What am I doing wrong? Exercise 2, chapter 2, section 3 from Guillemin and Pollack.












1












$begingroup$


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







share|cite|improve this question











$endgroup$












  • $begingroup$
    The problem is that $w$ and $c(t)$ are not necessarily orthogonal.
    $endgroup$
    – Mauro
    Dec 10 '18 at 19:45
















1












$begingroup$


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







share|cite|improve this question











$endgroup$












  • $begingroup$
    The problem is that $w$ and $c(t)$ are not necessarily orthogonal.
    $endgroup$
    – Mauro
    Dec 10 '18 at 19:45














1












1








1





$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 10 '18 at 19:42









Shaun

9,268113684




9,268113684










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


















  • $begingroup$
    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










1 Answer
1






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oldest

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2












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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











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1 Answer
1






active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









2












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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
















2












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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














2












2








2





$begingroup$

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.






share|cite|improve this answer









$endgroup$



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.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 10 '18 at 19:43









José Carlos SantosJosé Carlos Santos

163k22130233




163k22130233












  • $begingroup$
    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










  • $begingroup$
    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


















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