  We demonstrate direct evaporative cooling of $^{87}$Rb atoms confined in a
dimple trap produced by an atom chip. By changing the two chip currents and two
external bias fields, we show theoretically that the trap depth can be lowered
in a controlled way with no change in the trap frequencies or the value of the
field at the trap center. Experimentally, we maximized the decrease in trap
depth by allowing some loosening of the trap. In total, we reduced the trap
depth by a factor of 20. The geometric mean of the trap frequencies was reduced
by less than a factor of 6. The measured phase space density in the final two
stages increased by more than two orders of magnitude, and we estimate an
increase of four orders of magnitude over the entire sequence. A subsequent rf
evaporative sweep of only a few megahertz produced Bose-Einstein condensates.
We also produce condensates in which raising the trap bottom pushes hotter
atoms into an rf "knife" operating at a fixed frequency of 5\,MHz.
