Past exam of the mathematical methods II course of the University of Oxford / 2017 resit / 3 / b

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- Solution b

Solution

Using

r = (x, y, z)

and constant

a

, the divergence is

\partial_{i} (ϵ_{ijk} r_{j} a_{k}) = ϵ_{ijk} δ_{ij} a_{k} = 0.

For the curl, the curl of a cross product identity gives

\nabla \times (r \times a) = r (\nabla \cdot a) - a (\nabla \cdot r) + (a \cdot \nabla) r - (r \cdot \nabla) a = 0 - 3 a + a - 0.

Thus

\nabla \cdot (r \times a) = 0, \nabla \times (r \times a) = - 2 a .

For a quick sign check, taking

a = e_{z}

gives

r \times a = (y, - x, 0)

, whose curl is

(0, 0, - 2)

.

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