Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2003 / Paper 2 / 10

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

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.

b

Write

c = cos θ

s = sin θ

. Since

θ

is constant, the multivariable chain rule gives

\partial_{x} = c \partial_{u} - s \partial_{v}

and

\partial_{y} = s \partial_{u} + c \partial_{v}

. Applying these operators twice to a twice continuously differentiable function gives

g_{xx} = c^{2} g_{uu} - 2 cs g_{uv} + s^{2} g_{vv}, g_{yy} = s^{2} g_{uu} + 2 cs g_{uv} + c^{2} g_{vv} .

The mixed terms cancel and

c^{2} + s^{2} = 1

, so

g_{xx} + g_{yy} = g_{uu} + g_{vv} .

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