Context : 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 .

Aims : In this first paper , we investigate the correlation of H _ { 2 } , H i , and total gas surface densities with the star forming activity , derived from the radio continuum ( RC ) , along radial averages out to radii of 12 kpc .

Methods : We have created a complete map of M51 in ^ { 12 } CO 2–1 at a resolution of 450 kpc using HERA at the IRAM-30m telescope .
These data are combined with maps of H i and the radio-continuum at 20 cm 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 \Sigma _ { SFR } \propto \Sigma _ { gas } ^ { n } .
The velocity dispersion from CO is used to study the critical surface density and the gravitational stability of the disk .

Results : The total mass of molecular material derived from the integrated ^ { 12 } CO 2–1 intensities is 2 10 ^ { 9 } M _ { \odot } .
The 3 \sigma detection limit corresponds to a mass of 1.7 10 ^ { 5 } M _ { \odot } .
The global star formation rate is 2.56 M _ { \odot } yr ^ { -1 } and the global gas depletion time is 0.8 Gyr .
H i and RC emission are found to peak on the concave , downstream side of the outer south-western CO arm , outside the corotation radius .
The total gas surface density \Sigma _ { gas } drops by a factor of \sim 20 from 70 M _ { \odot } pc ^ { -2 } at the center to 3 M _ { \odot } pc ^ { -2 } in the outskirts at radii of 12 kpc .
The fraction of atomic gas gradually increases with radius .
The ratio of H i over H _ { 2 } surface densities , \Sigma _ { HI } / \Sigma _ { H 2 } , increases from \sim 0.1 near the center to \sim 20 in the outskirts without following a simple power-law . \Sigma _ { HI } starts to exceed \Sigma _ { H 2 } at a radius of \sim 4 kpc .
The star formation rate per unit area drops from \sim 400 M _ { \odot } pc ^ { -2 } Gyr ^ { -1 } in the starburst center to \sim 2 M _ { \odot } pc ^ { -2 } Gyr ^ { -1 } in the outskirts .
The gas depletion time varies between 0.1 Gyr in the center and 1 Gyr in the outskirts , and is shorter than in other non-interacting normal galaxies .
Neither the H i surface densities nor the H _ { 2 } surface densities show a simple power-law dependence on the star formation rate per unit area .
In contrast , \Sigma _ { gas } and \Sigma _ { SFR } are well characterized by a local Schmidt law with a power-law index of n = 1.4 \pm 0.6 .
The index equals the global Schmidt law derived from disk-averaged values of \Sigma _ { gas } and \Sigma _ { SFR } of large samples of normal and starburst galaxies .
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 kms ^ { -1 } .
Observed radially averaged dispersions derived from the CO data vary between 28 kms ^ { -1 } in the center and \sim 8 kms ^ { -1 } 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 .

Conclusions :