Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 32C / c / iii

Table of contents
- Solution iii

Solution

For

f (x) = x^{- 2}

,

pr^{(3)} V_{2} (S - x^{- 2}) = - 2 S - x \frac{d}{d x} (x^{- 2}) = - 2 S + 2 x^{- 2},

which vanishes when

S = x^{- 2}

. The corresponding expressions for

V_{1}

and

V_{3}

are nonzero on the equation. Thus $S [u] = x^{- 2}$ has the dilation symmetry $V_{2} = x \partial_{x}$ among the three listed generators.

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