  The specific star formation rate (sSFR) is commonly used to describe the
level of galaxy star formation (SF) and to select quenched galaxies. However,
being a relative measure of the young-to-old population, an ambiguity in its
interpretation may arise because a small sSFR can be either because of a
substantial previous mass build up, or because SF is low. We show, using large
samples spanning 0 < z < 2, that the normalization of SFR by the physical
extent over which SF is taking place (i.e., SFR surface density,
$\Sigma_{\mathrm{SFR}}$) overcomes this ambiguity. $\Sigma_{\mathrm{SFR}}$ has
a strong physical basis, being tied to the molecular gas density and the
effectiveness of stellar feedback, so we propose $\Sigma_{\mathrm{SFR}}$-M* as
an important galaxy evolution diagram to complement (s)SFR-M* diagrams. Using
the $\Sigma_{\mathrm{SFR}}$-M* diagram we confirm the Schiminovich et al.
(2007) result that the level of SF along the main sequence today is only weakly
mass dependent - high-mass galaxies, despite their redder colors, are as active
as blue, low-mass ones. At higher redshift, the slope of the
"$\Sigma_{\mathrm{SFR}}$ main sequence" steepens, signaling the epoch of bulge
build-up in massive galaxies. We also find that $\Sigma_{\mathrm{SFR}}$ based
on the optical isophotal radius more cleanly selects both the starbursting and
the spheroid-dominated (early-type) galaxies than sSFR. One implication of our
analysis is that the assessment of the inside-out vs. outside-in quenching
scenarios should consider both sSFR and $\Sigma_{\mathrm{SFR}}$ radial
profiles, because ample SF may be present in bulges with low sSFR (red color).
