Mixing
Given a smooth manifold $M$, how does local coordinates affect the basis of $T_p(M)$?
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Let's say I have a smooth manifold $M$ of dimension $n$ and a smooth chart $(U, phi)$ consisting of open set $U$ of $M$ along with a homeomorphism $phi : U to widehat{U} =phi[U] subseteq mathbb{R}^n$.
Recall that the component functions $x^i : U to mathbb{R}$ of $phi$ defined by $phi(p) = (x^1(p), dots, x^n(p))$ are called the local coordinates on $U$.
Now the basis for $T_p(M)$ is given by $$left{frac{partial}{partial x^1}bigg|_p, dots, frac{partial}{partial x^n}bigg|_pright}$$ where each $frac{partial}{partial x^i}bigg|_p$ is the derivation defined by $$frac{partial}{partial x^i}bigg|_p(f) = frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) (*)$$ for $f in C^{infty}(U)$.
Now in Introduction to Smooth Manifolds by John Lee, it's said that these tangent vectors $frac{partial}{partial x^i}bigg|_p$ are called the coordinate vectors at $p$ assosciated with the given coordinate system.
But I don't exactly see how the basis vectors $frac{partial}{partial x^i}bigg|_p$ depend on a given coordinate system other than simply the homeomorphism $phi$. To me it seems that the right hand side of $(*)$, that being $frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right)$, is just the usual partial derivative operator in $mathbb{R}^n$, so by that I mean that $$frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) = lim_{t to 0} frac{(f circ phi^{-1})(phi(p) + t e_i) - (fcirc phi^{-1})(phi(p))}{t}$$
where $e_i$ is the element $(0, dots, 1, dots, 0) in mathbb{R}^n$ with a $1$ in the $i$-th position of the $n$-tuple. Am I correct in saying that? I think I am wrong, because what if I have some $2$-dimensional manifold $N$, and a chart $(V, psi)$ where points of $widehat{V}$ are best expressed in polar coordiantes and I wanted to compute $frac{partial}{partial x^i}bigg|_{q}(f)$ for some $q in N$ and $f in C^{infty}(V)$. I don't see how my interpretation of $(*)$ would make sense then there.
Maybe I'm getting lost in notation but for example I don't understand what the relation between the symbolic $x^i$'s in the notation of the component functions $x^i$ of $phi$, the basis vectors $frac{partial}{partial x^i}bigg|_{p}$, and $frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) $ are other than they are simply a reminder for the latter two that we're taking the $i$-th partial derivative in $widehat{U}$. The notation used suggests there must be some stronger relationship between these.
differential-geometry
asked Jan 9 at 1:47
PerturbativePerturbative
4,29811551
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add a comment |
$begingroup$
Let's say I have a smooth manifold $M$ of dimension $n$ and a smooth chart $(U, phi)$ consisting of open set $U$ of $M$ along with a homeomorphism $phi : U to widehat{U} =phi[U] subseteq mathbb{R}^n$.
Recall that the component functions $x^i : U to mathbb{R}$ of $phi$ defined by $phi(p) = (x^1(p), dots, x^n(p))$ are called the local coordinates on $U$.
Now the basis for $T_p(M)$ is given by $$left{frac{partial}{partial x^1}bigg|_p, dots, frac{partial}{partial x^n}bigg|_pright}$$ where each $frac{partial}{partial x^i}bigg|_p$ is the derivation defined by $$frac{partial}{partial x^i}bigg|_p(f) = frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) (*)$$ for $f in C^{infty}(U)$.
Now in Introduction to Smooth Manifolds by John Lee, it's said that these tangent vectors $frac{partial}{partial x^i}bigg|_p$ are called the coordinate vectors at $p$ assosciated with the given coordinate system.
But I don't exactly see how the basis vectors $frac{partial}{partial x^i}bigg|_p$ depend on a given coordinate system other than simply the homeomorphism $phi$. To me it seems that the right hand side of $(*)$, that being $frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right)$, is just the usual partial derivative operator in $mathbb{R}^n$, so by that I mean that $$frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) = lim_{t to 0} frac{(f circ phi^{-1})(phi(p) + t e_i) - (fcirc phi^{-1})(phi(p))}{t}$$
where $e_i$ is the element $(0, dots, 1, dots, 0) in mathbb{R}^n$ with a $1$ in the $i$-th position of the $n$-tuple. Am I correct in saying that? I think I am wrong, because what if I have some $2$-dimensional manifold $N$, and a chart $(V, psi)$ where points of $widehat{V}$ are best expressed in polar coordiantes and I wanted to compute $frac{partial}{partial x^i}bigg|_{q}(f)$ for some $q in N$ and $f in C^{infty}(V)$. I don't see how my interpretation of $(*)$ would make sense then there.
Maybe I'm getting lost in notation but for example I don't understand what the relation between the symbolic $x^i$'s in the notation of the component functions $x^i$ of $phi$, the basis vectors $frac{partial}{partial x^i}bigg|_{p}$, and $frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) $ are other than they are simply a reminder for the latter two that we're taking the $i$-th partial derivative in $widehat{U}$. The notation used suggests there must be some stronger relationship between these.
differential-geometry
asked Jan 9 at 1:47
PerturbativePerturbative
4,29811551
$endgroup$
add a comment |
$begingroup$
Let's say I have a smooth manifold $M$ of dimension $n$ and a smooth chart $(U, phi)$ consisting of open set $U$ of $M$ along with a homeomorphism $phi : U to widehat{U} =phi[U] subseteq mathbb{R}^n$.
Recall that the component functions $x^i : U to mathbb{R}$ of $phi$ defined by $phi(p) = (x^1(p), dots, x^n(p))$ are called the local coordinates on $U$.
Now the basis for $T_p(M)$ is given by $$left{frac{partial}{partial x^1}bigg|_p, dots, frac{partial}{partial x^n}bigg|_pright}$$ where each $frac{partial}{partial x^i}bigg|_p$ is the derivation defined by $$frac{partial}{partial x^i}bigg|_p(f) = frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) (*)$$ for $f in C^{infty}(U)$.
Now in Introduction to Smooth Manifolds by John Lee, it's said that these tangent vectors $frac{partial}{partial x^i}bigg|_p$ are called the coordinate vectors at $p$ assosciated with the given coordinate system.
But I don't exactly see how the basis vectors $frac{partial}{partial x^i}bigg|_p$ depend on a given coordinate system other than simply the homeomorphism $phi$. To me it seems that the right hand side of $(*)$, that being $frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right)$, is just the usual partial derivative operator in $mathbb{R}^n$, so by that I mean that $$frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) = lim_{t to 0} frac{(f circ phi^{-1})(phi(p) + t e_i) - (fcirc phi^{-1})(phi(p))}{t}$$
where $e_i$ is the element $(0, dots, 1, dots, 0) in mathbb{R}^n$ with a $1$ in the $i$-th position of the $n$-tuple. Am I correct in saying that? I think I am wrong, because what if I have some $2$-dimensional manifold $N$, and a chart $(V, psi)$ where points of $widehat{V}$ are best expressed in polar coordiantes and I wanted to compute $frac{partial}{partial x^i}bigg|_{q}(f)$ for some $q in N$ and $f in C^{infty}(V)$. I don't see how my interpretation of $(*)$ would make sense then there.
Maybe I'm getting lost in notation but for example I don't understand what the relation between the symbolic $x^i$'s in the notation of the component functions $x^i$ of $phi$, the basis vectors $frac{partial}{partial x^i}bigg|_{p}$, and $frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) $ are other than they are simply a reminder for the latter two that we're taking the $i$-th partial derivative in $widehat{U}$. The notation used suggests there must be some stronger relationship between these.
differential-geometry
asked Jan 9 at 1:47
PerturbativePerturbative
4,29811551
$endgroup$
Let's say I have a smooth manifold $M$ of dimension $n$ and a smooth chart $(U, phi)$ consisting of open set $U$ of $M$ along with a homeomorphism $phi : U to widehat{U} =phi[U] subseteq mathbb{R}^n$.
Recall that the component functions $x^i : U to mathbb{R}$ of $phi$ defined by $phi(p) = (x^1(p), dots, x^n(p))$ are called the local coordinates on $U$.
Now the basis for $T_p(M)$ is given by $$left{frac{partial}{partial x^1}bigg|_p, dots, frac{partial}{partial x^n}bigg|_pright}$$ where each $frac{partial}{partial x^i}bigg|_p$ is the derivation defined by $$frac{partial}{partial x^i}bigg|_p(f) = frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) (*)$$ for $f in C^{infty}(U)$.
Now in Introduction to Smooth Manifolds by John Lee, it's said that these tangent vectors $frac{partial}{partial x^i}bigg|_p$ are called the coordinate vectors at $p$ assosciated with the given coordinate system.
But I don't exactly see how the basis vectors $frac{partial}{partial x^i}bigg|_p$ depend on a given coordinate system other than simply the homeomorphism $phi$. To me it seems that the right hand side of $(*)$, that being $frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right)$, is just the usual partial derivative operator in $mathbb{R}^n$, so by that I mean that $$frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) = lim_{t to 0} frac{(f circ phi^{-1})(phi(p) + t e_i) - (fcirc phi^{-1})(phi(p))}{t}$$
where $e_i$ is the element $(0, dots, 1, dots, 0) in mathbb{R}^n$ with a $1$ in the $i$-th position of the $n$-tuple. Am I correct in saying that? I think I am wrong, because what if I have some $2$-dimensional manifold $N$, and a chart $(V, psi)$ where points of $widehat{V}$ are best expressed in polar coordiantes and I wanted to compute $frac{partial}{partial x^i}bigg|_{q}(f)$ for some $q in N$ and $f in C^{infty}(V)$. I don't see how my interpretation of $(*)$ would make sense then there.
Maybe I'm getting lost in notation but for example I don't understand what the relation between the symbolic $x^i$'s in the notation of the component functions $x^i$ of $phi$, the basis vectors $frac{partial}{partial x^i}bigg|_{p}$, and $frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) $ are other than they are simply a reminder for the latter two that we're taking the $i$-th partial derivative in $widehat{U}$. The notation used suggests there must be some stronger relationship between these.
differential-geometry
differential-geometry
asked Jan 9 at 1:47
PerturbativePerturbative
4,29811551
asked Jan 9 at 1:47
PerturbativePerturbative
4,29811551
asked Jan 9 at 1:47
PerturbativePerturbative
4,29811551
asked Jan 9 at 1:47
PerturbativePerturbative
4,29811551
asked Jan 9 at 1:47
PerturbativePerturbative
4,29811551
4,29811551
add a comment |
add a comment |
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I'm not shure I have fully understood your question, but hopefully what I say can help.
1) When you say it's said that these tangent vectors $frac{partial}{partial x^i}bigg|_p$ are called the coordinate vectors at $p$ assosciated with the given coordinate system.
Here by the given coordinate system we mean simply the chart $phi$. If $phi=(x^1,dots,x^n)$ then $(x^1,dots,x^n)$ are exactly the coordinates we are referring to.
2) As regards the symbolic $x^i$'s, they are present in the notation of the basis vectors $frac{partial}{partial x^i}bigg|_{p}$, beacause this definition depends on $phi$ and so on the function $x^i$'s, but they should not be in the notation of $frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) $.
Beacuse, as you sad, $$frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) = lim_{t to 0} frac{(f circ phi^{-1})(phi(p) + t e_i) - (fcirc phi^{-1})(phi(p))}{t}$$ and in the right hand side we have no link to the coordinate functions of $phi$, i.e. to te $x^i$'s.
Maybe it could be better to write $$partial_ibigg|_{phi(p)}left(f circ phi^{-1} right) = lim_{t to 0} frac{(f circ phi^{-1})(phi(p) + t e_i) - (fcirc phi^{-1})(phi(p))}{t}$$
Definitely, we should define $frac{partial}{partial x^i}bigg|_{p}f=partial_ibigg|_{phi(p)}left(f circ phi^{-1} right)$
answered Jan 9 at 18:14
MinatoMinato
467213
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I'm not shure I have fully understood your question, but hopefully what I say can help.
1) When you say it's said that these tangent vectors $frac{partial}{partial x^i}bigg|_p$ are called the coordinate vectors at $p$ assosciated with the given coordinate system.
Here by the given coordinate system we mean simply the chart $phi$. If $phi=(x^1,dots,x^n)$ then $(x^1,dots,x^n)$ are exactly the coordinates we are referring to.
2) As regards the symbolic $x^i$'s, they are present in the notation of the basis vectors $frac{partial}{partial x^i}bigg|_{p}$, beacause this definition depends on $phi$ and so on the function $x^i$'s, but they should not be in the notation of $frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) $.
Beacuse, as you sad, $$frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) = lim_{t to 0} frac{(f circ phi^{-1})(phi(p) + t e_i) - (fcirc phi^{-1})(phi(p))}{t}$$ and in the right hand side we have no link to the coordinate functions of $phi$, i.e. to te $x^i$'s.
Maybe it could be better to write $$partial_ibigg|_{phi(p)}left(f circ phi^{-1} right) = lim_{t to 0} frac{(f circ phi^{-1})(phi(p) + t e_i) - (fcirc phi^{-1})(phi(p))}{t}$$
Definitely, we should define $frac{partial}{partial x^i}bigg|_{p}f=partial_ibigg|_{phi(p)}left(f circ phi^{-1} right)$
answered Jan 9 at 18:14
MinatoMinato
467213
$endgroup$
add a comment |
$begingroup$
I'm not shure I have fully understood your question, but hopefully what I say can help.
1) When you say it's said that these tangent vectors $frac{partial}{partial x^i}bigg|_p$ are called the coordinate vectors at $p$ assosciated with the given coordinate system.
Here by the given coordinate system we mean simply the chart $phi$. If $phi=(x^1,dots,x^n)$ then $(x^1,dots,x^n)$ are exactly the coordinates we are referring to.
2) As regards the symbolic $x^i$'s, they are present in the notation of the basis vectors $frac{partial}{partial x^i}bigg|_{p}$, beacause this definition depends on $phi$ and so on the function $x^i$'s, but they should not be in the notation of $frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) $.
Beacuse, as you sad, $$frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) = lim_{t to 0} frac{(f circ phi^{-1})(phi(p) + t e_i) - (fcirc phi^{-1})(phi(p))}{t}$$ and in the right hand side we have no link to the coordinate functions of $phi$, i.e. to te $x^i$'s.
Maybe it could be better to write $$partial_ibigg|_{phi(p)}left(f circ phi^{-1} right) = lim_{t to 0} frac{(f circ phi^{-1})(phi(p) + t e_i) - (fcirc phi^{-1})(phi(p))}{t}$$
Definitely, we should define $frac{partial}{partial x^i}bigg|_{p}f=partial_ibigg|_{phi(p)}left(f circ phi^{-1} right)$
answered Jan 9 at 18:14
MinatoMinato
467213
$endgroup$
add a comment |
$begingroup$
I'm not shure I have fully understood your question, but hopefully what I say can help.
1) When you say it's said that these tangent vectors $frac{partial}{partial x^i}bigg|_p$ are called the coordinate vectors at $p$ assosciated with the given coordinate system.
Here by the given coordinate system we mean simply the chart $phi$. If $phi=(x^1,dots,x^n)$ then $(x^1,dots,x^n)$ are exactly the coordinates we are referring to.
2) As regards the symbolic $x^i$'s, they are present in the notation of the basis vectors $frac{partial}{partial x^i}bigg|_{p}$, beacause this definition depends on $phi$ and so on the function $x^i$'s, but they should not be in the notation of $frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) $.
Beacuse, as you sad, $$frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) = lim_{t to 0} frac{(f circ phi^{-1})(phi(p) + t e_i) - (fcirc phi^{-1})(phi(p))}{t}$$ and in the right hand side we have no link to the coordinate functions of $phi$, i.e. to te $x^i$'s.
Maybe it could be better to write $$partial_ibigg|_{phi(p)}left(f circ phi^{-1} right) = lim_{t to 0} frac{(f circ phi^{-1})(phi(p) + t e_i) - (fcirc phi^{-1})(phi(p))}{t}$$
Definitely, we should define $frac{partial}{partial x^i}bigg|_{p}f=partial_ibigg|_{phi(p)}left(f circ phi^{-1} right)$
answered Jan 9 at 18:14
MinatoMinato
467213
$endgroup$
I'm not shure I have fully understood your question, but hopefully what I say can help.
1) When you say it's said that these tangent vectors $frac{partial}{partial x^i}bigg|_p$ are called the coordinate vectors at $p$ assosciated with the given coordinate system.
Here by the given coordinate system we mean simply the chart $phi$. If $phi=(x^1,dots,x^n)$ then $(x^1,dots,x^n)$ are exactly the coordinates we are referring to.
2) As regards the symbolic $x^i$'s, they are present in the notation of the basis vectors $frac{partial}{partial x^i}bigg|_{p}$, beacause this definition depends on $phi$ and so on the function $x^i$'s, but they should not be in the notation of $frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) $.
Beacuse, as you sad, $$frac{partial}{partial x^i}bigg|_{phi(p)}left(f circ phi^{-1} right) = lim_{t to 0} frac{(f circ phi^{-1})(phi(p) + t e_i) - (fcirc phi^{-1})(phi(p))}{t}$$ and in the right hand side we have no link to the coordinate functions of $phi$, i.e. to te $x^i$'s.
Maybe it could be better to write $$partial_ibigg|_{phi(p)}left(f circ phi^{-1} right) = lim_{t to 0} frac{(f circ phi^{-1})(phi(p) + t e_i) - (fcirc phi^{-1})(phi(p))}{t}$$
Definitely, we should define $frac{partial}{partial x^i}bigg|_{p}f=partial_ibigg|_{phi(p)}left(f circ phi^{-1} right)$
answered Jan 9 at 18:14
MinatoMinato
467213
answered Jan 9 at 18:14
MinatoMinato
467213
answered Jan 9 at 18:14
MinatoMinato
467213
answered Jan 9 at 18:14
MinatoMinato
467213
467213
add a comment |
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