Talk:Surface (geometry)

Article Checklist for "Surface (geometry)"

Workgroup category or categories	Mathematics Workgroup [Categories OK]
Article status	Stub: no more than a few sentences
Underlinked article?	Yes
Basic cleanup done?	Yes
Checklist last edited by	--AlekStos 14:46, 26 March 2007 (CDT)

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Are surfaces necessarily infinite?

Is a surface only something like a plane or sphere, which has no edges, or would one face of a cube count as a surface? If a surface necessarily has no edges, isn't it misleading to say it has length and breadth? --Catherine Woodgold 20:20, 27 April 2007 (CDT)

I think the point here is that surfaces are 2-dimensional. In algebraic geometry, you can consider surfaces defined over arbitrartry fields (even finite ones), but in differfdential geometry you're pretty much limited to R (or C). But even in the case of algebraic surfaces, you usually work over an algebraically closed field and then talk about points definable (or "rational") over a subfield. Greg Woodhouse 21:37, 27 April 2007 (CDT)

I think there are different definitions of surface used in different branches of mathematics, thus the confusion. What the article currently says doesn't seem to me to agree with what you're saying.

I think the article needs to be expanded to cover several different definitions, including at least one mathematically rigourous one (probably provided by someone other than me). It could mention definitions from physics or whatever, at least in order to clarify that that's not what's meant here.

At the moment it looks ambiguous to me. --Catherine Woodgold 10:26, 29 April 2007 (CDT)

That's a tough one. The usual approach is to require that there be mappings (functions) of the form $\scriptstyle \phi :U\subset \mathbb {R} ^{2}\rightarrow S$, sometimes called coorinate charts, such that if $\phi$ and $\psi$ are two charts about the same point, the composite $\scriptstyle \phi \circ \psi ^{-1}:\mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$ is a differentiable map with a differentiable inverse. This, in turn, is true if the Jacobian matrix has nonzdero determinant (by the inverse function theorem for functions of several variables). Intuitively, all this means is that the change of local coordinates doesn't "collapse" anything as would, say, $\scriptstyle (x,y)\mapsto (x,0)$. That's a lot of jargon. Feel fre to translate it all into English! Greg Woodhouse 11:49, 29 April 2007 (CDT)