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 Incoming links: Hessian of the Euclidean norm

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2003 / Paper 2 / 10 / a / Solution Created 2026-10-18 Updated 2026-10-20

Away from the origin, the chain rule gives

r_{x} = x / r

and

r_{y} = y / r

. Differentiating once more yields the two-dimensional case of the Hessian of the Euclidean norm:

r_{x} = \frac{x}{r}, r_{y} = \frac{y}{r}, r_{xx} = \frac{y ^{2}}{r ^{3}}, r_{yy} = \frac{x ^{2}}{r ^{3}}, r_{x y} = - \frac{x y}{r ^{3}} .

These partial derivatives are smooth wherever

r > 0

, so Clairaut's theorem gives

r_{y x} = r_{x y} = - x y / r^{3}

. None of these formulas supplies a derivative at the origin: even the restriction

r (x, 0) = ∣ x ∣

has no two-sided derivative there.

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