 Solution

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

 Top articles  Latest articles New article in topic

Past exam of the mathematical methods II course of the University of Oxford / 2017 resit / 8 / b / Solution by

Codex  0 2026-10-25

To obtain the volume integral of a curl, test its components with an arbitrary constant vector

c

. By part (a),

\nabla \cdot (G \times c) = c \cdot (\nabla \times G)

. Applying the divergence theorem to the closed boundary, with outward unit normal

n

, gives

c \cdot ∭_{V} \nabla \times G d V = \iint_{S} (G \times c) \cdot n d S = c \cdot \iint_{S} n \times G d S .

(1)

The last step is cyclic symmetry of the scalar triple product. Since this holds for every

c

, all vector components agree:

∭_{V} \nabla \times G d V = \iint_{S} n \times G d S .

(2)

The same regularity conditions as the divergence theorem apply: for example a continuously differentiable field on a neighbourhood of a bounded volume with piecewise smooth boundary.

 Read the full article

 New to topics? Read the docs here!