 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2011/ia/paper-3/12c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Use suffix notation, with repeated indices summed. Contracting the two Levi-Civita symbols gives

[u \times (\nabla \times A)]_{i} = ϵ_{ijk} u_{j} ϵ_{k l m} \partial_{l} A_{m} = u_{j} (\partial_{i} A_{j} - \partial_{j} A_{i}) .

Applying the product rule to

h = A_{i} B_{i}

and substituting the two evolution equations yields

\partial_{t} h = B_{i} u_{j} \partial_{i} A_{j} - B_{i} u_{j} \partial_{j} A_{i} + B_{i} \partial_{i} ψ + A_{i} B_{j} \partial_{j} u_{i} - A_{i} u_{j} \partial_{j} B_{i} = B_{i} \partial_{i} (u_{j} A_{j} + ψ) - u_{j} \partial_{j} (A_{i} B_{i}) .

In the second line we renamed dummy indices in

A_{i} B_{j} \partial_{j} u_{i}

and then combined product derivatives. Hence

f = u \cdot A + ψ, \partial_{t} h = B \cdot \nabla f - u \cdot \nabla h .

No zero-divergence assumption was used in this local identity.

\partial_{t} h = \nabla \cdot (f B - h u) .

Because the volume is fixed, differentiation under its integral and the divergence theorem give

\frac{d H}{d t} = \int_{\partial V} (f B - h u) \cdot n d S = 0

under the two tangency conditions. Thus the volume integral is conserved. This is helicity conservation by tangent boundary conditions; when

B = \nabla \times A

For the given axisymmetric field, the curl in spherical coordinates gives

B_{r} B_{θ} B_{ϕ} = \frac{1}{r sin θ} \partial_{θ} (sin θ A_{ϕ}) = 2 (a^{2} - r^{2}) cos θ, = - \frac{1}{r} \partial_{r} (r A_{ϕ}) = (4 r^{2} - 2 a^{2}) sin θ, = \frac{1}{r} \partial_{r} (r A_{θ}) = 3 a r sin θ .

There are no

ϕ

A_{r} = 0

. Consequently

h = A_{θ} B_{θ} + A_{ϕ} B_{ϕ} = a r^{2} (4 r^{2} - 2 a^{2}) sin^{2} θ + 3 a r^{2} (a^{2} - r^{2}) sin^{2} θ = a r^{2} (a^{2} + r^{2}) sin^{2} θ .

r^{2} sin θ d r d θ d ϕ

, so

H = 2 πa \int_{0}^{a} r^{4} (a^{2} + r^{2}) d r \int_{0}^{π} sin^{3} θ d θ = \frac{8 πa}{3} (\frac{a ^{7}}{5} + \frac{a ^{7}}{7}) = \frac{32 π a ^{8}}{35} .

Here

B_{r} = 0

r = a

, as required for the magnetic-field tangency condition in the conservation argument; tangency of

u

is a separate hypothesis.

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