Euclidean Laplacian - Knowlpedia

The Euclidean Laplacian of a $C^2$ scalar function is

\Delta f=\sum_{i=1}^n\partial_{x_i}^2 f.

For a vector field it acts on Cartesian components: $(\Delta u)_i=\Delta u_i$ . This convention has Fourier symbol $-|\xi|^2$ when derivatives have symbol $i\xi$ .

Cylindrical coordinates

For scalar functions on $r>0$ , set

L=\partial_r^2+r^{-1}\partial_r+r^{-2}\partial_\theta^2+\partial_z^2.

Then $\Delta f=Lf$ . For cylindrical vector components,

\begin{aligned} (\Delta u)_r&=Lu_r-r^{-2}u_r-2r^{-2}\partial_\theta u_\theta,\\ (\Delta u)_\theta&=Lu_\theta-r^{-2}u_\theta+2r^{-2}\partial_\theta u_r,\\ (\Delta u)_z&=Lu_z. \end{aligned}

These follow by applying $L$ to $u_r e_r+u_\theta e_\theta+u_z e_z$ and differentiating the frame. Applying the scalar formula separately to moving components would omit the displayed terms.

References

Arthur Mattuck, MIT 18.02SC, The Del Operator.
MIT 6.013, Differential operators in cylindrical and spherical coordinates.