Spatial mean of a periodic heat equation
= Spatial mean of a periodic heat equation
{title2=$\frac{d\overline u}{dt}=\overline Q$}
Integrate $u_t=\alpha\Delta u+Q$ over a periodic cell. Opposite boundary contributions to the <Laplacian> cancel, so the spatial <mean> obeys $\overline u'=\overline Q$. With no source the mean is conserved, although the local <temperature> changes. The mean is the <zero Fourier mode> in a normalized <Fourier series> convention.