 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-3/36b/c/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

p (t) = \int_{R^{3}} x ρ (x, t) d^{3} x .

Differentiate and use the continuity equation

\overset{ρ}{˙} + \nabla \cdot J = 0

\overset{p}{˙}_{i} = - \int x_{i} \partial_{j} J_{j} d^{3} x = \int J_{i} d^{3} x - \int_{\partial R^{3}} x_{i} J_{j} n_{j} d S .

Localization makes the boundary integral vanish. Component by component,

\dot{p} (t) = \int_{R^{3}} J (x, t) d^{3} x .

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