= Axisymmetric free-surface curvature
{title2=$\kappa=-r^{-1}\partial_r[rh'/\sqrt{1+h'^2}]$}
For an axisymmetric graph $z=h(r)$ with the normal pointing out of the liquid, the sum of principal <curvatures> is the displayed expression. In the small-slope limit it becomes $\kappa=-(h''+h'/r)$, giving the <capillary pressure> $p_{\mathrm{liquid}}-p_{\mathrm{air}}=\gamma\kappa$. Regularity at the axis gives $h'(0)=0$ and $h'/r\to h''(0)$: the azimuthal contribution does not disappear at the axis. For example, the paraboloid $h=H(1-r^2/R^2)$ has $\kappa\approx4H/R^2$, twice the meridional contribution.
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