 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2003/paper-2/4/a/solution

 Top articles  Latest articles New article in topic

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2003 / Paper 2 / 4 / a / Solution by

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

For a bounded volume

V

with piecewise smooth closed boundary

\partial V

, oriented by the outward unit normal vector, and a continuously differentiable vector field on a neighborhood of

\overline{V}

, the divergence theorem states

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

For an oriented piecewise smooth surface

S

with piecewise smooth boundary

C = \partial S

, and a continuously differentiable vector field on a neighborhood of

S

, Stokes theorem states

\int_{S} (\nabla \times F) \cdot n d S = \oint_{C} F \cdot d r .

The boundary orientation follows the right-hand rule relative to the chosen normal vector. Reversing that orientation reverses both sides of the latter identity.

 Read the full article

 New to topics? Read the docs here!