  We have investigated the spatial distribution of the electron temperature
generated in a two-dimensional electron system (2DES) subjected to a
perpendicular magnetic field. We measure thermoelectric voltages between Ohmic
contacts located at the end of the voltage-probe arms of a Hall bar fabricated
from a GaAs/AlGaAs 2DES wafer, immersed in the mixing chamber of a dilution
refrigerator held at 20 mK. Magneto-oscillations due to the Landau quantization
are examined for the thermoelectric voltages between the contact pairs
straddling the main bar (arrangement to measure the transverse component
$V_{yx}$), and between the pairs located along the same side of the main bar
(arrangement for the longitudinal component $V_{xx}$). For the former
arrangement, the oscillation amplitude diminishes with the distance from the
heater. For the latter arrangement, the pair on one side exhibits much larger
amplitude than the pair on the opposite side, and the relation becomes reversed
by inverting the magnetic field. The behaviours of the oscillation amplitude
are qualitatively explained by the spatial distribution of the electron
temperature numerically calculated taking into consideration the thermal
diffusion into the voltage contacts and the electron-phonon interaction. For
both arrangements, the oscillations are shown to derive predominantly from the
transverse (Nernst) component, $S_{yx}$, of the thermopower tensor. The
calculation also reveals that the voltage probes, introducing only minor
disturbance at zero magnetic field, substantially reduce the temperature once a
magnetic field is applied, and the thermoelectric voltages generated at the
voltage arms account for a significant part of the measured voltages.
