Why is the set of tangent vectors at 0 in R^m bijective with R^m itself?
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Barden & Thomas write in An Introduction to Differentiable Manifolds that the set of tangent vectors at 0 in R^m is bijective with R^m?
Why is this the case? Can 0 be replaced by an arbitrary point p of R^m?
manifolds smooth-manifolds
asked Dec 7 '18 at 16:09
gengen
4702521
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add a comment |
$begingroup$
Barden & Thomas write in An Introduction to Differentiable Manifolds that the set of tangent vectors at 0 in R^m is bijective with R^m?
Why is this the case? Can 0 be replaced by an arbitrary point p of R^m?
manifolds smooth-manifolds
asked Dec 7 '18 at 16:09
gengen
4702521
$endgroup$
$begingroup$
What is your definition of tangent vector? There are several different definitions--all equivalent--but the answer you get depends on your particular definition.
$endgroup$
– Randall
Dec 7 '18 at 16:14
add a comment |
$begingroup$
Barden & Thomas write in An Introduction to Differentiable Manifolds that the set of tangent vectors at 0 in R^m is bijective with R^m?
Why is this the case? Can 0 be replaced by an arbitrary point p of R^m?
manifolds smooth-manifolds
asked Dec 7 '18 at 16:09
gengen
4702521
$endgroup$
Barden & Thomas write in An Introduction to Differentiable Manifolds that the set of tangent vectors at 0 in R^m is bijective with R^m?
Why is this the case? Can 0 be replaced by an arbitrary point p of R^m?
manifolds smooth-manifolds
manifolds smooth-manifolds
asked Dec 7 '18 at 16:09
gengen
4702521
asked Dec 7 '18 at 16:09
gengen
4702521
asked Dec 7 '18 at 16:09
gengen
4702521
asked Dec 7 '18 at 16:09
gengen
4702521
asked Dec 7 '18 at 16:09
gengen
4702521
4702521
$begingroup$
What is your definition of tangent vector? There are several different definitions--all equivalent--but the answer you get depends on your particular definition.
$endgroup$
– Randall
Dec 7 '18 at 16:14
add a comment |
$begingroup$
What is your definition of tangent vector? There are several different definitions--all equivalent--but the answer you get depends on your particular definition.
$endgroup$
– Randall
Dec 7 '18 at 16:14
$begingroup$
What is your definition of tangent vector? There are several different definitions--all equivalent--but the answer you get depends on your particular definition.
$endgroup$
– Randall
Dec 7 '18 at 16:14
$begingroup$
What is your definition of tangent vector? There are several different definitions--all equivalent--but the answer you get depends on your particular definition.
$endgroup$
– Randall
Dec 7 '18 at 16:14
add a comment |
2 Answers
2
active
oldest
votes
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Intuitively:
The tangent vector space of a line ($mathbb{R^1}$) at a point of the line (that can be the origin) is the line itself.
The tangent space of a plane($mathbb{R^2}$) is the plane itself.
ans so for $mathbb{R^n}$.....
Any formal definition of the tangent vector space of a variety conserve this intuition.
answered Dec 7 '18 at 16:21
Emilio NovatiEmilio Novati
52.2k43474
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add a comment |
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Yes you can replace $0$ with any $pinmathbb R^m$.
By definition the tangent space, $T_pM$, is an $m$-dimensional vector space. But any two $m$-dimensional vector spaces are isomorphic.
answered Dec 7 '18 at 16:24
Chris CusterChris Custer
14.2k3827
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add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Intuitively:
The tangent vector space of a line ($mathbb{R^1}$) at a point of the line (that can be the origin) is the line itself.
The tangent space of a plane($mathbb{R^2}$) is the plane itself.
ans so for $mathbb{R^n}$.....
Any formal definition of the tangent vector space of a variety conserve this intuition.
answered Dec 7 '18 at 16:21
Emilio NovatiEmilio Novati
52.2k43474
$endgroup$
add a comment |
$begingroup$
Intuitively:
The tangent vector space of a line ($mathbb{R^1}$) at a point of the line (that can be the origin) is the line itself.
The tangent space of a plane($mathbb{R^2}$) is the plane itself.
ans so for $mathbb{R^n}$.....
Any formal definition of the tangent vector space of a variety conserve this intuition.
answered Dec 7 '18 at 16:21
Emilio NovatiEmilio Novati
52.2k43474
$endgroup$
add a comment |
$begingroup$
Intuitively:
The tangent vector space of a line ($mathbb{R^1}$) at a point of the line (that can be the origin) is the line itself.
The tangent space of a plane($mathbb{R^2}$) is the plane itself.
ans so for $mathbb{R^n}$.....
Any formal definition of the tangent vector space of a variety conserve this intuition.
answered Dec 7 '18 at 16:21
Emilio NovatiEmilio Novati
52.2k43474
$endgroup$
Intuitively:
The tangent vector space of a line ($mathbb{R^1}$) at a point of the line (that can be the origin) is the line itself.
The tangent space of a plane($mathbb{R^2}$) is the plane itself.
ans so for $mathbb{R^n}$.....
Any formal definition of the tangent vector space of a variety conserve this intuition.
answered Dec 7 '18 at 16:21
Emilio NovatiEmilio Novati
52.2k43474
answered Dec 7 '18 at 16:21
Emilio NovatiEmilio Novati
52.2k43474
answered Dec 7 '18 at 16:21
Emilio NovatiEmilio Novati
52.2k43474
answered Dec 7 '18 at 16:21
Emilio NovatiEmilio Novati
52.2k43474
52.2k43474
add a comment |
add a comment |
$begingroup$
Yes you can replace $0$ with any $pinmathbb R^m$.
By definition the tangent space, $T_pM$, is an $m$-dimensional vector space. But any two $m$-dimensional vector spaces are isomorphic.
answered Dec 7 '18 at 16:24
Chris CusterChris Custer
14.2k3827
$endgroup$
add a comment |
$begingroup$
Yes you can replace $0$ with any $pinmathbb R^m$.
By definition the tangent space, $T_pM$, is an $m$-dimensional vector space. But any two $m$-dimensional vector spaces are isomorphic.
answered Dec 7 '18 at 16:24
Chris CusterChris Custer
14.2k3827
$endgroup$
add a comment |
$begingroup$
Yes you can replace $0$ with any $pinmathbb R^m$.
By definition the tangent space, $T_pM$, is an $m$-dimensional vector space. But any two $m$-dimensional vector spaces are isomorphic.
answered Dec 7 '18 at 16:24
Chris CusterChris Custer
14.2k3827
$endgroup$
Yes you can replace $0$ with any $pinmathbb R^m$.
By definition the tangent space, $T_pM$, is an $m$-dimensional vector space. But any two $m$-dimensional vector spaces are isomorphic.
answered Dec 7 '18 at 16:24
Chris CusterChris Custer
14.2k3827
answered Dec 7 '18 at 16:24
Chris CusterChris Custer
14.2k3827
answered Dec 7 '18 at 16:24
Chris CusterChris Custer
14.2k3827
answered Dec 7 '18 at 16:24
Chris CusterChris Custer
14.2k3827
14.2k3827
add a comment |
add a comment |
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$begingroup$
What is your definition of tangent vector? There are several different definitions--all equivalent--but the answer you get depends on your particular definition.
$endgroup$
– Randall
Dec 7 '18 at 16:14