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The Mount of Olives is a large hill, east of the city, separated from it by the Kidron Valley. Kidron is a wadi, it only has water during part of the year, in this case, the winter. So Jesus crosses this winter flow.
Gethsemane (lit. oil press) means a place where there were oil vats. There are olive trees about, and they need oil presses to press out the oil. Gethsemane is a place where this is done. This seems to be part of the earliest tradition, that there was this place called Gethsemane. Mount of Olives has interesting theologizing around it, and it’s mentioned in Luke that on Easter Sunday night Jesus ascends to heaven there, and in Acts, he again does this after 40 days. It’s usually inferred that he will come back to that spot in the future.