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 Incoming links: Second derivatives under a two-variable coordinate change

Cartesian second derivatives in polar coordinates 2026-10-20

For

x = r cos θ

,

y = r sin θ

and

r > 0

c = cos θ

,

s = sin θ

. The differential operators are

\partial_{x} = c \partial_{r} - (s / r) \partial_{θ}

and

\partial_{y} = s \partial_{r} + (c / r) \partial_{θ}

. Applying them twice gives

f_{xx} = c^{2} f_{rr} - \frac{2 sc}{r} f_{r θ} + \frac{s ^{2}}{r ^{2}} f_{θθ} + \frac{s ^{2}}{r} f_{r} + \frac{2 sc}{r ^{2}} f_{θ},

f_{yy} = s^{2} f_{rr} + \frac{2 sc}{r} f_{r θ} + \frac{c ^{2}}{r ^{2}} f_{θθ} + \frac{c ^{2}}{r} f_{r} - \frac{2 sc}{r ^{2}} f_{θ} .

Their sum is the Laplacian in polar coordinates. The first-derivative terms come from differentiating the coordinate coefficients, as required for second derivatives under a two-variable coordinate change. The origin is excluded from the coordinate formulas, even when the underlying Cartesian function is smooth there.

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 2 / 11 / b / Solution Created 2026-10-18 Updated 2026-10-20

Differentiate the first-order chain rule again, including derivatives of the coordinate coefficients. The second derivatives under a two-variable coordinate change are

f_{xx} = f_{uu} u_{x}^{2} + 2 f_{uv} u_{x} v_{x} + f_{vv} v_{x}^{2} + f_{u} u_{xx} + f_{v} v_{xx},

and

f_{yy} = f_{uu} u_{y}^{2} + 2 f_{uv} u_{y} v_{y} + f_{vv} v_{y}^{2} + f_{u} u_{yy} + f_{v} v_{yy} .

The last two terms account for derivatives of the coordinate functions; they vanish for an affine map. Smoothness justifies equality of the two mixed partial derivatives.

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