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 February 10th, 2018, 09:57 PM #1 Newbie   Joined: Oct 2017 From: sweden Posts: 24 Thanks: 0 Problems on Surfaces I have seeked in other forums as well, but I get no replies, and I really need some solutions to these kind of problems as the examples in my book are just not good enough to help me further solve problems at the end of the chapter. The problem goes like this: If we have a unit sphere denoted by $S^2$ , in $\mathbb{R}^3$, and some function $z$ that maps $z:S^2 \rightarrow (0,\infty)$ which is defined to be smooth and has $+$ value. Now given some defined set $\Lambda$: $$\Lambda=\{{z(x)x: x\in S^2}\}$$ Question 1) If $Z(u_1,u_2):U\rightarrow S^2$ is a parametrisation of $S^2$ (smooth), how can one show that another parametrisation on $\Lambda$ defined by $F:U \rightarrow \Lambda$, $(u_1,u_2) \rightarrow$ $z(Z(u_1,u_2))Z(u_1,u_2)$, then $\Lambda$ is a regular surface? Question 2) Now, we have coinciding parametrisations of $S^2$ which are given by $Z_i(u_1,u_2):U_i \rightarrow S^2$, and $F_i:U_i\rightarrow \Lambda$ the parametrization of $\Lambda$ caused by $Z_i$. Prove that $F_1^{-1}(F_2)=Z_1^{-1}(Z_2)$ for $i=1,2$. Question 3) Is $\Lambda$ and $S^2$ diffeomorhpic? If so give an explicit form of the diffeomorphism. Question 4) Let us now define some map $\Psi: \mathbb{R}^3 -\{(0,0,0)\} \rightarrow$ $\mathbb{R}^3 - \{(0,0,0)\}$, which is given by: $$\Psi(x) = \frac{x}{|x|^2}$$ Denote $\Lambda^*=\Psi(\Lambda)$. a) Is $\Lambda^*$ a regular surface? b) For some map $\psi:\Lambda \rightarrow \Lambda ^*$which is some restriction on $\Psi$ on $\Lambda$, prove that $\psi$ is a diffeomorphism. c) For any $c \in \Lambda$, the tangent map $d\psi_c:T_c \Lambda \rightarrow T_{\psi(c)} \Lambda^*$ at $c$ (which is a map from the tangent plane of $\Lambda$ at $c$ to the tangent plane at $\Lambda ^*$ at $\psi(c)$), prove that: $$d\psi_c(W) =\frac{|c|^2 W-2(c.W)c}{|c|^4}$$ where $c.W$ is the dot product in $\mathbb{R}^3$. February 11th, 2018, 01:34 PM #2 Global Moderator   Joined: May 2007 Posts: 6,834 Thanks: 733 If you haven't tried it already, try Mathematics Stack Exchange. https://math.stackexchange.com/ February 11th, 2018, 05:11 PM #3 Newbie   Joined: Oct 2017 From: sweden Posts: 24 Thanks: 0 I also have, no reply yet. Maybe I will try mathoverflow. Tags problems, surfaces Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post bigli Real Analysis 1 July 19th, 2013 05:27 PM johng23 Complex Analysis 0 March 9th, 2013 05:41 AM alex_0245 Algebra 0 May 15th, 2012 10:17 PM aaron-math Calculus 6 February 12th, 2012 07:19 PM remeday86 Calculus 1 February 17th, 2009 04:40 AM

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