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I'm studying the "Elementary Differential Geometry" from O'Neil and he mentions what he calls the "Natural Coordinate Functions". In particular, he says that if $p = (p_1 ... p_n) \in \mathbb{R}^n$ is a point in the $n$-space then we define the $i$-th coordinate function $x_i : \mathbb{R}^n \to \mathbb{R}$ as $x_i(p) = p_i$

My question is: why it's needed to define such functions ? I mean why is it relevant to define this ? He says that in $n$-space it's very important this definition to avoid serious ambiguities, but I couldn't identify any special case when there's a ambiguity for not defining those functions.

Can someone please explain this and show an example where it causes ambiguities for not defining those functions ?

Thanks a lot in advance.

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2 Answers

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The reason to introduce these functions is clear from calculus III. We don't even need to consider anything too fancy. Consider the following:

$$ \frac{\partial x}{\partial y}=0 $$

How is this shown? We need to introduce a function $f(x,y) = x$ and then calculate from the definition for partial derivative with respect to $y$ at $(a,b)$

$$ \frac{\partial x}{\partial y}(a,b) = \lim_{h \rightarrow 0} \frac{f(a,b+h)-f(a,b)}{h} = \lim_{h \rightarrow 0} \frac{a-a}{h} = \lim_{h \rightarrow 0} 0 = 0 $$

Of course this holds for all $(a,b)$ so we can drop it without ambiguity.

Fun exercise: do the same calculation for $f(\vec{x}) = e_j \cdot \vec{x} = x_j$ to prove that $\frac{\partial x_j}{\partial x_i} = \delta_{ij}$. Here the partial derivative with respect to $x_i$ would be defined by $$ \frac{\partial f}{\partial x_i}(\vec{p}) = \lim_{h \rightarrow 0}\frac{f(\vec{p}+he_j)-f(\vec{p})}{h} $$

Oneil's point is that without this notation it makes many statements relating maps on the surface to maps on $\mathbb{R}^n$ awkward. Often an embedded surface can be covered by restricting the ambient maps $x,y,z$ on $\mathbb{R}^3$. For example, a plane with $z=2$ natrually inherits the $x,y$ coordinate maps by simply restricting their domain from the natural $\mathbb{R}^3$. In short, it's about having a notation for formulas you need to write.

If coordinates are abhorrent then perhaps you should read Lang's differential geometry, I believe he makes an effort to avoid unnecessary coordinated formulas. Elementary Differential Geometry by Oneil is written at a very different level. It is meant to be a natural extension from calculus III.

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Hi James S. Cook, I understood what's the point now. Thanks for your aid. The book of differential geometry you speak of is Fundamentals of Differential Geometry from Serge Lang? Thanks again for your answer. –  user1620696 Sep 17 '12 at 17:14
    
Correct, although be careful of which edition you get. There is a big difference between the 3rd and earlier editions. I just bought the third ed. and it is much bigger than the earlier ed. You can look at the reviews to decide if it's worth it... –  James S. Cook Sep 17 '12 at 20:06
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It is possible to reduce the use of coordinates in differential geometry.

By definition, on an $n$-manifold, for every point there is given an open neighborhood with homeomorphism to an open subset of $\mathbb R^n$. Now, the coordinates of these given homeomorphisms can be set as the coordinate functions- Its use is that differentials and integrals are then locally computable on the manifold, using these in $\mathbb R^n$.

For example, the surface of the sphere, as 2-manifold, can be given by 2 "maps", one circle of a plain stereographic-projected to the upper part of the sphere, and another circle for the lower part.

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Berci Pecsi, I know that idea, it's the idea of using charts to descibre a manifold right? Then, if $(U, \varphi)$ is a chart on a manifold $M$ and if $p \in M$ I'll have $\varphi(p) \in \mathbb{R}^n$. If I understood you, I'll use the coordinate functions to write $p_i = (x_i \circ \varphi)(p)$ the $i$-th coordinate of the point in the manifold? I mean, I'll use it to assign coordinates to points on the manifold using a chart? Thanks a lot for your answer. –  user1620696 Sep 16 '12 at 20:55
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