  We determine the variation in the nonthermal radio spectral index in the
nearby spiral galaxy M33 at a linear resolution of 360 pc. We separate the
thermal and nonthermal components of the radio continuum emission without the
assumption of a constant nonthermal spectral index. Using the Spitzer FIR data
at 70 and 160 $\mu$m and a standard dust model, we deredden the H$\alpha$
emission. The extinction corrected H$\alpha$ emission serves as a template for
the thermal free-free radio emission. Subtracting from the observed 3.6 cm and
20 cm emission (Effelsberg and the VLA) this free-free emission, we obtain the
nonthermal maps. A constant electron temperature used to obtain the thermal
radio intensity seems appropriate for M~33 which, unlike the Milky Way, has a
shallow metallicity gradient. For the first time, we derive the distribution of
the nonthermal spectral index across a galaxy, M33. We detect strong nonthermal
emission from the spiral arms and star-forming regions. Wavelet analysis shows
that at 3.6 cm the nonthermal emission is dominated by contributions from
star-forming regions, while it is smoothly distributed at 20 cm. For the whole
galaxy, we obtain thermal fractions of 51% and 18% at 3.6 cm and 20 cm,
respectively. The thermal emission is slightly stronger in the southern than in
the northern half of the galaxy. We find a clear radial gradient of mean
extinction in the galactic plane. The nonthermal spectral index map indicates
that the relativistic electrons suffer energy-loss when diffusing from their
origin in star-forming regions towards interarm regions and the outer parts of
the galaxy. We also conclude that the radio emission is mostly nonthermal at R
$>$ 5 kpc in M33.
