Problem 16.4 on related vector fields Differentiable Manifolds Marcut
$begingroup$
In the course I am taking on differentiable manifolds, we have been using the notes written by Marcut . This problem caught my attention as it seemed pretty straightforward.
Consider the map:
$$phi: mathbb{R}^2 to mathbb{R}^2,~ phi(x,y):=(e^xcos(y),e^xsin(y))$$
- Find two vector fields $X,~Y~in~mathfrak{X}(mathbb{R}^2$ such that $X$ is $phi$-related to $partial_x$ and $Y$ is $phi$-related to $partial_y$. Are these vector fields uniquely determined by this condition?. Calculate $[X,Y]$
- Find two vector fields $U,~V~in~mathfrak{X}(mathbb{R}^2$ such that $partial_x$ is $phi$-related to $U$ and $partial_y$ is $phi$-related to $V$. Are these vector fields uniquely determined by this condition?. Calculate $[U,V]$
Solution
I have found and checked the solutions for both parts.
For part a), I used the fact that the required condition is equivalent to $ partial_x(phi^*(f))=phi^*(X( f)) label{eq:related}$. I find $X$. Using the same procedure for $Y$, I get that:
$$X = xpartial_x+ypartial_y,~ Y= -ypartial_x+xpartial_y$$
$$[X,Y]=0$$
For part b), I noticed that $U$ can be written as $U=u_1(x,y)partial_x+u_2(x,y)partial_y$. Using the condition that $U(phi^*(f))_p=(partial_u( f))_{phi(p)}$, I find $U$. Using the same procedure for $V$, I get:
$$U= e^{-x}cos(y)partial_x-e^{-x}sin(y)partial_y,~V= e^{-x}sin(y)partial_x+e^{-x}cos(y)partial_y$$
$$[U,V]=0$$
So in summary:
- In part a), we are finding the pushforward of the coordinate directions: $$X=phi_*(partial_x),~Y=phi_*(partial_y) $$
- In part b), we are finding vector fields that correspond to the pushforward of he coordinate vector fields:
$$U,~V~ text{such that}~ phi_*(U)=partial_x,~phi_*(V)=partial_y$$.
Because the pushfoward has the property that $F_*[X,Y]=[F_*X,F_*Y]$, I know that the lie bracket in both cases should be 0, since $[partial_x, partial_y]=0$
My questions are:
Is my summary of the problem correct?
I am not sure if the vector fields in both cases are uniquely determined by these conditions. I think they are uniquely determined by these conditions if $phi$ was bijective. But that is not the case. How can I know if those fields are uniquely determined?
differential-geometry vector-fields
$endgroup$
add a comment |
$begingroup$
In the course I am taking on differentiable manifolds, we have been using the notes written by Marcut . This problem caught my attention as it seemed pretty straightforward.
Consider the map:
$$phi: mathbb{R}^2 to mathbb{R}^2,~ phi(x,y):=(e^xcos(y),e^xsin(y))$$
- Find two vector fields $X,~Y~in~mathfrak{X}(mathbb{R}^2$ such that $X$ is $phi$-related to $partial_x$ and $Y$ is $phi$-related to $partial_y$. Are these vector fields uniquely determined by this condition?. Calculate $[X,Y]$
- Find two vector fields $U,~V~in~mathfrak{X}(mathbb{R}^2$ such that $partial_x$ is $phi$-related to $U$ and $partial_y$ is $phi$-related to $V$. Are these vector fields uniquely determined by this condition?. Calculate $[U,V]$
Solution
I have found and checked the solutions for both parts.
For part a), I used the fact that the required condition is equivalent to $ partial_x(phi^*(f))=phi^*(X( f)) label{eq:related}$. I find $X$. Using the same procedure for $Y$, I get that:
$$X = xpartial_x+ypartial_y,~ Y= -ypartial_x+xpartial_y$$
$$[X,Y]=0$$
For part b), I noticed that $U$ can be written as $U=u_1(x,y)partial_x+u_2(x,y)partial_y$. Using the condition that $U(phi^*(f))_p=(partial_u( f))_{phi(p)}$, I find $U$. Using the same procedure for $V$, I get:
$$U= e^{-x}cos(y)partial_x-e^{-x}sin(y)partial_y,~V= e^{-x}sin(y)partial_x+e^{-x}cos(y)partial_y$$
$$[U,V]=0$$
So in summary:
- In part a), we are finding the pushforward of the coordinate directions: $$X=phi_*(partial_x),~Y=phi_*(partial_y) $$
- In part b), we are finding vector fields that correspond to the pushforward of he coordinate vector fields:
$$U,~V~ text{such that}~ phi_*(U)=partial_x,~phi_*(V)=partial_y$$.
Because the pushfoward has the property that $F_*[X,Y]=[F_*X,F_*Y]$, I know that the lie bracket in both cases should be 0, since $[partial_x, partial_y]=0$
My questions are:
Is my summary of the problem correct?
I am not sure if the vector fields in both cases are uniquely determined by these conditions. I think they are uniquely determined by these conditions if $phi$ was bijective. But that is not the case. How can I know if those fields are uniquely determined?
differential-geometry vector-fields
$endgroup$
$begingroup$
Whoa. $phi$ is far from bijective.
$endgroup$
– Ted Shifrin
Jan 5 at 22:21
$begingroup$
That is right (corrected that in the question). The map$phi$ is not bijective, it will cover $mathbb{R}^2$ many times as $sin(y)$ varies. Then, I have no idea about why those fields are uniquely determined.
$endgroup$
– Amphiaraos
Jan 6 at 3:10
add a comment |
$begingroup$
In the course I am taking on differentiable manifolds, we have been using the notes written by Marcut . This problem caught my attention as it seemed pretty straightforward.
Consider the map:
$$phi: mathbb{R}^2 to mathbb{R}^2,~ phi(x,y):=(e^xcos(y),e^xsin(y))$$
- Find two vector fields $X,~Y~in~mathfrak{X}(mathbb{R}^2$ such that $X$ is $phi$-related to $partial_x$ and $Y$ is $phi$-related to $partial_y$. Are these vector fields uniquely determined by this condition?. Calculate $[X,Y]$
- Find two vector fields $U,~V~in~mathfrak{X}(mathbb{R}^2$ such that $partial_x$ is $phi$-related to $U$ and $partial_y$ is $phi$-related to $V$. Are these vector fields uniquely determined by this condition?. Calculate $[U,V]$
Solution
I have found and checked the solutions for both parts.
For part a), I used the fact that the required condition is equivalent to $ partial_x(phi^*(f))=phi^*(X( f)) label{eq:related}$. I find $X$. Using the same procedure for $Y$, I get that:
$$X = xpartial_x+ypartial_y,~ Y= -ypartial_x+xpartial_y$$
$$[X,Y]=0$$
For part b), I noticed that $U$ can be written as $U=u_1(x,y)partial_x+u_2(x,y)partial_y$. Using the condition that $U(phi^*(f))_p=(partial_u( f))_{phi(p)}$, I find $U$. Using the same procedure for $V$, I get:
$$U= e^{-x}cos(y)partial_x-e^{-x}sin(y)partial_y,~V= e^{-x}sin(y)partial_x+e^{-x}cos(y)partial_y$$
$$[U,V]=0$$
So in summary:
- In part a), we are finding the pushforward of the coordinate directions: $$X=phi_*(partial_x),~Y=phi_*(partial_y) $$
- In part b), we are finding vector fields that correspond to the pushforward of he coordinate vector fields:
$$U,~V~ text{such that}~ phi_*(U)=partial_x,~phi_*(V)=partial_y$$.
Because the pushfoward has the property that $F_*[X,Y]=[F_*X,F_*Y]$, I know that the lie bracket in both cases should be 0, since $[partial_x, partial_y]=0$
My questions are:
Is my summary of the problem correct?
I am not sure if the vector fields in both cases are uniquely determined by these conditions. I think they are uniquely determined by these conditions if $phi$ was bijective. But that is not the case. How can I know if those fields are uniquely determined?
differential-geometry vector-fields
$endgroup$
In the course I am taking on differentiable manifolds, we have been using the notes written by Marcut . This problem caught my attention as it seemed pretty straightforward.
Consider the map:
$$phi: mathbb{R}^2 to mathbb{R}^2,~ phi(x,y):=(e^xcos(y),e^xsin(y))$$
- Find two vector fields $X,~Y~in~mathfrak{X}(mathbb{R}^2$ such that $X$ is $phi$-related to $partial_x$ and $Y$ is $phi$-related to $partial_y$. Are these vector fields uniquely determined by this condition?. Calculate $[X,Y]$
- Find two vector fields $U,~V~in~mathfrak{X}(mathbb{R}^2$ such that $partial_x$ is $phi$-related to $U$ and $partial_y$ is $phi$-related to $V$. Are these vector fields uniquely determined by this condition?. Calculate $[U,V]$
Solution
I have found and checked the solutions for both parts.
For part a), I used the fact that the required condition is equivalent to $ partial_x(phi^*(f))=phi^*(X( f)) label{eq:related}$. I find $X$. Using the same procedure for $Y$, I get that:
$$X = xpartial_x+ypartial_y,~ Y= -ypartial_x+xpartial_y$$
$$[X,Y]=0$$
For part b), I noticed that $U$ can be written as $U=u_1(x,y)partial_x+u_2(x,y)partial_y$. Using the condition that $U(phi^*(f))_p=(partial_u( f))_{phi(p)}$, I find $U$. Using the same procedure for $V$, I get:
$$U= e^{-x}cos(y)partial_x-e^{-x}sin(y)partial_y,~V= e^{-x}sin(y)partial_x+e^{-x}cos(y)partial_y$$
$$[U,V]=0$$
So in summary:
- In part a), we are finding the pushforward of the coordinate directions: $$X=phi_*(partial_x),~Y=phi_*(partial_y) $$
- In part b), we are finding vector fields that correspond to the pushforward of he coordinate vector fields:
$$U,~V~ text{such that}~ phi_*(U)=partial_x,~phi_*(V)=partial_y$$.
Because the pushfoward has the property that $F_*[X,Y]=[F_*X,F_*Y]$, I know that the lie bracket in both cases should be 0, since $[partial_x, partial_y]=0$
My questions are:
Is my summary of the problem correct?
I am not sure if the vector fields in both cases are uniquely determined by these conditions. I think they are uniquely determined by these conditions if $phi$ was bijective. But that is not the case. How can I know if those fields are uniquely determined?
differential-geometry vector-fields
differential-geometry vector-fields
edited Jan 6 at 3:11
Amphiaraos
asked Jan 5 at 20:43
AmphiaraosAmphiaraos
301211
301211
$begingroup$
Whoa. $phi$ is far from bijective.
$endgroup$
– Ted Shifrin
Jan 5 at 22:21
$begingroup$
That is right (corrected that in the question). The map$phi$ is not bijective, it will cover $mathbb{R}^2$ many times as $sin(y)$ varies. Then, I have no idea about why those fields are uniquely determined.
$endgroup$
– Amphiaraos
Jan 6 at 3:10
add a comment |
$begingroup$
Whoa. $phi$ is far from bijective.
$endgroup$
– Ted Shifrin
Jan 5 at 22:21
$begingroup$
That is right (corrected that in the question). The map$phi$ is not bijective, it will cover $mathbb{R}^2$ many times as $sin(y)$ varies. Then, I have no idea about why those fields are uniquely determined.
$endgroup$
– Amphiaraos
Jan 6 at 3:10
$begingroup$
Whoa. $phi$ is far from bijective.
$endgroup$
– Ted Shifrin
Jan 5 at 22:21
$begingroup$
Whoa. $phi$ is far from bijective.
$endgroup$
– Ted Shifrin
Jan 5 at 22:21
$begingroup$
That is right (corrected that in the question). The map$phi$ is not bijective, it will cover $mathbb{R}^2$ many times as $sin(y)$ varies. Then, I have no idea about why those fields are uniquely determined.
$endgroup$
– Amphiaraos
Jan 6 at 3:10
$begingroup$
That is right (corrected that in the question). The map$phi$ is not bijective, it will cover $mathbb{R}^2$ many times as $sin(y)$ varies. Then, I have no idea about why those fields are uniquely determined.
$endgroup$
– Amphiaraos
Jan 6 at 3:10
add a comment |
1 Answer
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oldest
votes
$begingroup$
I believe that the vector fields you found are correct, so I will only comment on the uniqueness.
Note that the differential of the map $phi$ is
begin{equation}tag{1}label{1}
d_{(x,y)}phi=begin{pmatrix} e^{x}cos y & -e^{x}sin y\ e^{x}sin y & e^{x}cos yend{pmatrix},
end{equation}
which is a linear isomorphism (its determinant is nonzero).
You found $X,Yinmathfrak{X}(mathbb{R}^{2})$ such that for all $pinmathbb{R}^{2}$:
begin{cases}
d_{p}phi[left(partial_{x}right)_{p}]=X_{phi(p)}\
d_{p}phi[left(partial_{y}right)_{p}]=Y_{phi(p)}.
end{cases}
These equations determine $X,Y$ uniquely on the image of $phi$. Since $text{Im}(phi)=mathbb{R}^{2}setminus{(0,0)}$ is dense in $mathbb{R}^{2}$, it follows by continuity that $X$ and $Y$ are uniquely determined.Now you found $U,Vinmathfrak{X}(mathbb{R}^{2})$ such that for all $pinmathbb{R}^{2}$:
begin{cases}
d_{p}phi(U_{p})=left(partial_{x}right)_{phi(p)}\
d_{p}phi(V_{p})=left(partial_{y}right)_{phi(p)}.
end{cases}
Again such $U,V$ are unique, since the differential eqref{1} is injective.
$endgroup$
add a comment |
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$begingroup$
I believe that the vector fields you found are correct, so I will only comment on the uniqueness.
Note that the differential of the map $phi$ is
begin{equation}tag{1}label{1}
d_{(x,y)}phi=begin{pmatrix} e^{x}cos y & -e^{x}sin y\ e^{x}sin y & e^{x}cos yend{pmatrix},
end{equation}
which is a linear isomorphism (its determinant is nonzero).
You found $X,Yinmathfrak{X}(mathbb{R}^{2})$ such that for all $pinmathbb{R}^{2}$:
begin{cases}
d_{p}phi[left(partial_{x}right)_{p}]=X_{phi(p)}\
d_{p}phi[left(partial_{y}right)_{p}]=Y_{phi(p)}.
end{cases}
These equations determine $X,Y$ uniquely on the image of $phi$. Since $text{Im}(phi)=mathbb{R}^{2}setminus{(0,0)}$ is dense in $mathbb{R}^{2}$, it follows by continuity that $X$ and $Y$ are uniquely determined.Now you found $U,Vinmathfrak{X}(mathbb{R}^{2})$ such that for all $pinmathbb{R}^{2}$:
begin{cases}
d_{p}phi(U_{p})=left(partial_{x}right)_{phi(p)}\
d_{p}phi(V_{p})=left(partial_{y}right)_{phi(p)}.
end{cases}
Again such $U,V$ are unique, since the differential eqref{1} is injective.
$endgroup$
add a comment |
$begingroup$
I believe that the vector fields you found are correct, so I will only comment on the uniqueness.
Note that the differential of the map $phi$ is
begin{equation}tag{1}label{1}
d_{(x,y)}phi=begin{pmatrix} e^{x}cos y & -e^{x}sin y\ e^{x}sin y & e^{x}cos yend{pmatrix},
end{equation}
which is a linear isomorphism (its determinant is nonzero).
You found $X,Yinmathfrak{X}(mathbb{R}^{2})$ such that for all $pinmathbb{R}^{2}$:
begin{cases}
d_{p}phi[left(partial_{x}right)_{p}]=X_{phi(p)}\
d_{p}phi[left(partial_{y}right)_{p}]=Y_{phi(p)}.
end{cases}
These equations determine $X,Y$ uniquely on the image of $phi$. Since $text{Im}(phi)=mathbb{R}^{2}setminus{(0,0)}$ is dense in $mathbb{R}^{2}$, it follows by continuity that $X$ and $Y$ are uniquely determined.Now you found $U,Vinmathfrak{X}(mathbb{R}^{2})$ such that for all $pinmathbb{R}^{2}$:
begin{cases}
d_{p}phi(U_{p})=left(partial_{x}right)_{phi(p)}\
d_{p}phi(V_{p})=left(partial_{y}right)_{phi(p)}.
end{cases}
Again such $U,V$ are unique, since the differential eqref{1} is injective.
$endgroup$
add a comment |
$begingroup$
I believe that the vector fields you found are correct, so I will only comment on the uniqueness.
Note that the differential of the map $phi$ is
begin{equation}tag{1}label{1}
d_{(x,y)}phi=begin{pmatrix} e^{x}cos y & -e^{x}sin y\ e^{x}sin y & e^{x}cos yend{pmatrix},
end{equation}
which is a linear isomorphism (its determinant is nonzero).
You found $X,Yinmathfrak{X}(mathbb{R}^{2})$ such that for all $pinmathbb{R}^{2}$:
begin{cases}
d_{p}phi[left(partial_{x}right)_{p}]=X_{phi(p)}\
d_{p}phi[left(partial_{y}right)_{p}]=Y_{phi(p)}.
end{cases}
These equations determine $X,Y$ uniquely on the image of $phi$. Since $text{Im}(phi)=mathbb{R}^{2}setminus{(0,0)}$ is dense in $mathbb{R}^{2}$, it follows by continuity that $X$ and $Y$ are uniquely determined.Now you found $U,Vinmathfrak{X}(mathbb{R}^{2})$ such that for all $pinmathbb{R}^{2}$:
begin{cases}
d_{p}phi(U_{p})=left(partial_{x}right)_{phi(p)}\
d_{p}phi(V_{p})=left(partial_{y}right)_{phi(p)}.
end{cases}
Again such $U,V$ are unique, since the differential eqref{1} is injective.
$endgroup$
I believe that the vector fields you found are correct, so I will only comment on the uniqueness.
Note that the differential of the map $phi$ is
begin{equation}tag{1}label{1}
d_{(x,y)}phi=begin{pmatrix} e^{x}cos y & -e^{x}sin y\ e^{x}sin y & e^{x}cos yend{pmatrix},
end{equation}
which is a linear isomorphism (its determinant is nonzero).
You found $X,Yinmathfrak{X}(mathbb{R}^{2})$ such that for all $pinmathbb{R}^{2}$:
begin{cases}
d_{p}phi[left(partial_{x}right)_{p}]=X_{phi(p)}\
d_{p}phi[left(partial_{y}right)_{p}]=Y_{phi(p)}.
end{cases}
These equations determine $X,Y$ uniquely on the image of $phi$. Since $text{Im}(phi)=mathbb{R}^{2}setminus{(0,0)}$ is dense in $mathbb{R}^{2}$, it follows by continuity that $X$ and $Y$ are uniquely determined.Now you found $U,Vinmathfrak{X}(mathbb{R}^{2})$ such that for all $pinmathbb{R}^{2}$:
begin{cases}
d_{p}phi(U_{p})=left(partial_{x}right)_{phi(p)}\
d_{p}phi(V_{p})=left(partial_{y}right)_{phi(p)}.
end{cases}
Again such $U,V$ are unique, since the differential eqref{1} is injective.
answered Jan 7 at 16:42
studiosusstudiosus
2,124714
2,124714
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$begingroup$
Whoa. $phi$ is far from bijective.
$endgroup$
– Ted Shifrin
Jan 5 at 22:21
$begingroup$
That is right (corrected that in the question). The map$phi$ is not bijective, it will cover $mathbb{R}^2$ many times as $sin(y)$ varies. Then, I have no idea about why those fields are uniquely determined.
$endgroup$
– Amphiaraos
Jan 6 at 3:10