= Solution
Use the equivalent density form of <harmonic slicing> from part (ii). With $\sqrt{-g}=\alpha\sqrt\gamma$, positive <lapse function> $\alpha$, and the corrected inverse <metric tensor>, its time component becomes
$$
0=-\partial_t\left(\frac{\sqrt\gamma}{\alpha}\right)
+\partial_i\left(\frac{\sqrt\gamma}{\alpha}\beta^i\right).
$$
Thus $D_t(\sqrt\gamma/\alpha)=(\sqrt\gamma/\alpha)\partial_i\beta^i$, or
$$
D_t\log\sqrt\gamma-D_t\log\alpha=\partial_i\beta^i.
$$
The <spatial volume evolution identity> gives $D_t\log\sqrt\gamma=\partial_i\beta^i-\alpha K$. Subtraction yields $D_t\log\alpha=-\alpha K$, hence \b[the <harmonic lapse evolution> is]
$$
\boxed{\partial_t\alpha-\beta^i\partial_i\alpha=-\alpha^2K,\qquad f(\alpha)=-\alpha^2.}
$$
The <shift vector> need not vanish; its transport term is essential in a general <3+1 decomposition of spacetime>.
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