 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/ia/paper-3/12a/a/solution

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

r = (x_{1}^{2} + x_{2}^{2} + x_{3}^{2})^{1/2}

. Away from the origin the partial derivatives of

ψ = 1/ r

are

\partial_{i} ψ = - \frac{x _{i}}{r ^{3}}, \partial_{i}^{2} ψ = - \frac{1}{r ^{3}} + \frac{3 x _{i}^{2}}{r ^{5}} .

\nabla^{2} ψ = - \frac{3}{r ^{3}} + \frac{3 ( x _{1}^{2} + x _{2}^{2} + x _{3}^{2} )}{r ^{5}} = 0 (r > 0) .

Thus

1/ r

is a harmonic function on the punctured space and solves Laplace equation there. The origin is excluded: the singularity cannot be treated as an ordinary harmonic point.

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