Mixing
Computing the differential of a certain smooth map
$begingroup$
Let $M subseteq mathbb{R}^k$ be an embedded submanifold of $mathbb{R}^k$, with dim$M=n$.
Let $v$ be in $mathbb{S}^{k-1}$, and let $P_v:mathbb{R}^kto(mathbb{R}v)^{bot}$ defined by $P_v(x)=x-<x,v>v$ where $<x,v>=x^1v^1+dots+x^kv^k$ is the Euclidean dot product and $(mathbb{R}v)^{bot}$ is the vector space of the vectors in $mathbb{R}^k$ orthogonal to $v$. We know that $(mathbb{R}v)^{bot}$ has dimension $k-1$ as a real vector space, so it is isomorphic to $mathbb{R}^{k-1}$.
Question 1) If I want to consider $(mathbb{R}v)^{bot}$ as a smooth manifold, I can choose an isomorphism between $(mathbb{R}v)^{bot}$ and $mathbb{R}^{k-1}$ and declare this to be a diffeomorphism, right? Or, better, is there a canonical identification between $(mathbb{R}v)^{bot}$ and $mathbb{R}^{k-1}$ ?
Now, (assuming the answer to Question 1 is affirmative), I have that $P_v:mathbb{R}^kto(mathbb{R}v)^{bot}simeq mathbb{R}^{k-1}$ is a map between two smooth manifolds.
Question 2) How can I show that this map is smooth? Do I have to calculate it in local coordinates? So do I have to explictly choose an isomorphism between $(mathbb{R}v)^{bot}$ and $mathbb{R}^{k-1}$ and then calculate the map in local coordinates?
Let $Phi_v:Mto (mathbb{R}v)^{bot}$ be $P_vcirc iota_M$ with $iota_M$ the inclusion map of $M$ in $mathbb{R}^k$. So, since $P_v$ is smooth, then also $Phi_v$ is smooth. Let $pin M$ and $Xin T_pM$. Then $d(iota_M)_p(X)=sum_iw^i partial_i|_pin T_pmathbb{R}^k$ for some $win mathbb{R}^k$.
My notes say that $$d(Phi_v)_p(X)=sum_iP_v(w)^ipartial_i|_p$$
Question 3) How can I prove the above equation? How should I imagine $T_{Phi_v(p)}((mathbb{R}v)^{bot})$? Who is a (canonical) basis for $T_{Phi_v(p)}((mathbb{R}v)^{bot})$?
All I can see is that $$d(Phi_v)_p(X)=d(P_v)_p(sum_iw^ipartial_i|_p)=sum_iw^id(P_v)_p(partial_i|_p)$$ where the last $=$ is by the linearity of the differential.
Please use simple language since I'm a beginner in this subject, do full calculation if needed, and also, if you think I lack some knowledge of some topic of smooth manifold theory (useful to better understand your answer to my question), please let me know.
multivariable-calculus differential-geometry smooth-manifolds smooth-functions
asked Jan 2 at 12:06
MinatoMinato
445212
$endgroup$
add a comment |
$begingroup$
Let $M subseteq mathbb{R}^k$ be an embedded submanifold of $mathbb{R}^k$, with dim$M=n$.
Let $v$ be in $mathbb{S}^{k-1}$, and let $P_v:mathbb{R}^kto(mathbb{R}v)^{bot}$ defined by $P_v(x)=x-<x,v>v$ where $<x,v>=x^1v^1+dots+x^kv^k$ is the Euclidean dot product and $(mathbb{R}v)^{bot}$ is the vector space of the vectors in $mathbb{R}^k$ orthogonal to $v$. We know that $(mathbb{R}v)^{bot}$ has dimension $k-1$ as a real vector space, so it is isomorphic to $mathbb{R}^{k-1}$.
Question 1) If I want to consider $(mathbb{R}v)^{bot}$ as a smooth manifold, I can choose an isomorphism between $(mathbb{R}v)^{bot}$ and $mathbb{R}^{k-1}$ and declare this to be a diffeomorphism, right? Or, better, is there a canonical identification between $(mathbb{R}v)^{bot}$ and $mathbb{R}^{k-1}$ ?
Now, (assuming the answer to Question 1 is affirmative), I have that $P_v:mathbb{R}^kto(mathbb{R}v)^{bot}simeq mathbb{R}^{k-1}$ is a map between two smooth manifolds.
Question 2) How can I show that this map is smooth? Do I have to calculate it in local coordinates? So do I have to explictly choose an isomorphism between $(mathbb{R}v)^{bot}$ and $mathbb{R}^{k-1}$ and then calculate the map in local coordinates?
Let $Phi_v:Mto (mathbb{R}v)^{bot}$ be $P_vcirc iota_M$ with $iota_M$ the inclusion map of $M$ in $mathbb{R}^k$. So, since $P_v$ is smooth, then also $Phi_v$ is smooth. Let $pin M$ and $Xin T_pM$. Then $d(iota_M)_p(X)=sum_iw^i partial_i|_pin T_pmathbb{R}^k$ for some $win mathbb{R}^k$.
My notes say that $$d(Phi_v)_p(X)=sum_iP_v(w)^ipartial_i|_p$$
Question 3) How can I prove the above equation? How should I imagine $T_{Phi_v(p)}((mathbb{R}v)^{bot})$? Who is a (canonical) basis for $T_{Phi_v(p)}((mathbb{R}v)^{bot})$?
All I can see is that $$d(Phi_v)_p(X)=d(P_v)_p(sum_iw^ipartial_i|_p)=sum_iw^id(P_v)_p(partial_i|_p)$$ where the last $=$ is by the linearity of the differential.
Please use simple language since I'm a beginner in this subject, do full calculation if needed, and also, if you think I lack some knowledge of some topic of smooth manifold theory (useful to better understand your answer to my question), please let me know.
multivariable-calculus differential-geometry smooth-manifolds smooth-functions
asked Jan 2 at 12:06
MinatoMinato
445212
$endgroup$
1
$begingroup$
You need to work through this yourself. You are trying to build fluency in a language. The details you are asking for are primitive and they can only flow if you first squeeze them out, step by step.
$endgroup$
– Charlie Frohman
Jan 2 at 13:59
1
$begingroup$
I fully agree with you, but self studing this subject is not easy, and since I'm studing alone ,it is also difficult to be self aware if I misunderstood something or I wrongly believed to have understood something. For example, in this situation, I'm not even shure if my problem is that I lack knowledge, or if misunderstood something or do I have all the ingredients, it's only a matter of doing the exact obeservation in the right way
$endgroup$
– Minato
Jan 2 at 15:00
$begingroup$
I should add that if think my notes are not "linearly progressive". For example, in my question math.stackexchange.com/questions/2873996/… I think I need the notion of vector field, since I (think) I want to consider a map $phi(U)to Tphi(U)$ defined by $xmapsto d(phi^{-1})_x$ as smooth, in order to write that map $G$ as a composition of smooth maps. But in my notes the chapter of vector field is only after!
$endgroup$
– Minato
Jan 2 at 15:18
add a comment |
$begingroup$
Let $M subseteq mathbb{R}^k$ be an embedded submanifold of $mathbb{R}^k$, with dim$M=n$.
Let $v$ be in $mathbb{S}^{k-1}$, and let $P_v:mathbb{R}^kto(mathbb{R}v)^{bot}$ defined by $P_v(x)=x-<x,v>v$ where $<x,v>=x^1v^1+dots+x^kv^k$ is the Euclidean dot product and $(mathbb{R}v)^{bot}$ is the vector space of the vectors in $mathbb{R}^k$ orthogonal to $v$. We know that $(mathbb{R}v)^{bot}$ has dimension $k-1$ as a real vector space, so it is isomorphic to $mathbb{R}^{k-1}$.
Question 1) If I want to consider $(mathbb{R}v)^{bot}$ as a smooth manifold, I can choose an isomorphism between $(mathbb{R}v)^{bot}$ and $mathbb{R}^{k-1}$ and declare this to be a diffeomorphism, right? Or, better, is there a canonical identification between $(mathbb{R}v)^{bot}$ and $mathbb{R}^{k-1}$ ?
Now, (assuming the answer to Question 1 is affirmative), I have that $P_v:mathbb{R}^kto(mathbb{R}v)^{bot}simeq mathbb{R}^{k-1}$ is a map between two smooth manifolds.
Question 2) How can I show that this map is smooth? Do I have to calculate it in local coordinates? So do I have to explictly choose an isomorphism between $(mathbb{R}v)^{bot}$ and $mathbb{R}^{k-1}$ and then calculate the map in local coordinates?
Let $Phi_v:Mto (mathbb{R}v)^{bot}$ be $P_vcirc iota_M$ with $iota_M$ the inclusion map of $M$ in $mathbb{R}^k$. So, since $P_v$ is smooth, then also $Phi_v$ is smooth. Let $pin M$ and $Xin T_pM$. Then $d(iota_M)_p(X)=sum_iw^i partial_i|_pin T_pmathbb{R}^k$ for some $win mathbb{R}^k$.
My notes say that $$d(Phi_v)_p(X)=sum_iP_v(w)^ipartial_i|_p$$
Question 3) How can I prove the above equation? How should I imagine $T_{Phi_v(p)}((mathbb{R}v)^{bot})$? Who is a (canonical) basis for $T_{Phi_v(p)}((mathbb{R}v)^{bot})$?
All I can see is that $$d(Phi_v)_p(X)=d(P_v)_p(sum_iw^ipartial_i|_p)=sum_iw^id(P_v)_p(partial_i|_p)$$ where the last $=$ is by the linearity of the differential.
Please use simple language since I'm a beginner in this subject, do full calculation if needed, and also, if you think I lack some knowledge of some topic of smooth manifold theory (useful to better understand your answer to my question), please let me know.
multivariable-calculus differential-geometry smooth-manifolds smooth-functions
asked Jan 2 at 12:06
MinatoMinato
445212
$endgroup$
Let $M subseteq mathbb{R}^k$ be an embedded submanifold of $mathbb{R}^k$, with dim$M=n$.
Let $v$ be in $mathbb{S}^{k-1}$, and let $P_v:mathbb{R}^kto(mathbb{R}v)^{bot}$ defined by $P_v(x)=x-<x,v>v$ where $<x,v>=x^1v^1+dots+x^kv^k$ is the Euclidean dot product and $(mathbb{R}v)^{bot}$ is the vector space of the vectors in $mathbb{R}^k$ orthogonal to $v$. We know that $(mathbb{R}v)^{bot}$ has dimension $k-1$ as a real vector space, so it is isomorphic to $mathbb{R}^{k-1}$.
Question 1) If I want to consider $(mathbb{R}v)^{bot}$ as a smooth manifold, I can choose an isomorphism between $(mathbb{R}v)^{bot}$ and $mathbb{R}^{k-1}$ and declare this to be a diffeomorphism, right? Or, better, is there a canonical identification between $(mathbb{R}v)^{bot}$ and $mathbb{R}^{k-1}$ ?
Now, (assuming the answer to Question 1 is affirmative), I have that $P_v:mathbb{R}^kto(mathbb{R}v)^{bot}simeq mathbb{R}^{k-1}$ is a map between two smooth manifolds.
Question 2) How can I show that this map is smooth? Do I have to calculate it in local coordinates? So do I have to explictly choose an isomorphism between $(mathbb{R}v)^{bot}$ and $mathbb{R}^{k-1}$ and then calculate the map in local coordinates?
Let $Phi_v:Mto (mathbb{R}v)^{bot}$ be $P_vcirc iota_M$ with $iota_M$ the inclusion map of $M$ in $mathbb{R}^k$. So, since $P_v$ is smooth, then also $Phi_v$ is smooth. Let $pin M$ and $Xin T_pM$. Then $d(iota_M)_p(X)=sum_iw^i partial_i|_pin T_pmathbb{R}^k$ for some $win mathbb{R}^k$.
My notes say that $$d(Phi_v)_p(X)=sum_iP_v(w)^ipartial_i|_p$$
Question 3) How can I prove the above equation? How should I imagine $T_{Phi_v(p)}((mathbb{R}v)^{bot})$? Who is a (canonical) basis for $T_{Phi_v(p)}((mathbb{R}v)^{bot})$?
All I can see is that $$d(Phi_v)_p(X)=d(P_v)_p(sum_iw^ipartial_i|_p)=sum_iw^id(P_v)_p(partial_i|_p)$$ where the last $=$ is by the linearity of the differential.
Please use simple language since I'm a beginner in this subject, do full calculation if needed, and also, if you think I lack some knowledge of some topic of smooth manifold theory (useful to better understand your answer to my question), please let me know.
multivariable-calculus differential-geometry smooth-manifolds smooth-functions
multivariable-calculus differential-geometry smooth-manifolds smooth-functions
asked Jan 2 at 12:06
MinatoMinato
445212
asked Jan 2 at 12:06
MinatoMinato
445212
edited Jan 2 at 13:47
Minato
asked Jan 2 at 12:06
MinatoMinato
445212
asked Jan 2 at 12:06
MinatoMinato
445212
asked Jan 2 at 12:06
MinatoMinato
445212
445212
1
$begingroup$
You need to work through this yourself. You are trying to build fluency in a language. The details you are asking for are primitive and they can only flow if you first squeeze them out, step by step.
$endgroup$
– Charlie Frohman
Jan 2 at 13:59
1
$begingroup$
I fully agree with you, but self studing this subject is not easy, and since I'm studing alone ,it is also difficult to be self aware if I misunderstood something or I wrongly believed to have understood something. For example, in this situation, I'm not even shure if my problem is that I lack knowledge, or if misunderstood something or do I have all the ingredients, it's only a matter of doing the exact obeservation in the right way
$endgroup$
– Minato
Jan 2 at 15:00
$begingroup$
I should add that if think my notes are not "linearly progressive". For example, in my question math.stackexchange.com/questions/2873996/… I think I need the notion of vector field, since I (think) I want to consider a map $phi(U)to Tphi(U)$ defined by $xmapsto d(phi^{-1})_x$ as smooth, in order to write that map $G$ as a composition of smooth maps. But in my notes the chapter of vector field is only after!
$endgroup$
– Minato
Jan 2 at 15:18
add a comment |
1
$begingroup$
You need to work through this yourself. You are trying to build fluency in a language. The details you are asking for are primitive and they can only flow if you first squeeze them out, step by step.
$endgroup$
– Charlie Frohman
Jan 2 at 13:59
1
$begingroup$
I fully agree with you, but self studing this subject is not easy, and since I'm studing alone ,it is also difficult to be self aware if I misunderstood something or I wrongly believed to have understood something. For example, in this situation, I'm not even shure if my problem is that I lack knowledge, or if misunderstood something or do I have all the ingredients, it's only a matter of doing the exact obeservation in the right way
$endgroup$
– Minato
Jan 2 at 15:00
$begingroup$
I should add that if think my notes are not "linearly progressive". For example, in my question math.stackexchange.com/questions/2873996/… I think I need the notion of vector field, since I (think) I want to consider a map $phi(U)to Tphi(U)$ defined by $xmapsto d(phi^{-1})_x$ as smooth, in order to write that map $G$ as a composition of smooth maps. But in my notes the chapter of vector field is only after!
$endgroup$
– Minato
Jan 2 at 15:18
1
1
$begingroup$
You need to work through this yourself. You are trying to build fluency in a language. The details you are asking for are primitive and they can only flow if you first squeeze them out, step by step.
$endgroup$
– Charlie Frohman
Jan 2 at 13:59
$begingroup$
You need to work through this yourself. You are trying to build fluency in a language. The details you are asking for are primitive and they can only flow if you first squeeze them out, step by step.
$endgroup$
– Charlie Frohman
Jan 2 at 13:59
1
1
$begingroup$
I fully agree with you, but self studing this subject is not easy, and since I'm studing alone ,it is also difficult to be self aware if I misunderstood something or I wrongly believed to have understood something. For example, in this situation, I'm not even shure if my problem is that I lack knowledge, or if misunderstood something or do I have all the ingredients, it's only a matter of doing the exact obeservation in the right way
$endgroup$
– Minato
Jan 2 at 15:00
$begingroup$
I fully agree with you, but self studing this subject is not easy, and since I'm studing alone ,it is also difficult to be self aware if I misunderstood something or I wrongly believed to have understood something. For example, in this situation, I'm not even shure if my problem is that I lack knowledge, or if misunderstood something or do I have all the ingredients, it's only a matter of doing the exact obeservation in the right way
$endgroup$
– Minato
Jan 2 at 15:00
$begingroup$
I should add that if think my notes are not "linearly progressive". For example, in my question math.stackexchange.com/questions/2873996/… I think I need the notion of vector field, since I (think) I want to consider a map $phi(U)to Tphi(U)$ defined by $xmapsto d(phi^{-1})_x$ as smooth, in order to write that map $G$ as a composition of smooth maps. But in my notes the chapter of vector field is only after!
$endgroup$
– Minato
Jan 2 at 15:18
$begingroup$
I should add that if think my notes are not "linearly progressive". For example, in my question math.stackexchange.com/questions/2873996/… I think I need the notion of vector field, since I (think) I want to consider a map $phi(U)to Tphi(U)$ defined by $xmapsto d(phi^{-1})_x$ as smooth, in order to write that map $G$ as a composition of smooth maps. But in my notes the chapter of vector field is only after!
$endgroup$
– Minato
Jan 2 at 15:18
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
There is no canonical choice. Probably the best way to deal with it is to treat it as a regular submanifold. You can then treat its tangent space as a subspace of the ambient manifold, and only choose coordinates once.
The definition of smooth mapping involves local coordinates so... yes. However having chosen slice coordinates the map is just cutting off the last few coordinates, and since the coordinate functions are smoot you are in.
- Once again this is just a statement in local coordinates, depending on the first question.
Loring Tu’s Book on manifolds is the one most beginners find easiest to read. The exercises are straightforward plug and chug based on the material being studied.
answered Jan 2 at 16:30
Charlie FrohmanCharlie Frohman
1,468913
$endgroup$
add a comment |
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$begingroup$
There is no canonical choice. Probably the best way to deal with it is to treat it as a regular submanifold. You can then treat its tangent space as a subspace of the ambient manifold, and only choose coordinates once.
The definition of smooth mapping involves local coordinates so... yes. However having chosen slice coordinates the map is just cutting off the last few coordinates, and since the coordinate functions are smoot you are in.
- Once again this is just a statement in local coordinates, depending on the first question.
Loring Tu’s Book on manifolds is the one most beginners find easiest to read. The exercises are straightforward plug and chug based on the material being studied.
answered Jan 2 at 16:30
Charlie FrohmanCharlie Frohman
1,468913
$endgroup$
add a comment |
$begingroup$
There is no canonical choice. Probably the best way to deal with it is to treat it as a regular submanifold. You can then treat its tangent space as a subspace of the ambient manifold, and only choose coordinates once.
The definition of smooth mapping involves local coordinates so... yes. However having chosen slice coordinates the map is just cutting off the last few coordinates, and since the coordinate functions are smoot you are in.
- Once again this is just a statement in local coordinates, depending on the first question.
Loring Tu’s Book on manifolds is the one most beginners find easiest to read. The exercises are straightforward plug and chug based on the material being studied.
answered Jan 2 at 16:30
Charlie FrohmanCharlie Frohman
1,468913
$endgroup$
add a comment |
$begingroup$
There is no canonical choice. Probably the best way to deal with it is to treat it as a regular submanifold. You can then treat its tangent space as a subspace of the ambient manifold, and only choose coordinates once.
The definition of smooth mapping involves local coordinates so... yes. However having chosen slice coordinates the map is just cutting off the last few coordinates, and since the coordinate functions are smoot you are in.
- Once again this is just a statement in local coordinates, depending on the first question.
Loring Tu’s Book on manifolds is the one most beginners find easiest to read. The exercises are straightforward plug and chug based on the material being studied.
answered Jan 2 at 16:30
Charlie FrohmanCharlie Frohman
1,468913
$endgroup$
There is no canonical choice. Probably the best way to deal with it is to treat it as a regular submanifold. You can then treat its tangent space as a subspace of the ambient manifold, and only choose coordinates once.
The definition of smooth mapping involves local coordinates so... yes. However having chosen slice coordinates the map is just cutting off the last few coordinates, and since the coordinate functions are smoot you are in.
- Once again this is just a statement in local coordinates, depending on the first question.
Loring Tu’s Book on manifolds is the one most beginners find easiest to read. The exercises are straightforward plug and chug based on the material being studied.
answered Jan 2 at 16:30
Charlie FrohmanCharlie Frohman
1,468913
answered Jan 2 at 16:30
Charlie FrohmanCharlie Frohman
1,468913
answered Jan 2 at 16:30
Charlie FrohmanCharlie Frohman
1,468913
answered Jan 2 at 16:30
Charlie FrohmanCharlie Frohman
1,468913
1,468913
add a comment |
add a comment |
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$begingroup$
You need to work through this yourself. You are trying to build fluency in a language. The details you are asking for are primitive and they can only flow if you first squeeze them out, step by step.
$endgroup$
– Charlie Frohman
Jan 2 at 13:59
1
$begingroup$
I fully agree with you, but self studing this subject is not easy, and since I'm studing alone ,it is also difficult to be self aware if I misunderstood something or I wrongly believed to have understood something. For example, in this situation, I'm not even shure if my problem is that I lack knowledge, or if misunderstood something or do I have all the ingredients, it's only a matter of doing the exact obeservation in the right way
$endgroup$
– Minato
Jan 2 at 15:00
$begingroup$
I should add that if think my notes are not "linearly progressive". For example, in my question math.stackexchange.com/questions/2873996/… I think I need the notion of vector field, since I (think) I want to consider a map $phi(U)to Tphi(U)$ defined by $xmapsto d(phi^{-1})_x$ as smooth, in order to write that map $G$ as a composition of smooth maps. But in my notes the chapter of vector field is only after!
$endgroup$
– Minato
Jan 2 at 15:18