This is a question about the definition of a smooth function in Guillemin and Pollack's "Differential Topology". G&P define all manifolds as objects embedded in $\mathbb{R}^N$ for some $N$, which (as a zillion people have reminded me) is not how mathematicians usually think of manifolds - nevertheless, this is a question about how things are done in G&P:


Right on the bottom of page 2 and top of page 3, G&P say: "a mapping $f$ is smooth if it has continuous partial derivatives of all orders. However, when the domain of $f$ is not open, on usually cannot speak of partial derivatives...So we adapt the open situation to more general spaces. A map $f:X \to \mathbb{R}^m$ defined on an arbitrary subset $X$ of $\mathbb{R}^n$ is called smooth if it may be locally extended to a smooth map on open sets; that is, if around each point $x$ in $X$ there is an open set $U$ in $\mathbb{R}^n$ and a smooth map $F: U\to\mathbb{R}^m$ such that $F$ equals $f$ on $U \cap X$".


So consider the map $f$ from the $x$-axis in $\mathbb{R}^2$ to $\mathbb{R}$ defined by $f(x,0) = 0$. Since the $x$-axis is not open in $\mathbb{R}^2$, we need to extend $f$ to a smooth function $F$ around (say) $(0,0)$. There are TWO ways to do this (at least): $F(x,y) = y$ and $F(x,y) = 0$ for all points in the plane. These two different extensions give different derivatives around $(0,0)$, so it would seem that when specifying an extension we also need to specify which onee, if we want to talk about "THE" derivative at a point in the domain of $f$.


Clearly I'm misunderstanding something basic - what is it?



On The Definition of Smoothness in "Differential Topology" by Guillemin & Pollack
when you read a little further, page 8 or so, you will find that the derivative of a smooth map is a linear map defined only on vectors tangent to the manifold. it will turn out that all extensions of the smooth function act the same on vectors tangent to the manifold. In your example both extensions y and 0 act as the zero map on vectors tangent to the axis. so it works out ok. – roy smith 2 days ago


