Mixing
Show that if $left|Tleft(xright)-Tleft(yright)right|geq cleft|x-yright|$ then $vleft(Tleft(Eright)right)geq...
$begingroup$
Let $c>0$ be a real positive number and let $Tcolonmathbb{R}^{n}tomathbb{R}^{n}$
be a diffeomorphism such that $left|Tleft(xright)-Tleft(yright)right|geq cleft|x-yright|$.
I wish to show that for every Jordan Measurable set $E$ we have $vleft(Tleft(Eright)right)geq c^{n}vleft(Eright)$.
In my attempt I realized that using change of variables we get
$$
vleft(Tleft(Eright)right)=int_{Tleft(Eright)}1=int_{E}left|J_{T}left(xright)right|
$$
where $J_{T}$ is the Jacobian of $T$, and therefore it is sufficient to show
that for every $xin E$ we have $left|J_{T}left(xright)right|geq c^{n}$.
I couldn't see how to prove that so I was thinking about looking first
on the case where $c=1$, but also there I couldn't figure how to
do it. Any suggestions?
EDIT:
More I know is that for every $x,hinmathbb{R}^{n}$ we have $Tleft(x+hright)=Tleft(xright)+D_{T}left(xright)left(hright)+oleft(left|hright|right)$
and therefore
$$
begin{align*}
cleft|hright| & leqleft|Tleft(x+hright)-Tleft(xright)right|=left|D_{T}left(xright)left(hright)+oleft(left|hright|right)right|leq\
& leqleft|D_{T}left(xright)left(hright)right|+left|oleft(left|hright|right)right|leqleftVert D_{T}left(xright)rightVert left|hright|+oleft(left|hright|right)
end{align*}
$$
so $cleqleftVert D_{T}left(xright)rightVert +frac{oleft(left|hright|right)}{left|hright|}$
and when $hto0$ we get $leftVert D_{T}left(xright)rightVert geq c$
multivariable-calculus diffeomorphism
asked Jan 18 at 19:31
JonJon
565414
$endgroup$
add a comment |
$begingroup$
Let $c>0$ be a real positive number and let $Tcolonmathbb{R}^{n}tomathbb{R}^{n}$
be a diffeomorphism such that $left|Tleft(xright)-Tleft(yright)right|geq cleft|x-yright|$.
I wish to show that for every Jordan Measurable set $E$ we have $vleft(Tleft(Eright)right)geq c^{n}vleft(Eright)$.
In my attempt I realized that using change of variables we get
$$
vleft(Tleft(Eright)right)=int_{Tleft(Eright)}1=int_{E}left|J_{T}left(xright)right|
$$
where $J_{T}$ is the Jacobian of $T$, and therefore it is sufficient to show
that for every $xin E$ we have $left|J_{T}left(xright)right|geq c^{n}$.
I couldn't see how to prove that so I was thinking about looking first
on the case where $c=1$, but also there I couldn't figure how to
do it. Any suggestions?
EDIT:
More I know is that for every $x,hinmathbb{R}^{n}$ we have $Tleft(x+hright)=Tleft(xright)+D_{T}left(xright)left(hright)+oleft(left|hright|right)$
and therefore
$$
begin{align*}
cleft|hright| & leqleft|Tleft(x+hright)-Tleft(xright)right|=left|D_{T}left(xright)left(hright)+oleft(left|hright|right)right|leq\
& leqleft|D_{T}left(xright)left(hright)right|+left|oleft(left|hright|right)right|leqleftVert D_{T}left(xright)rightVert left|hright|+oleft(left|hright|right)
end{align*}
$$
so $cleqleftVert D_{T}left(xright)rightVert +frac{oleft(left|hright|right)}{left|hright|}$
and when $hto0$ we get $leftVert D_{T}left(xright)rightVert geq c$
multivariable-calculus diffeomorphism
asked Jan 18 at 19:31
JonJon
565414
$endgroup$
add a comment |
$begingroup$
Let $c>0$ be a real positive number and let $Tcolonmathbb{R}^{n}tomathbb{R}^{n}$
be a diffeomorphism such that $left|Tleft(xright)-Tleft(yright)right|geq cleft|x-yright|$.
I wish to show that for every Jordan Measurable set $E$ we have $vleft(Tleft(Eright)right)geq c^{n}vleft(Eright)$.
In my attempt I realized that using change of variables we get
$$
vleft(Tleft(Eright)right)=int_{Tleft(Eright)}1=int_{E}left|J_{T}left(xright)right|
$$
where $J_{T}$ is the Jacobian of $T$, and therefore it is sufficient to show
that for every $xin E$ we have $left|J_{T}left(xright)right|geq c^{n}$.
I couldn't see how to prove that so I was thinking about looking first
on the case where $c=1$, but also there I couldn't figure how to
do it. Any suggestions?
EDIT:
More I know is that for every $x,hinmathbb{R}^{n}$ we have $Tleft(x+hright)=Tleft(xright)+D_{T}left(xright)left(hright)+oleft(left|hright|right)$
and therefore
$$
begin{align*}
cleft|hright| & leqleft|Tleft(x+hright)-Tleft(xright)right|=left|D_{T}left(xright)left(hright)+oleft(left|hright|right)right|leq\
& leqleft|D_{T}left(xright)left(hright)right|+left|oleft(left|hright|right)right|leqleftVert D_{T}left(xright)rightVert left|hright|+oleft(left|hright|right)
end{align*}
$$
so $cleqleftVert D_{T}left(xright)rightVert +frac{oleft(left|hright|right)}{left|hright|}$
and when $hto0$ we get $leftVert D_{T}left(xright)rightVert geq c$
multivariable-calculus diffeomorphism
asked Jan 18 at 19:31
JonJon
565414
$endgroup$
Let $c>0$ be a real positive number and let $Tcolonmathbb{R}^{n}tomathbb{R}^{n}$
be a diffeomorphism such that $left|Tleft(xright)-Tleft(yright)right|geq cleft|x-yright|$.
I wish to show that for every Jordan Measurable set $E$ we have $vleft(Tleft(Eright)right)geq c^{n}vleft(Eright)$.
In my attempt I realized that using change of variables we get
$$
vleft(Tleft(Eright)right)=int_{Tleft(Eright)}1=int_{E}left|J_{T}left(xright)right|
$$
where $J_{T}$ is the Jacobian of $T$, and therefore it is sufficient to show
that for every $xin E$ we have $left|J_{T}left(xright)right|geq c^{n}$.
I couldn't see how to prove that so I was thinking about looking first
on the case where $c=1$, but also there I couldn't figure how to
do it. Any suggestions?
EDIT:
More I know is that for every $x,hinmathbb{R}^{n}$ we have $Tleft(x+hright)=Tleft(xright)+D_{T}left(xright)left(hright)+oleft(left|hright|right)$
and therefore
$$
begin{align*}
cleft|hright| & leqleft|Tleft(x+hright)-Tleft(xright)right|=left|D_{T}left(xright)left(hright)+oleft(left|hright|right)right|leq\
& leqleft|D_{T}left(xright)left(hright)right|+left|oleft(left|hright|right)right|leqleftVert D_{T}left(xright)rightVert left|hright|+oleft(left|hright|right)
end{align*}
$$
so $cleqleftVert D_{T}left(xright)rightVert +frac{oleft(left|hright|right)}{left|hright|}$
and when $hto0$ we get $leftVert D_{T}left(xright)rightVert geq c$
multivariable-calculus diffeomorphism
multivariable-calculus diffeomorphism
asked Jan 18 at 19:31
JonJon
565414
asked Jan 18 at 19:31
JonJon
565414
edited Jan 18 at 20:47
Jon
asked Jan 18 at 19:31
JonJon
565414
asked Jan 18 at 19:31
JonJon
565414
asked Jan 18 at 19:31
JonJon
565414
565414
add a comment |
add a comment |
1 Answer
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$begingroup$
$T(x+tv)=T(x)+dT_x(tv)+tO(t)$. We deduce that $|{{|T(x+tv)-T(x)|}over{|tv|}}=|dT_x({vover |v|})+O(t)|geq c$. This implies that $|dT_x(v)|geq c|v|$.
Let $e$ be an eigenvalue of $dT_x$ associated to the eigenvector $u$ (I will eventually work with the complexification), we have:
$T(x+tv)=T(x)+dT_x(tv)+tO(t)=T(x)+etv+tO(t)$, we deduce that:
$|{{|T(x+tv)-T(x)|}over{|tv|}}=|e({vover |v|})+O(t)|geq c$.
answered Jan 18 at 20:04
Tsemo AristideTsemo Aristide
58.8k11445
$endgroup$
$begingroup$
How can I use that to prove what I need?
$endgroup$
– Jon
Jan 18 at 20:26
$begingroup$
This implies that the norm of the eigenvalues of the Jacobian are superior to $c$.
$endgroup$
– Tsemo Aristide
Jan 18 at 20:52
$begingroup$
How it implies that?
$endgroup$
– Jon
Jan 18 at 20:56
add a comment |
Your Answer
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1 Answer
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$begingroup$
$T(x+tv)=T(x)+dT_x(tv)+tO(t)$. We deduce that $|{{|T(x+tv)-T(x)|}over{|tv|}}=|dT_x({vover |v|})+O(t)|geq c$. This implies that $|dT_x(v)|geq c|v|$.
Let $e$ be an eigenvalue of $dT_x$ associated to the eigenvector $u$ (I will eventually work with the complexification), we have:
$T(x+tv)=T(x)+dT_x(tv)+tO(t)=T(x)+etv+tO(t)$, we deduce that:
$|{{|T(x+tv)-T(x)|}over{|tv|}}=|e({vover |v|})+O(t)|geq c$.
answered Jan 18 at 20:04
Tsemo AristideTsemo Aristide
58.8k11445
$endgroup$
$begingroup$
How can I use that to prove what I need?
$endgroup$
– Jon
Jan 18 at 20:26
$begingroup$
This implies that the norm of the eigenvalues of the Jacobian are superior to $c$.
$endgroup$
– Tsemo Aristide
Jan 18 at 20:52
$begingroup$
How it implies that?
$endgroup$
– Jon
Jan 18 at 20:56
add a comment |
$begingroup$
$T(x+tv)=T(x)+dT_x(tv)+tO(t)$. We deduce that $|{{|T(x+tv)-T(x)|}over{|tv|}}=|dT_x({vover |v|})+O(t)|geq c$. This implies that $|dT_x(v)|geq c|v|$.
Let $e$ be an eigenvalue of $dT_x$ associated to the eigenvector $u$ (I will eventually work with the complexification), we have:
$T(x+tv)=T(x)+dT_x(tv)+tO(t)=T(x)+etv+tO(t)$, we deduce that:
$|{{|T(x+tv)-T(x)|}over{|tv|}}=|e({vover |v|})+O(t)|geq c$.
answered Jan 18 at 20:04
Tsemo AristideTsemo Aristide
58.8k11445
$endgroup$
$begingroup$
How can I use that to prove what I need?
$endgroup$
– Jon
Jan 18 at 20:26
$begingroup$
This implies that the norm of the eigenvalues of the Jacobian are superior to $c$.
$endgroup$
– Tsemo Aristide
Jan 18 at 20:52
$begingroup$
How it implies that?
$endgroup$
– Jon
Jan 18 at 20:56
add a comment |
$begingroup$
$T(x+tv)=T(x)+dT_x(tv)+tO(t)$. We deduce that $|{{|T(x+tv)-T(x)|}over{|tv|}}=|dT_x({vover |v|})+O(t)|geq c$. This implies that $|dT_x(v)|geq c|v|$.
Let $e$ be an eigenvalue of $dT_x$ associated to the eigenvector $u$ (I will eventually work with the complexification), we have:
$T(x+tv)=T(x)+dT_x(tv)+tO(t)=T(x)+etv+tO(t)$, we deduce that:
$|{{|T(x+tv)-T(x)|}over{|tv|}}=|e({vover |v|})+O(t)|geq c$.
answered Jan 18 at 20:04
Tsemo AristideTsemo Aristide
58.8k11445
$endgroup$
$T(x+tv)=T(x)+dT_x(tv)+tO(t)$. We deduce that $|{{|T(x+tv)-T(x)|}over{|tv|}}=|dT_x({vover |v|})+O(t)|geq c$. This implies that $|dT_x(v)|geq c|v|$.
Let $e$ be an eigenvalue of $dT_x$ associated to the eigenvector $u$ (I will eventually work with the complexification), we have:
$T(x+tv)=T(x)+dT_x(tv)+tO(t)=T(x)+etv+tO(t)$, we deduce that:
$|{{|T(x+tv)-T(x)|}over{|tv|}}=|e({vover |v|})+O(t)|geq c$.
answered Jan 18 at 20:04
Tsemo AristideTsemo Aristide
58.8k11445
edited Jan 18 at 23:18
answered Jan 18 at 20:04
Tsemo AristideTsemo Aristide
58.8k11445
answered Jan 18 at 20:04
Tsemo AristideTsemo Aristide
58.8k11445
answered Jan 18 at 20:04
Tsemo AristideTsemo Aristide
58.8k11445
58.8k11445
$begingroup$
How can I use that to prove what I need?
$endgroup$
– Jon
Jan 18 at 20:26
$begingroup$
This implies that the norm of the eigenvalues of the Jacobian are superior to $c$.
$endgroup$
– Tsemo Aristide
Jan 18 at 20:52
$begingroup$
How it implies that?
$endgroup$
– Jon
Jan 18 at 20:56
add a comment |
$begingroup$
How can I use that to prove what I need?
$endgroup$
– Jon
Jan 18 at 20:26
$begingroup$
This implies that the norm of the eigenvalues of the Jacobian are superior to $c$.
$endgroup$
– Tsemo Aristide
Jan 18 at 20:52
$begingroup$
How it implies that?
$endgroup$
– Jon
Jan 18 at 20:56
$begingroup$
How can I use that to prove what I need?
$endgroup$
– Jon
Jan 18 at 20:26
$begingroup$
How can I use that to prove what I need?
$endgroup$
– Jon
Jan 18 at 20:26
$begingroup$
This implies that the norm of the eigenvalues of the Jacobian are superior to $c$.
$endgroup$
– Tsemo Aristide
Jan 18 at 20:52
$begingroup$
This implies that the norm of the eigenvalues of the Jacobian are superior to $c$.
$endgroup$
– Tsemo Aristide
Jan 18 at 20:52
$begingroup$
How it implies that?
$endgroup$
– Jon
Jan 18 at 20:56
$begingroup$
How it implies that?
$endgroup$
– Jon
Jan 18 at 20:56
add a comment |
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