- #1

facenian

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This book difines a coordinate patch as: A coordinate patch of class ##C^m## in ##S\subset R^3## is a mapping ##\vec{x}=\vec{x}(u,v)## of an open set ##U\subset R^2## into S such that:

(i) ##\vec{x}:U\subset R^2\rightarrow S\subset R^3## is of class ##C^m## on U

(ii) ##\vec{x}_u\wedge\vec{x}_v\neq 0## for all (u,v) in U.

(iii) ##\vec{x}## is 1-1 and bicontinuous on U, that is, its inverse is also continous.

Then he defines a simple surface in ##R^3## as a set S of points and a collection ##\mathcal{B}## of coordinates patches satisfying:

(1) ##\mathcal{B}## covers S

(2) Every coordinate pacht is the intersecction of an open set O in ##E^3## with S, ##\vec{x}(U)=O\cap S##

Notice that condition (2) is not demanded for a coordinate patch. Later in the book one can find the following satatement:

(3) It can be shown that if ##\vec{x}=\vec{x}(u,v)## is a coordinate patch on a simple surface S and P is a point on ##\vec{x}=\vec{x}(u,v)##, then there exists a spherical neighborhood S(P) in ##E^3## such that the intersecction of S(P) with the surface S is contained in the pathc ##\vec{x}=\vec{x}(u,v)##.

Then he uses (3) to show that every patch on a simple surface is the intersection of an open set in ##R^3## with S which is easy to prove. However he does not provide a proof for (3).

Does someone know how (3) can be proven in the context of these definitions?