  We measure the star formation rate (SFR) as a function of redshift z up to z
\~4.5, based on B, I and (I+B) selected galaxy catalogues from the FORS Deep
Field (FDF) and the K-selected catalogue from the GOODS-South field. Distances
are computed from spectroscopically calibrated photometric redshifts accurate
to (Delta_z / (z_spec+1)) ~0.03 for the FDF and ~0.056 for the GOODS-South
field. The SFRs are derived from the luminosities at 1500 Angstroem. We find
that the total SFR estimates derived from B, I and I+B catalogues agree very
well ($\lsim 0.1$ dex) while the SFR from the K catalogue is lower by ~0.2 dex.
We show that the latter is solely due to the lower star-forming activity of
K-selected intermediate and low luminosity (L<L_*) galaxies. The SFR of bright
(L>L_*) galaxies is independent of the selection band, i.e. the same for B, I,
(I+B), and K-selected galaxy samples. At all redshifts, luminous galaxies
(L>L_*) contribute only ~1/3 to the total SFR. There is no evidence for
significant cosmic variance between the SFRs in the FDF and GOODs-South field,
  ~0.1 dex, consistent with theoretical expectations. The SFRs derived here are
in excellent agreement with previous measurements provided we assume the same
faint-end slope of the luminosity function as previous works (alpha ~ -1.6).
However, our deep FDF data indicate a shallower slope of alpha=-1.07, implying
a SFR lower by ~0.3 dex. We find the SFR to be roughly constant up to z ~4 and
then to decline slowly beyond, if dust extinctions are assumed to be constant
with redshift.
