Past exam of the mathematics course of the University of Cambridge / 2006 / ia / Paper 3 / 4A (Vector Calculus)

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- Solution 4A

Solution

Use suffix notation, summing repeated indices, and write

\partial_{i} = \partial / \partial x_{i}

. For a twice continuously differentiable vector field, mixed derivatives commute. Thus the divergence of its curl satisfies

\partial_{i} (ϵ_{ijk} \partial_{j} v_{k}) = ϵ_{ijk} \partial_{i} \partial_{j} v_{k} = 0,

because the derivatives are symmetric in

i, j

, while the Levi-Civita symbol is antisymmetric in those indices. Hence

\nabla \cdot (\nabla \times v) = 0 .

For the second identity, use the contraction of two Levi-Civita symbols:

[v \times (\nabla \times v)]_{i} = ϵ_{ijk} v_{j} ϵ_{k l m} \partial_{l} v_{m} = (δ_{i l} δ_{jm} - δ_{im} δ_{j l}) v_{j} \partial_{l} v_{m} = v_{j} \partial_{i} v_{j} - v_{j} \partial_{j} v_{i} = \partial_{i} (\frac{1}{2} v_{j} v_{j}) - [(v \cdot \nabla) v]_{i} .

Rearranging each component proves the vorticity cross-product identity

(v \cdot \nabla) v = \nabla (\frac{1}{2} ∣ v ∣^{2}) - v \times (\nabla \times v) .

The second identity needs first derivatives; the first additionally uses equality of the mixed second derivatives.

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