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

$\Sigma_{\mathrm{SFR}}$-M* Diagram: A Valuable Galaxy Evolution Diagnostic to Complement (s)SFR-M* Diagrams