 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2019-resit/8/a/solution

For a bounded volume

V

with outward-oriented piecewise smooth boundary

S

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

Here

G

is a continuously differentiable vector field on a neighborhood of

V

n

the outward unit normal vector, and

d V

the volume element. The left side is its outward flux.

For a twice continuously differentiable field

F

, equality of mixed partial derivatives gives

\nabla \cdot (\nabla \times F) = (F_{z, y x} - F_{y, z x}) + (F_{x, zy} - F_{z, x y}) + (F_{y, x z} - F_{x, yz}) = 0.

G = \nabla \times F

to obtain

\iint_{S} (\nabla \times F) \cdot d S = 0.

The regularity hypotheses justify differentiating and integrating the curl; a field singularity cannot simply be ignored in a volume argument.

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