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According to Presley (Elementary differential geometry),
"A smooth surface is a surface [itex]\mathbf{\sigma}[/itex] whose atlas consists of regular surface patches."
(The atlas of a surface is a collection of homeomorphisms that "cover" it. A surface patch is just another word for an homeomorphism in the atlas. Finally, a surface patch is regular if 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)
"A smooth surface is a surface [itex]\mathbf{\sigma}[/itex] whose atlas consists of regular surface patches."
(The atlas of a surface is a collection of homeomorphisms that "cover" it. A surface patch is just another word for an homeomorphism in the atlas. Finally, a surface patch is regular if 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)
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