Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 3 / 12A / d

Table of contents
- i d
  - Solution i
- ii d
  - Solution ii
- iii d
  - Solution iii

i

Solution

Rotational invariance makes this an isotropic vector, hence it is

0

; oddness gives the same result.

ii

Solution

It equals

C δ_{ij}

. Taking the trace gives

3 C = 4 π a^{4} e^{- a^{2}}

, hence

C = 4 π a^{4} e^{- a^{2}} /3

.

iii

Solution

This isotropic rank-three integral is

0

by oddness under

x \mapsto - x

.

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