  We have studied the relationship between the star formation rate (SFR)
surface density and gas surface density in the spiral galaxy M51a (NGC 5194),
using multi-wavelength data obtained as part of the Spitzer Infrared Nearby
Galaxies Survey (SINGS). We introduce a new SFR index based on a linear
combination of H-alpha emission-line and 24 micron continuum luminosities, that
provides reliable extinction-corrected ionizing fluxes and SFR densities over a
wide range of dust attenuations. The combination of these extinction-corrected
SFR densities with aperture synthesis HI and CO maps has allowed us to probe
the form of the spatially-resolved star formation law on scales of 0.5 to 2
kpc. We find that the resolved SFR vs gas surface density relation is well
represented by a Schmidt power law, which is similar in form and dispersion to
the disk-averaged Schmidt law. We observe a comparably strong correlation of
the SFR surface density with the molecular gas surface density, but no
significant correlation with the surface density of atomic gas. The
best-fitting slope of the Schmidt law varies from N = 1.37 to 1.56, with
zeropoint and slope that change systematically with the spatial sampling scale.
We tentatively attribute these variations to the effects of areal sampling and
averaging of a nonlinear intrinsic star formation law. Our data can also be
fitted by an alternative parametrization of the SFR surface density in terms of
the ratio of gas surface density to local dynamical time, but with a
considerable dispersion.
