11.1 The Laplacian in curvilinear coordinates

In polar coordinates \((r, \theta )\) and cylindrical coordinates \((r, \theta , z)\), \[ \Delta u = u_{rr} + \frac 1ru_r + \frac 1{r^2}u_{\theta \theta } \;(+\,u_{zz}) , \] and in spherical coordinates \((r, \theta , \varphi )\), with \(\theta \) the angle from the north pole, \[ \Delta u = u_{rr} + \frac 2ru_r + \frac 1{r^2}\Delta _Su, \qquad \Delta _Su = \frac 1{\sin \theta }\frac {\partial }{\partial \theta }\Bigl (\sin \theta \,\frac {\partial u}{\partial \theta }\Bigr ) + \frac 1{\sin ^2\theta }\frac {\partial ^2u}{\partial \varphi ^2} , \] where \(\Delta _S\) is the Laplacian on the unit sphere. These follow from the chain rule, or more quickly from the divergence theorem applied to a small coordinate box. The term \(\frac {n - 1}ru_r\) in the radial part is the one met for radial solutions in Section 8.

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