  We borrow the general idea of renormalization-group equations (RGEs) to
understand how neutrino masses and flavor mixing parameters evolve when
neutrinos propagate in a medium, highlighting a meaningful possibility that the
genuine flavor quantities in vacuum can be extrapolated from their
matter-corrected counterparts to be measured in some realistic neutrino
oscillation experiments. Taking the matter parameter $a \equiv 2\sqrt{2} \
G^{}_{\rm F} N^{}_e E$ to be an arbitrary scale-like variable with $N^{}_e$
being the net electron number density and $E$ being the neutrino beam energy,
we derive a complete set of differential equations for the effective neutrino
mixing matrix $V$ and the effective neutrino masses $\widetilde{m}^{}_i$ (for
$i = 1, 2, 3$). Given the standard parametrization of $V$, the RGEs for
$\{\widetilde{\theta}^{}_{12}, \widetilde{\theta}^{}_{13},
\widetilde{\theta}^{}_{23}, \widetilde{\delta}\}$ in matter are formulated for
the first time. We demonstrate some useful differential invariants which retain
the same form from vacuum to matter, including the well-known Naumov and Toshev
relations. The RGEs of the partial $\mu$-$\tau$ asymmetries, the off-diagonal
asymmetries and the sides of unitarity triangles of $V$ are also obtained as a
by-product.
