kaya, 1 month ago (edited 1 month ago)

@gregeganSF B appears to be the answer that generalizes most immediately to higher dimensions: hold all coordinates except u constant and you still get a curve. To generalize A, we have to talk about "the u-hypersurfaces".

And this way, the x-axis is an x-curve.

fgrosshans, 1 month ago

@gregeganSF I obviously am in the minority here, but I'm a physicist, not a mathematician.I don't know if I'm influenced by iso-u curves, which is the case where I've seen this kind of things (e.g. in thermodynamics) or by my native language, French, where contour lines on maps are literally called “level curves” (courbes de niveau)

gregeganSF, 1 month ago

@fgrosshans I understand the attraction of level curves too!

narain, 1 month ago

@gregeganSF Surely "u-curve" is analogous to "x-axis"?

gregeganSF, 1 month ago

@narain I suspect that is what people usually mean, but I’m not so sure that the intention is obvious.

I can imagine being drawn towards either meaning: curves of varying u if I’m thinking of the coordinate tangent vectors, which are tangent to those curves, or curves of constant u if I’m thinking of the coordinate 1-forms.

narain, 1 month ago

@gregeganSF Makes sense. In a ≥3-manifold the u-curves would be unambiguous, and transverse to the u-surfaces.