Thank you, Gene. Perhaps even my poor keyboard is still too fast for my brain. So, the surface of the Earth -- and indeed the surface of an unconstrained liquid (e.g., the ocean?) -- at the South Pole is concave downward. Now consider again our liquid _in a fixed container_ at the South Pole. Unless the Earth spins very fast, the Earth retains a nearly spherical shape, and therefore appears to the container and its contents as if the entire mass of the Earth were concentrated at a point at the Earth's center. So the container at 4000 miles feels a really "flat" parallel force field. It would take a pretty large container to "see" a large angle relative to the earth's center. So it would appear that the surface of our liquid in a container would be concave upwards, after all, unless its radius were quite large. Q: what is the size of such a container at the South Pole which would produce a truly flat liquid surface? At 09:50 AM 3/5/2015, Eugene Salamin via math-fun wrote:
A lake at the South Pole would have the shape of the geoid, the equipotential surface due to gravity and centrifugal force.
Would you expect the lake to somehow have a concave surface?