Difference between coordinate map and standard coordinates?
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I quote from W. Tu's introduction on manifolds:
«On a manifold $M$ of dimension $n$, let $(U, varphi )$ be a chart and $f$ a $C^infty$ function. As a
function into $mathbb{R}^n$ , $varphi$ has $n$ components $x^1, cdots, x^n$. This means that if $r^1 , cdots , r^n$ are the standard coordinates on $mathbb{R}^n$, then $x^i = r^i circ φ$.»
My problem is that I do not understand why $x^i$ are not standard coordinates on $mathbb{R}^n$. I suppose that the author has implicitly defined $x^i = pi_icirc varphi$, where $pi_i : mathbb{R}^n to mathbb{R}$ is the function given by $pi_i(x^1,cdots,x^n) = x^i$. Could you explain this to me? Is the author confusing notation, i.e. $pi_i = r^i$, or is he insisting that there exists some function $r^i neq pi_i$ for a $1 leq i leq n$?
differential-geometry manifolds smooth-manifolds coordinate-systems
asked Jan 25 at 15:53
Marius JonssonMarius Jonsson
705414
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add a comment |
$begingroup$
I quote from W. Tu's introduction on manifolds:
«On a manifold $M$ of dimension $n$, let $(U, varphi )$ be a chart and $f$ a $C^infty$ function. As a
function into $mathbb{R}^n$ , $varphi$ has $n$ components $x^1, cdots, x^n$. This means that if $r^1 , cdots , r^n$ are the standard coordinates on $mathbb{R}^n$, then $x^i = r^i circ φ$.»
My problem is that I do not understand why $x^i$ are not standard coordinates on $mathbb{R}^n$. I suppose that the author has implicitly defined $x^i = pi_icirc varphi$, where $pi_i : mathbb{R}^n to mathbb{R}$ is the function given by $pi_i(x^1,cdots,x^n) = x^i$. Could you explain this to me? Is the author confusing notation, i.e. $pi_i = r^i$, or is he insisting that there exists some function $r^i neq pi_i$ for a $1 leq i leq n$?
differential-geometry manifolds smooth-manifolds coordinate-systems
asked Jan 25 at 15:53
Marius JonssonMarius Jonsson
705414
$endgroup$
$begingroup$
The $x^i$ are coordinates on $Usubset M$.
$endgroup$
– John Douma
Jan 25 at 15:56
add a comment |
$begingroup$
I quote from W. Tu's introduction on manifolds:
«On a manifold $M$ of dimension $n$, let $(U, varphi )$ be a chart and $f$ a $C^infty$ function. As a
function into $mathbb{R}^n$ , $varphi$ has $n$ components $x^1, cdots, x^n$. This means that if $r^1 , cdots , r^n$ are the standard coordinates on $mathbb{R}^n$, then $x^i = r^i circ φ$.»
My problem is that I do not understand why $x^i$ are not standard coordinates on $mathbb{R}^n$. I suppose that the author has implicitly defined $x^i = pi_icirc varphi$, where $pi_i : mathbb{R}^n to mathbb{R}$ is the function given by $pi_i(x^1,cdots,x^n) = x^i$. Could you explain this to me? Is the author confusing notation, i.e. $pi_i = r^i$, or is he insisting that there exists some function $r^i neq pi_i$ for a $1 leq i leq n$?
differential-geometry manifolds smooth-manifolds coordinate-systems
asked Jan 25 at 15:53
Marius JonssonMarius Jonsson
705414
$endgroup$
I quote from W. Tu's introduction on manifolds:
«On a manifold $M$ of dimension $n$, let $(U, varphi )$ be a chart and $f$ a $C^infty$ function. As a
function into $mathbb{R}^n$ , $varphi$ has $n$ components $x^1, cdots, x^n$. This means that if $r^1 , cdots , r^n$ are the standard coordinates on $mathbb{R}^n$, then $x^i = r^i circ φ$.»
My problem is that I do not understand why $x^i$ are not standard coordinates on $mathbb{R}^n$. I suppose that the author has implicitly defined $x^i = pi_icirc varphi$, where $pi_i : mathbb{R}^n to mathbb{R}$ is the function given by $pi_i(x^1,cdots,x^n) = x^i$. Could you explain this to me? Is the author confusing notation, i.e. $pi_i = r^i$, or is he insisting that there exists some function $r^i neq pi_i$ for a $1 leq i leq n$?
differential-geometry manifolds smooth-manifolds coordinate-systems
differential-geometry manifolds smooth-manifolds coordinate-systems
asked Jan 25 at 15:53
Marius JonssonMarius Jonsson
705414
asked Jan 25 at 15:53
Marius JonssonMarius Jonsson
705414
asked Jan 25 at 15:53
Marius JonssonMarius Jonsson
705414
asked Jan 25 at 15:53
Marius JonssonMarius Jonsson
705414
asked Jan 25 at 15:53
Marius JonssonMarius Jonsson
705414
705414
$begingroup$
The $x^i$ are coordinates on $Usubset M$.
$endgroup$
– John Douma
Jan 25 at 15:56
add a comment |
$begingroup$
The $x^i$ are coordinates on $Usubset M$.
$endgroup$
– John Douma
Jan 25 at 15:56
$begingroup$
The $x^i$ are coordinates on $Usubset M$.
$endgroup$
– John Douma
Jan 25 at 15:56
$begingroup$
The $x^i$ are coordinates on $Usubset M$.
$endgroup$
– John Douma
Jan 25 at 15:56
add a comment |
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$begingroup$
The $x^i$ are coordinates on $Usubset M$.
$endgroup$
– John Douma
Jan 25 at 15:56