 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-2/6/b/ii/solution

Under inversion

x \mapsto - x

, the spherical shell and its volume element are unchanged, while the product of three scalar products changes sign. Its integral is therefore its own negative:

\int_{V} (x \cdot n_{1}) (x \cdot n_{2}) (x \cdot n_{3}) d^{3} x = 0 .

Equivalently, its rank-three moment tensor is completely symmetric and cannot have a nonzero multiple of the antisymmetric Levi-Civita symbol.

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