\documentclass[12pt]{article} \usepackage{amsmath,amssymb,graphicx,txfonts,multicol,picins} \usepackage[margin=0.75in]{geometry} \usepackage[usenames,dvipsnames]{xcolor} % Following http://tex.stackexchange.com/a/847/22475: \usepackage[setpagesize=false]{hyperref}\hypersetup{colorlinks, linkcolor={blue!50!black}, citecolor={blue!50!black}, urlcolor=blue } \def\blue{\color{blue}} \def\red{\color{red}} \pagestyle{empty} \def\myurl{http://www.math.toronto.edu/~drorbn/} \begin{document} \noindent{\small \href{\myurl}{Dror Bar-Natan}: \href{\myurl/classes/}{Classes}: \href{\myurl/classes/\#1617}{2016-17}: \hfill\url{http://drorbn.net/?title=1617-257} } \begin{center} {\LARGE Homework Assignment 11} \end{center} \noindent{\large\bf Reading.} Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really understand everything in them. Do the same every week! Also, read, reread and rereread sections 17 and 20-24 of Munkres' book to the same standard of understanding. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, preread section 25, just to get a feel for the future. \noindent{\large\bf Doing.} {\bf Solve} problems \underline{6} and 7 in section 17, problem \underline{2} in section 20, problems \underline{3} and 5 in section 21, and problems 1 and \underline{2} in section 22, but submit only your solutions of the underlined problems. In addition, solve the following two exercises, but submit only your solution of part 1 of exercise 2. \noindent{\bf Hint for Problem 2 in Section 22.} In some approaches, it may be possible to simplify the end result using the formula $\det(I_{n\times n}+vw^T)=1+v^Tw$, which holds for vectors $v,w\in{\mathbb R}^n$. \noindent{\textbf{Exercise 1: Curves.}} Consider a simple curve given by a smooth map $\gamma:[0,T]\rightarrow\mathbb{R}^{n}$, and let $\mathcal{C}:=\mbox{Im}(\gamma)$. \\ 1) Prove that the arclength $\lambda(\mathcal{C})$ of the curve $\mathcal{C}$ is given by $ \lambda(\mathcal{C})=\int_{0}^{T}||\dot{\gamma}(t)||dt $, where $||\cdot||$ is the Euclidian norm on $\mathbb{R}^{n}$ and $\dot{\gamma}=\frac{d}{dt}\gamma$.\\ 2) Consider the length at time $t\in[0,T]$ of $\gamma$, $ s(t)=\int_{0}^{t}||\dot{\gamma}(u)||du $. Let $\hat{\gamma}:[0,\lambda(\mathcal{C})]\rightarrow\mathbb{R}^{n}$ be the reparametrization of $\mathcal{C}$ in terms of arclength (i.e. $\mathcal{C}=\mbox{Im}(\hat{\gamma})$). Supposing that $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is a function that is continuous in a neighborhood of $\mathcal{C}$, express the integral $\int_{\mathcal{C}}f$ in terms of the parametrizations $\gamma$ and $\hat{\gamma}$.\\ 3) Consider the helix $\mathcal{C}\subset\mathbb{R}^{3}$ parametrized as follows: \[ \gamma(t)=\left(\begin{array}{c} r\cos t\\ r\sin t\\ 2t \end{array}\right)\mbox{, }t\in[-2\pi,2\pi]. \] Compute the arclength $\lambda(\mathcal{C})$ and the integral $\int_{\mathcal{C}}f$ where $ f(x,y,z)=xy\sin z$. \parpic[r]{\includegraphics[width=2in]{WiperMotion.png}} \noindent{\textbf{Exercise 2: Surfaces.}} A parametrization of an orientable surface $\mathcal{S}\subset\mathbb{R}^{3}$ is a $\mathcal{C}^{r}$ map $ \sigma:\mathcal{U}\longrightarrow\mathbb{R}^{3}\mbox{, }(u,v)\longmapsto\left(x(u,v),y(u,v),z(u,v)\right), $ where $\mathcal{U}\subset\mathbb{R}^{2}$ is an open subset.\\ \underline{1)} Compute the area $\mathcal{A}(\mathcal{S})$ of the surface $\mathcal{S}\subset\mathbb{R}^{3}$ given by the parametrization: \[ \sigma(u,v)=\left(u\cos v,u\sin v,v\right)\mbox{, }\forall(u,v)\in(0,1)\times(0,\pi). \] 2) Let $\mathcal{S}$ be the part of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ lying inside the cylinder $x^{2}+y^{2}=b^{2}$ where $0