= Product rule for the positive Laplace-Beltrami operator
{title2=$\Delta_+(fh)=f\Delta_+h+h\Delta_+f-2\langle df,dh\rangle_g$}
For a function $f$ and one-form $\alpha$, the <codifferential> obeys $\delta(f\alpha)=f\delta\alpha-\langle df,\alpha\rangle_g$. This follows from the <graded Leibniz rule> and the definition of the <Hodge star operator>. Applying it to $d(fh)=f\,dh+h\,df$ gives the product formula. In particular $\Delta_+(f^2)=2f\Delta_+f-2|df|_g^2$. The cross-term sign reverses for the opposite <Laplace-Beltrami operator> convention.
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