= Cartesian formula for the angular Laplacian
{c}
{title2=$-\Delta_{S^2}=-r^2\Delta+(\mathbf x\cdot\nabla)^2+\mathbf x\cdot\nabla$}
Writing $D_r=\mathbf x\cdot\nabla=r\partial_r$, the spherical-coordinate <Laplacian> gives $r^2\Delta=D_r^2+D_r+\Delta_{S^2}$. Thus the positive angular <Laplace-Beltrami operator> has the displayed Cartesian expression and <eigenvalues> $l(l+1)$ on <spherical harmonics>. Omitting $D_r$ is incorrect when the square means composition: on $x_1$, the truncated expression gives $x_1$, whereas the degree-one angular <eigenvalue> is two.
// Target: fluid-mechanics.bigb
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