 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2005/paper-1/6/a/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

Let

V \subset R^{3}

be a bounded region with a piecewise smooth closed boundary

S = \partial V

, and let

F

have continuous first partial derivatives on a neighborhood of its closure. The divergence theorem states

\int_{V} \nabla \cdot F d V = \int_{S} F \cdot n d S,

where

n

is the outward unit normal vector. The right side is the outward flux; reversing the orientation reverses its sign.

 New to topics? Read the docs here!