According to Presley (Elementary differential geometry),(adsbygoogle = window.adsbygoogle || []).push({});

"Asmooth surfaceis a surface [itex]\mathbf{\sigma}[/itex] whose atlas consists of regular surface patches."

(Theatlasof a surface is a collection of homeomorphisms that "cover" it. Asurface patchis just another word for an homeomorphism in the atlas. Finally, a surface patch isregularif it is smooth and its first partial derivatives are linearly independant at all points (u,v) of its domain.)

Generally, there are several possible distinct atlases for a given surface.A priori, I see no reason to say that if a surface is smooth under some atlas, it is under every atlas.

So, is it really so that smoothness is not an intrinsic property of surfaces, but rather a "bonus" that comes with a proper choice of parametrization? (much like regularness of a curve is a property of the parametrization, not of the trace itself)

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Smoothness of surfaces

**Physics Forums | Science Articles, Homework Help, Discussion**