions; the outward orientation is especially important on the inner wall of the drilled solid.

 New to topics? Read the docs here!