= Laplacian of a restricted ambient function
{title2=$\Delta_{\mathrm{geom}}=\operatorname{div}\operatorname{grad}$}
Use the divergence-of-gradient convention $\Delta_{\mathrm{geom}}=\operatorname{div}\operatorname{grad}$. For an <embedded submanifold> $M\subset\mathbb R^{n+m}$ and a smooth ambient <function> $f$,
$$
\Delta_{\mathrm{geom},M}(f|_M)=\operatorname{tr}_{TM}(\overline{\operatorname{Hess}}f)+\langle\overline\nabla f,\mathbf H\rangle,
$$
where $\mathbf H=\operatorname{tr}II$ is the vector-valued <mean curvature>.
At a <point>, choose a local <orthonormal frame> whose intrinsic covariant <derivatives> vanish there. Differentiate $f|_M$ twice and insert the <Gauss formula>. The tangent <derivative> terms give the ambient <Hessian matrix> trace, and the normal terms sum to $\langle\overline\nabla f,\mathbf H\rangle$.
For a <coordinate> <function> $x^j$, the ambient <Hessian matrix> is zero, so $\Delta_{\mathrm{geom},M}x^j=H^j$. <coordinates> of a <minimal submanifold> are therefore harmonic. On the round unit <sphere>, $\mathbf H=-nX$, giving $\Delta_{\mathrm{geom}}x^j=-nx^j$. The nonnegative <Laplace-Beltrami operator> uses the opposite sign; declaring the convention fixes the signs in these formulas.
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