= Positive Laplace-Beltrami operator
{title2=$\Delta_+=-\operatorname{div}_g\operatorname{grad}_g$}
The positive convention for the <Laplace-Beltrami operator> is $\Delta_+f=\delta df$, where $\delta$ is the <codifferential>. It is the restriction of the <Hodge Laplacian> to functions. In coordinates,
$$
\Delta_+f=-\frac1{\sqrt{\det g}}\partial_i\bigl(\sqrt{\det g}\,g^{ij}\partial_jf\bigr).
$$
On a compact boundaryless <Riemannian manifold>, or for compactly supported functions, $\langle f,\Delta_+f\rangle=\|df\|^2\ge0$. The opposite $\operatorname{div}\operatorname{grad}$ convention is also standard; formulas involving the sign must specify which is used.
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