  In the Local Group spiral galaxy M33, we investigate the correlation between
the star formation rate (SFR) surface density, Sigma_SFR, and the gas density
Sigma_gas (molecular, atomic, and total). We also explore whether there are
other physical quantities, such as the hydrostatic pressure and dust optical
depth, which establish a good correlation with Sigma_SFR. We use the Ha,
far-ultraviolet (FUV), and bolometric emission maps to infer the SFR locally at
different spatial scales, and in radial bins using azimuthally averaged values.
Most of the local analysis is done using the highest spatial resolution allowed
by gas surveys, 180 pc. The Kennicutt-Schmidt (KS) law, Sigma_SFR \propto
(Sigma_gas)^n is analyzed by three statistical methods. At all spatial scales,
with Ha emission as a SFR tracer, the KS indices n are always steeper than
those derived with the FUV and bolometric emissions. We attribute this to the
lack of Ha emission in low luminosity regions where most stars form in small
clusters with an incomplete initial mass function at their high mass end. For
azimuthally averaged values the depletion timescale for the molecular gas is
constant, and the KS index is n_H2 = 1.1 +- 0.1. Locally, at a spatial
resolution of 180 pc, the correlation between Sigma_SFR and Sigma_gas is
generally poor, even though it is tighter with the molecular and total gas than
with the atomic gas alone. Considering only positions where the CO J=1-0 line
is above the 2-sigma detection threshold and taking into account uncertainties
in Sigma_H2 and Sigma_SFR, we obtain a steeper KS index than obtained with
radial averages: n_H2 = 2.22 +- 0.07 (for FUV and bolometric SFR tracers),
flatter than that relative to the total gas (n_Htot = 2.59 +- 0.05). [abridged]
