Mixing
Diffeomorphism between the unit ball to itself
$begingroup$
Let $T:Bto B$ be a diffeomorphism , where $Bsubsetmathbb{R}^n$ is the unit ball.
I want to show there exists $xin B$ such that $|J_T(x)|=1$ ($|J_T|$ is the absolute value of the Jacobian of $T$)
Intuitively speaking, since $T$ maps the ball to itself, I`d expect that $|J_T(x)|≡1$, Since the map does not inflate the ball, so it might only perform rotations or reflections. However my intuition here must be false.
My line of reasoning was as following:
$Volume(B):=int_B1dt=int_{T^-1(B)}|J_T(x)|dx=int_B|J_T(x)|dx$
Where the first equality is how we defined volume, the second is by changing variables (Possible since $T$ is diffeomorphism) and the last equality because $T$ maps $B$ to itself, so $T^{-1}$ must also map $B$ to itself.
From here one can conclude that either $|J_T(x)|≡1$ or that if $|J_T(x)|neq1$ (Say $|J_T(x)|<1$ for example) there must exist a point where $|J_T(x)|>1$ or we`d get $Volume(B)>Volume(B)$ as a contradiciton.Now simply using continuouity of $|J_T|$ along with the I.V.T gets the desired result.
From this proof, it seems like there might be such a map that isn`t a combination of rotations or reflections, and that $T$ might also "inflate" areas of the ball. This is very counter-intuitive to me, and I could not imagine such mapping. I'd be glad if someone could provide with an example, and perhaps some better intuition on the behaviour of diffeomorphisms.
If such an example does not exist, I'd love to see a proof.
real-analysis integration intuition diffeomorphism
asked Jan 8 at 21:52
SarSar
513113
$endgroup$
add a comment |
$begingroup$
Let $T:Bto B$ be a diffeomorphism , where $Bsubsetmathbb{R}^n$ is the unit ball.
I want to show there exists $xin B$ such that $|J_T(x)|=1$ ($|J_T|$ is the absolute value of the Jacobian of $T$)
Intuitively speaking, since $T$ maps the ball to itself, I`d expect that $|J_T(x)|≡1$, Since the map does not inflate the ball, so it might only perform rotations or reflections. However my intuition here must be false.
My line of reasoning was as following:
$Volume(B):=int_B1dt=int_{T^-1(B)}|J_T(x)|dx=int_B|J_T(x)|dx$
Where the first equality is how we defined volume, the second is by changing variables (Possible since $T$ is diffeomorphism) and the last equality because $T$ maps $B$ to itself, so $T^{-1}$ must also map $B$ to itself.
From here one can conclude that either $|J_T(x)|≡1$ or that if $|J_T(x)|neq1$ (Say $|J_T(x)|<1$ for example) there must exist a point where $|J_T(x)|>1$ or we`d get $Volume(B)>Volume(B)$ as a contradiciton.Now simply using continuouity of $|J_T|$ along with the I.V.T gets the desired result.
From this proof, it seems like there might be such a map that isn`t a combination of rotations or reflections, and that $T$ might also "inflate" areas of the ball. This is very counter-intuitive to me, and I could not imagine such mapping. I'd be glad if someone could provide with an example, and perhaps some better intuition on the behaviour of diffeomorphisms.
If such an example does not exist, I'd love to see a proof.
real-analysis integration intuition diffeomorphism
asked Jan 8 at 21:52
SarSar
513113
$endgroup$
$begingroup$
At each $xin B, J_T(x):mathbb R^nto mathbb R^n$ is an isomorphism. If $det J_T(x)=1, $ then $J_T(x)$ is a rotation; if $det J_T(x)=-1, $ then $J_T(x)$ is a reflection. If the only possibilities were these, then $every$ diffeomorphism would be locally a rotation or reflection, but clearly (as the answer shows), there are other possibilties.
$endgroup$
– Matematleta
Jan 9 at 0:35
add a comment |
$begingroup$
Let $T:Bto B$ be a diffeomorphism , where $Bsubsetmathbb{R}^n$ is the unit ball.
I want to show there exists $xin B$ such that $|J_T(x)|=1$ ($|J_T|$ is the absolute value of the Jacobian of $T$)
Intuitively speaking, since $T$ maps the ball to itself, I`d expect that $|J_T(x)|≡1$, Since the map does not inflate the ball, so it might only perform rotations or reflections. However my intuition here must be false.
My line of reasoning was as following:
$Volume(B):=int_B1dt=int_{T^-1(B)}|J_T(x)|dx=int_B|J_T(x)|dx$
Where the first equality is how we defined volume, the second is by changing variables (Possible since $T$ is diffeomorphism) and the last equality because $T$ maps $B$ to itself, so $T^{-1}$ must also map $B$ to itself.
From here one can conclude that either $|J_T(x)|≡1$ or that if $|J_T(x)|neq1$ (Say $|J_T(x)|<1$ for example) there must exist a point where $|J_T(x)|>1$ or we`d get $Volume(B)>Volume(B)$ as a contradiciton.Now simply using continuouity of $|J_T|$ along with the I.V.T gets the desired result.
From this proof, it seems like there might be such a map that isn`t a combination of rotations or reflections, and that $T$ might also "inflate" areas of the ball. This is very counter-intuitive to me, and I could not imagine such mapping. I'd be glad if someone could provide with an example, and perhaps some better intuition on the behaviour of diffeomorphisms.
If such an example does not exist, I'd love to see a proof.
real-analysis integration intuition diffeomorphism
asked Jan 8 at 21:52
SarSar
513113
$endgroup$
Let $T:Bto B$ be a diffeomorphism , where $Bsubsetmathbb{R}^n$ is the unit ball.
I want to show there exists $xin B$ such that $|J_T(x)|=1$ ($|J_T|$ is the absolute value of the Jacobian of $T$)
Intuitively speaking, since $T$ maps the ball to itself, I`d expect that $|J_T(x)|≡1$, Since the map does not inflate the ball, so it might only perform rotations or reflections. However my intuition here must be false.
My line of reasoning was as following:
$Volume(B):=int_B1dt=int_{T^-1(B)}|J_T(x)|dx=int_B|J_T(x)|dx$
Where the first equality is how we defined volume, the second is by changing variables (Possible since $T$ is diffeomorphism) and the last equality because $T$ maps $B$ to itself, so $T^{-1}$ must also map $B$ to itself.
From here one can conclude that either $|J_T(x)|≡1$ or that if $|J_T(x)|neq1$ (Say $|J_T(x)|<1$ for example) there must exist a point where $|J_T(x)|>1$ or we`d get $Volume(B)>Volume(B)$ as a contradiciton.Now simply using continuouity of $|J_T|$ along with the I.V.T gets the desired result.
From this proof, it seems like there might be such a map that isn`t a combination of rotations or reflections, and that $T$ might also "inflate" areas of the ball. This is very counter-intuitive to me, and I could not imagine such mapping. I'd be glad if someone could provide with an example, and perhaps some better intuition on the behaviour of diffeomorphisms.
If such an example does not exist, I'd love to see a proof.
real-analysis integration intuition diffeomorphism
real-analysis integration intuition diffeomorphism
asked Jan 8 at 21:52
SarSar
513113
asked Jan 8 at 21:52
SarSar
513113
edited Jan 8 at 21:59
Sar
asked Jan 8 at 21:52
SarSar
513113
asked Jan 8 at 21:52
SarSar
513113
asked Jan 8 at 21:52
SarSar
513113
513113
$begingroup$
At each $xin B, J_T(x):mathbb R^nto mathbb R^n$ is an isomorphism. If $det J_T(x)=1, $ then $J_T(x)$ is a rotation; if $det J_T(x)=-1, $ then $J_T(x)$ is a reflection. If the only possibilities were these, then $every$ diffeomorphism would be locally a rotation or reflection, but clearly (as the answer shows), there are other possibilties.
$endgroup$
– Matematleta
Jan 9 at 0:35
add a comment |
$begingroup$
At each $xin B, J_T(x):mathbb R^nto mathbb R^n$ is an isomorphism. If $det J_T(x)=1, $ then $J_T(x)$ is a rotation; if $det J_T(x)=-1, $ then $J_T(x)$ is a reflection. If the only possibilities were these, then $every$ diffeomorphism would be locally a rotation or reflection, but clearly (as the answer shows), there are other possibilties.
$endgroup$
– Matematleta
Jan 9 at 0:35
$begingroup$
At each $xin B, J_T(x):mathbb R^nto mathbb R^n$ is an isomorphism. If $det J_T(x)=1, $ then $J_T(x)$ is a rotation; if $det J_T(x)=-1, $ then $J_T(x)$ is a reflection. If the only possibilities were these, then $every$ diffeomorphism would be locally a rotation or reflection, but clearly (as the answer shows), there are other possibilties.
$endgroup$
– Matematleta
Jan 9 at 0:35
$begingroup$
At each $xin B, J_T(x):mathbb R^nto mathbb R^n$ is an isomorphism. If $det J_T(x)=1, $ then $J_T(x)$ is a rotation; if $det J_T(x)=-1, $ then $J_T(x)$ is a reflection. If the only possibilities were these, then $every$ diffeomorphism would be locally a rotation or reflection, but clearly (as the answer shows), there are other possibilties.
$endgroup$
– Matematleta
Jan 9 at 0:35
add a comment |
1 Answer
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$begingroup$
One possible diffeomorphism from the unit ball in $mathbb{R}^2$ to itself, which isn't rigid in the way you expect, could be described in polar coordinates as
$$(r, theta) mapsto left(frac{r + r^3}{2}, theta right),$$
or in rectangular coordinates as
$$(x, y) mapsto frac{1 + x^2 + y^2}{2} (x, y).$$
(The first representation was what I came up with to make it easy to show the function is bijective, whereas the second representation makes it easier to show the differentiability of the function and its inverse at the origin.)
answered Jan 8 at 23:27
Daniel ScheplerDaniel Schepler
8,5041618
$endgroup$
$begingroup$
I'm not sure if my new intuition following this example is right: We started out with a ball with its volume evenly spread out. After applying this transformation- We have a new set of points, they fill a new unit ball on the plane, however the volume of this new ball isn't evenly spread- but might be more dense in some places and thin on others, overall maintaining the volume of the original ball ? And the det of the Jacobian could be looked at as "weights" to tell us when we're integrating for the volume - when the volume is dense or thin ?
$endgroup$
– Sar
Jan 9 at 6:44
add a comment |
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$begingroup$
One possible diffeomorphism from the unit ball in $mathbb{R}^2$ to itself, which isn't rigid in the way you expect, could be described in polar coordinates as
$$(r, theta) mapsto left(frac{r + r^3}{2}, theta right),$$
or in rectangular coordinates as
$$(x, y) mapsto frac{1 + x^2 + y^2}{2} (x, y).$$
(The first representation was what I came up with to make it easy to show the function is bijective, whereas the second representation makes it easier to show the differentiability of the function and its inverse at the origin.)
answered Jan 8 at 23:27
Daniel ScheplerDaniel Schepler
8,5041618
$endgroup$
$begingroup$
I'm not sure if my new intuition following this example is right: We started out with a ball with its volume evenly spread out. After applying this transformation- We have a new set of points, they fill a new unit ball on the plane, however the volume of this new ball isn't evenly spread- but might be more dense in some places and thin on others, overall maintaining the volume of the original ball ? And the det of the Jacobian could be looked at as "weights" to tell us when we're integrating for the volume - when the volume is dense or thin ?
$endgroup$
– Sar
Jan 9 at 6:44
add a comment |
$begingroup$
One possible diffeomorphism from the unit ball in $mathbb{R}^2$ to itself, which isn't rigid in the way you expect, could be described in polar coordinates as
$$(r, theta) mapsto left(frac{r + r^3}{2}, theta right),$$
or in rectangular coordinates as
$$(x, y) mapsto frac{1 + x^2 + y^2}{2} (x, y).$$
(The first representation was what I came up with to make it easy to show the function is bijective, whereas the second representation makes it easier to show the differentiability of the function and its inverse at the origin.)
answered Jan 8 at 23:27
Daniel ScheplerDaniel Schepler
8,5041618
$endgroup$
$begingroup$
I'm not sure if my new intuition following this example is right: We started out with a ball with its volume evenly spread out. After applying this transformation- We have a new set of points, they fill a new unit ball on the plane, however the volume of this new ball isn't evenly spread- but might be more dense in some places and thin on others, overall maintaining the volume of the original ball ? And the det of the Jacobian could be looked at as "weights" to tell us when we're integrating for the volume - when the volume is dense or thin ?
$endgroup$
– Sar
Jan 9 at 6:44
add a comment |
$begingroup$
One possible diffeomorphism from the unit ball in $mathbb{R}^2$ to itself, which isn't rigid in the way you expect, could be described in polar coordinates as
$$(r, theta) mapsto left(frac{r + r^3}{2}, theta right),$$
or in rectangular coordinates as
$$(x, y) mapsto frac{1 + x^2 + y^2}{2} (x, y).$$
(The first representation was what I came up with to make it easy to show the function is bijective, whereas the second representation makes it easier to show the differentiability of the function and its inverse at the origin.)
answered Jan 8 at 23:27
Daniel ScheplerDaniel Schepler
8,5041618
$endgroup$
One possible diffeomorphism from the unit ball in $mathbb{R}^2$ to itself, which isn't rigid in the way you expect, could be described in polar coordinates as
$$(r, theta) mapsto left(frac{r + r^3}{2}, theta right),$$
or in rectangular coordinates as
$$(x, y) mapsto frac{1 + x^2 + y^2}{2} (x, y).$$
(The first representation was what I came up with to make it easy to show the function is bijective, whereas the second representation makes it easier to show the differentiability of the function and its inverse at the origin.)
answered Jan 8 at 23:27
Daniel ScheplerDaniel Schepler
8,5041618
answered Jan 8 at 23:27
Daniel ScheplerDaniel Schepler
8,5041618
answered Jan 8 at 23:27
Daniel ScheplerDaniel Schepler
8,5041618
answered Jan 8 at 23:27
Daniel ScheplerDaniel Schepler
8,5041618
8,5041618
$begingroup$
I'm not sure if my new intuition following this example is right: We started out with a ball with its volume evenly spread out. After applying this transformation- We have a new set of points, they fill a new unit ball on the plane, however the volume of this new ball isn't evenly spread- but might be more dense in some places and thin on others, overall maintaining the volume of the original ball ? And the det of the Jacobian could be looked at as "weights" to tell us when we're integrating for the volume - when the volume is dense or thin ?
$endgroup$
– Sar
Jan 9 at 6:44
add a comment |
$begingroup$
I'm not sure if my new intuition following this example is right: We started out with a ball with its volume evenly spread out. After applying this transformation- We have a new set of points, they fill a new unit ball on the plane, however the volume of this new ball isn't evenly spread- but might be more dense in some places and thin on others, overall maintaining the volume of the original ball ? And the det of the Jacobian could be looked at as "weights" to tell us when we're integrating for the volume - when the volume is dense or thin ?
$endgroup$
– Sar
Jan 9 at 6:44
$begingroup$
I'm not sure if my new intuition following this example is right: We started out with a ball with its volume evenly spread out. After applying this transformation- We have a new set of points, they fill a new unit ball on the plane, however the volume of this new ball isn't evenly spread- but might be more dense in some places and thin on others, overall maintaining the volume of the original ball ? And the det of the Jacobian could be looked at as "weights" to tell us when we're integrating for the volume - when the volume is dense or thin ?
$endgroup$
– Sar
Jan 9 at 6:44
$begingroup$
I'm not sure if my new intuition following this example is right: We started out with a ball with its volume evenly spread out. After applying this transformation- We have a new set of points, they fill a new unit ball on the plane, however the volume of this new ball isn't evenly spread- but might be more dense in some places and thin on others, overall maintaining the volume of the original ball ? And the det of the Jacobian could be looked at as "weights" to tell us when we're integrating for the volume - when the volume is dense or thin ?
$endgroup$
– Sar
Jan 9 at 6:44
add a comment |
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$begingroup$
At each $xin B, J_T(x):mathbb R^nto mathbb R^n$ is an isomorphism. If $det J_T(x)=1, $ then $J_T(x)$ is a rotation; if $det J_T(x)=-1, $ then $J_T(x)$ is a reflection. If the only possibilities were these, then $every$ diffeomorphism would be locally a rotation or reflection, but clearly (as the answer shows), there are other possibilties.
$endgroup$
– Matematleta
Jan 9 at 0:35