 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-3/13c/e/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

For

P (x) = Z^{- 1} exp (- \frac{1}{2} D_{ij} x_{i} x_{j})

with symmetric

D

\frac{\partial P}{\partial x _{j}} = - D_{jk} x_{k} P .

Substitution into the current from part (d) yields

J_{i} = (a_{ik} + \frac{1}{2} b_{ij} D_{jk}) x_{k} P .

It vanishes for all

x

exactly when

a + \frac{1}{2} b D = 0, D = - 2 b^{- 1} a .

Thus the stated multivariate normal distribution is a stationary solution whenever the zero-current assumption makes

D

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