Reading through Introduction to Manifolds, and quite often the term "Standard Coordinates" appears.
In Linear Algebra these are usually the weights of the standard basis (such weights are denoted by $r^1,\ldots,r^n$. However as soon as we get to smooth manifolds these seems to denote some functions. Suppose we have a manifold $M$ of dimension $n$ with some chart $(U,\phi) = (U,x^1,\ldots,x^n)$ centered around $p \in M$. The standard coordinates seem to be functions such that $x^i = r^i \circ \phi$, which to me this would translate in the projection operator. Can anyone clarify what the meaning of standard coordinates is in this context? It's really hard to work out the details of the proofs sometimes because of this subtle detail.
First of all, there's a lot of abuse of notation in differential geometry, so "standard coordinates" may have different meanings in different contexts. This is the first paragraph of Tu where I found the expression:
"The Euclidean space $\mathbb{R}^n$ is the prototype of all manifolds. Not only is it the simplest, but locally every manifold looks like $\mathbb{R}^n$. A good understanding of $\mathbb{R}^n$ is essential in generalizing differential and integral calculus to a manifold. Euclidean space is special in having a set of standard global coordinates."
This is what I understood from the paragraph:
Considering $\mathbb{R}^n$ as a manifold, you can give $\mathbb{R}^n$ the following smooth structure (which is usually called the standard structure of $\mathbb{R}^n$) with the following atlas, consisting on a single chart (this is what I think Tu means by "global"):
$$\{U, \varphi\}=\{\mathbb{R}^n, \text{Id}_\mathbb{{R}^n}\}$$
This chart provides you with an homeomorphism between $\mathbb{R}^n$ regarded as a manifold and the euclidean space $\mathbb{R}^n$, and its coordinates will be:
\begin{array}{cccl} \varphi \equiv (x^1,...,x^n): &U &\longrightarrow &\mathbb{R}^n \\\ &p &\longmapsto & (x^1(p),...,x^n(p)) = (p_1,...,p_n) \end{array}
So these "standard coordinates" will map a point $p$ of $\mathbb{R}^n$ considered as a manifold to the same point $p$ of $\mathbb{R}^n$ considered as a vector space.