  The mechanisms governing the star formation rate in spiral galaxies are not
yet clear. The nearby, almost face-on, and interacting galaxy M51 offers an
excellent opportunity to study at high spatial resolutions the local star
formation laws. In this first paper, we investigate the correlation of H2, HI,
and total gas surface densities with the star forming activity, derived from
the radio continuum (RC), along radial averages out to radii of 12kpc.
  We have created a complete map of M51 in 12CO 2-1 at a resolution of 450kpc
using HERA at the IRAM-30m telescope. These data are combined with maps of HI
and the radio-continuum at 20cm wavelength. The latter is used to estimate the
star formation rate (SFR), thus allowing to study the star formation efficiency
and the local Schmidt law. The velocity dispersion from CO is used to study the
critical surface density and the gravitational stability of the disk.
  The critical gas velocity dispersions needed to stabilize the gas against
gravitational collapse in the differentially rotating disk of M51 using the
Toomre criterion, vary with radius between 1.7 and 6.8 km/s. Observed radially
averaged dispersions derived from the CO data vary between 28 km/s in the
center and 8 km/s at radii of 7 to 9 kpc. They exceed the critical dispersions
by factors Q_gas of 1 to 5. We speculate that the gravitational potential of
stars leads to a critically stable disk.
