What am I doing wrong? Exercise 2, chapter 2, section 3 from Guillemin and Pollack.
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I am doing exercise 2, chapter 2, section 3 from Guillemin and Pollack's ''Differential Topology''. Part of the excercise is to prove that given a compact manifold $Y subset mathbb{R}^m$, and a pont $w in mathbb{R}^m$, there exist a point (not necessarily unique) $y in Y$ closest to $w$. This part I have done, the next part is to prove that $w-y in N_y(Y)$.
($N_y(Y)$ is the orthogonal complement of $T_yY$)
I followed the hint, and since any element in $T_yY$ is the velocity vector of a curve $c$ in $Y$ such that $c(0)=y$, then all i got to prove is that $(w-y) cdot dot{c(0)}=0$ for all of these curves.
Since the function $ g(t) =mid w-c(t) mid^2 = sum_{i=1}^m w_i^2+c(t)_i^2 $ has a minimum at $0$, deriving you get:
$0=g'(0)= 2sum_{i=1}^m c(0)_ic'(0)_i = ycdot dot{c(0)}$, which would mean that $y$ is in $N_y(Y)$. Which is in many cases false.
What is my error here?
differential-topology
edited Dec 10 '18 at 19:42
Shaun
9,789113684
asked Dec 10 '18 at 19:35
Bajo FondoBajo Fondo
410315
$endgroup$
add a comment |
$begingroup$
I am doing exercise 2, chapter 2, section 3 from Guillemin and Pollack's ''Differential Topology''. Part of the excercise is to prove that given a compact manifold $Y subset mathbb{R}^m$, and a pont $w in mathbb{R}^m$, there exist a point (not necessarily unique) $y in Y$ closest to $w$. This part I have done, the next part is to prove that $w-y in N_y(Y)$.
($N_y(Y)$ is the orthogonal complement of $T_yY$)
I followed the hint, and since any element in $T_yY$ is the velocity vector of a curve $c$ in $Y$ such that $c(0)=y$, then all i got to prove is that $(w-y) cdot dot{c(0)}=0$ for all of these curves.
Since the function $ g(t) =mid w-c(t) mid^2 = sum_{i=1}^m w_i^2+c(t)_i^2 $ has a minimum at $0$, deriving you get:
$0=g'(0)= 2sum_{i=1}^m c(0)_ic'(0)_i = ycdot dot{c(0)}$, which would mean that $y$ is in $N_y(Y)$. Which is in many cases false.
What is my error here?
differential-topology
edited Dec 10 '18 at 19:42
Shaun
9,789113684
asked Dec 10 '18 at 19:35
Bajo FondoBajo Fondo
410315
$endgroup$
$begingroup$
The problem is that $w$ and $c(t)$ are not necessarily orthogonal.
$endgroup$
– Mauro
Dec 10 '18 at 19:45
add a comment |
$begingroup$
I am doing exercise 2, chapter 2, section 3 from Guillemin and Pollack's ''Differential Topology''. Part of the excercise is to prove that given a compact manifold $Y subset mathbb{R}^m$, and a pont $w in mathbb{R}^m$, there exist a point (not necessarily unique) $y in Y$ closest to $w$. This part I have done, the next part is to prove that $w-y in N_y(Y)$.
($N_y(Y)$ is the orthogonal complement of $T_yY$)
I followed the hint, and since any element in $T_yY$ is the velocity vector of a curve $c$ in $Y$ such that $c(0)=y$, then all i got to prove is that $(w-y) cdot dot{c(0)}=0$ for all of these curves.
Since the function $ g(t) =mid w-c(t) mid^2 = sum_{i=1}^m w_i^2+c(t)_i^2 $ has a minimum at $0$, deriving you get:
$0=g'(0)= 2sum_{i=1}^m c(0)_ic'(0)_i = ycdot dot{c(0)}$, which would mean that $y$ is in $N_y(Y)$. Which is in many cases false.
What is my error here?
differential-topology
edited Dec 10 '18 at 19:42
Shaun
9,789113684
asked Dec 10 '18 at 19:35
Bajo FondoBajo Fondo
410315
$endgroup$
I am doing exercise 2, chapter 2, section 3 from Guillemin and Pollack's ''Differential Topology''. Part of the excercise is to prove that given a compact manifold $Y subset mathbb{R}^m$, and a pont $w in mathbb{R}^m$, there exist a point (not necessarily unique) $y in Y$ closest to $w$. This part I have done, the next part is to prove that $w-y in N_y(Y)$.
($N_y(Y)$ is the orthogonal complement of $T_yY$)
I followed the hint, and since any element in $T_yY$ is the velocity vector of a curve $c$ in $Y$ such that $c(0)=y$, then all i got to prove is that $(w-y) cdot dot{c(0)}=0$ for all of these curves.
Since the function $ g(t) =mid w-c(t) mid^2 = sum_{i=1}^m w_i^2+c(t)_i^2 $ has a minimum at $0$, deriving you get:
$0=g'(0)= 2sum_{i=1}^m c(0)_ic'(0)_i = ycdot dot{c(0)}$, which would mean that $y$ is in $N_y(Y)$. Which is in many cases false.
What is my error here?
differential-topology
differential-topology
edited Dec 10 '18 at 19:42
Shaun
9,789113684
asked Dec 10 '18 at 19:35
Bajo FondoBajo Fondo
410315
edited Dec 10 '18 at 19:42
Shaun
9,789113684
asked Dec 10 '18 at 19:35
Bajo FondoBajo Fondo
410315
edited Dec 10 '18 at 19:42
Shaun
9,789113684
edited Dec 10 '18 at 19:42
Shaun
9,789113684
edited Dec 10 '18 at 19:42
Shaun
9,789113684
9,789113684
asked Dec 10 '18 at 19:35
Bajo FondoBajo Fondo
410315
asked Dec 10 '18 at 19:35
Bajo FondoBajo Fondo
410315
asked Dec 10 '18 at 19:35
Bajo FondoBajo Fondo
410315
410315
$begingroup$
The problem is that $w$ and $c(t)$ are not necessarily orthogonal.
$endgroup$
– Mauro
Dec 10 '18 at 19:45
add a comment |
$begingroup$
The problem is that $w$ and $c(t)$ are not necessarily orthogonal.
$endgroup$
– Mauro
Dec 10 '18 at 19:45
$begingroup$
The problem is that $w$ and $c(t)$ are not necessarily orthogonal.
$endgroup$
– Mauro
Dec 10 '18 at 19:45
$begingroup$
The problem is that $w$ and $c(t)$ are not necessarily orthogonal.
$endgroup$
– Mauro
Dec 10 '18 at 19:45
add a comment |
1 Answer
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votes
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Note that $displaystylelVert w-c(t)rVert^2=sum_{i=1}^m{w_i}^2-2w_ic_i(t)+{c_i}^2(t)$. It looks like you forgot the middle term.
answered Dec 10 '18 at 19:43
José Carlos SantosJosé Carlos Santos
170k23132238
$endgroup$
$begingroup$
Yes! Thank you, and it is a mistake that at this point in my studies I should not be making. The worst part is that I had a feeling it was an error of that sort. I feel ashamed. Thank you.
$endgroup$
– Bajo Fondo
Dec 10 '18 at 19:45
$begingroup$
the site did not let me accept it until 6 minutes had passed from the posting of the question. Thank you again.
$endgroup$
– Bajo Fondo
Dec 10 '18 at 19:56
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
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votes
$begingroup$
Note that $displaystylelVert w-c(t)rVert^2=sum_{i=1}^m{w_i}^2-2w_ic_i(t)+{c_i}^2(t)$. It looks like you forgot the middle term.
answered Dec 10 '18 at 19:43
José Carlos SantosJosé Carlos Santos
170k23132238
$endgroup$
$begingroup$
Yes! Thank you, and it is a mistake that at this point in my studies I should not be making. The worst part is that I had a feeling it was an error of that sort. I feel ashamed. Thank you.
$endgroup$
– Bajo Fondo
Dec 10 '18 at 19:45
$begingroup$
the site did not let me accept it until 6 minutes had passed from the posting of the question. Thank you again.
$endgroup$
– Bajo Fondo
Dec 10 '18 at 19:56
add a comment |
$begingroup$
Note that $displaystylelVert w-c(t)rVert^2=sum_{i=1}^m{w_i}^2-2w_ic_i(t)+{c_i}^2(t)$. It looks like you forgot the middle term.
answered Dec 10 '18 at 19:43
José Carlos SantosJosé Carlos Santos
170k23132238
$endgroup$
$begingroup$
Yes! Thank you, and it is a mistake that at this point in my studies I should not be making. The worst part is that I had a feeling it was an error of that sort. I feel ashamed. Thank you.
$endgroup$
– Bajo Fondo
Dec 10 '18 at 19:45
$begingroup$
the site did not let me accept it until 6 minutes had passed from the posting of the question. Thank you again.
$endgroup$
– Bajo Fondo
Dec 10 '18 at 19:56
add a comment |
$begingroup$
Note that $displaystylelVert w-c(t)rVert^2=sum_{i=1}^m{w_i}^2-2w_ic_i(t)+{c_i}^2(t)$. It looks like you forgot the middle term.
answered Dec 10 '18 at 19:43
José Carlos SantosJosé Carlos Santos
170k23132238
$endgroup$
Note that $displaystylelVert w-c(t)rVert^2=sum_{i=1}^m{w_i}^2-2w_ic_i(t)+{c_i}^2(t)$. It looks like you forgot the middle term.
answered Dec 10 '18 at 19:43
José Carlos SantosJosé Carlos Santos
170k23132238
answered Dec 10 '18 at 19:43
José Carlos SantosJosé Carlos Santos
170k23132238
answered Dec 10 '18 at 19:43
José Carlos SantosJosé Carlos Santos
170k23132238
answered Dec 10 '18 at 19:43
José Carlos SantosJosé Carlos Santos
170k23132238
170k23132238
$begingroup$
Yes! Thank you, and it is a mistake that at this point in my studies I should not be making. The worst part is that I had a feeling it was an error of that sort. I feel ashamed. Thank you.
$endgroup$
– Bajo Fondo
Dec 10 '18 at 19:45
$begingroup$
the site did not let me accept it until 6 minutes had passed from the posting of the question. Thank you again.
$endgroup$
– Bajo Fondo
Dec 10 '18 at 19:56
add a comment |
$begingroup$
Yes! Thank you, and it is a mistake that at this point in my studies I should not be making. The worst part is that I had a feeling it was an error of that sort. I feel ashamed. Thank you.
$endgroup$
– Bajo Fondo
Dec 10 '18 at 19:45
$begingroup$
the site did not let me accept it until 6 minutes had passed from the posting of the question. Thank you again.
$endgroup$
– Bajo Fondo
Dec 10 '18 at 19:56
$begingroup$
Yes! Thank you, and it is a mistake that at this point in my studies I should not be making. The worst part is that I had a feeling it was an error of that sort. I feel ashamed. Thank you.
$endgroup$
– Bajo Fondo
Dec 10 '18 at 19:45
$begingroup$
Yes! Thank you, and it is a mistake that at this point in my studies I should not be making. The worst part is that I had a feeling it was an error of that sort. I feel ashamed. Thank you.
$endgroup$
– Bajo Fondo
Dec 10 '18 at 19:45
$begingroup$
the site did not let me accept it until 6 minutes had passed from the posting of the question. Thank you again.
$endgroup$
– Bajo Fondo
Dec 10 '18 at 19:56
$begingroup$
the site did not let me accept it until 6 minutes had passed from the posting of the question. Thank you again.
$endgroup$
– Bajo Fondo
Dec 10 '18 at 19:56
add a comment |
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$begingroup$
The problem is that $w$ and $c(t)$ are not necessarily orthogonal.
$endgroup$
– Mauro
Dec 10 '18 at 19:45