\documentclass[11pt, letterpaper, onecolumn, conference, final]{IEEEtran}

\usepackage[margin = .5in]{geometry}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{subfigure}
\usepackage[utf8]{inputenc}
\usepackage{amsthm}

\newfam\hebfam
\font\tmp=rcjhbltx at10pt \textfont\hebfam=\tmp
\font\tmp=rcjhbltx at7pt  \scriptfont\hebfam=\tmp
\font\tmp=rcjhbltx at5pt  \scriptscriptfont\hebfam=\tmp

\edef\declfam{\ifcase\hebfam 
     0\or1\or2\or3\or4\or5\or6\or7\or8\or9\or A\or B\or C\or D\or E\or F\fi}

\mathchardef\aleph  = "0\declfam 27

\DeclareMathOperator*{\argmin}{arg\,min}

\theoremstyle{definition}
\newtheorem{problem}{Problem}
\newtheorem{solution}{Solution}

\title{MATH 208 - HW \# 4 - Corrections}
\author{Nathan Marianovsky \\ October 20, 2017}

\begin{document}

\maketitle

\begin{problem}
View $x$ as the affine coordinate for $\mathbb{RP}^1$ so as to identify $\mathbb{RP}^1$ with $\mathbb{R} \cup \{\infty\}$. Show that the vector field $\frac{\partial}{\partial x}$ on the open set $\mathbb{R} \subset \mathbb{RP}^1$ extends to a smooth vector field on all of $\mathbb{RP}^1$. Does this vector field vanish at $\infty$?
\end{problem}

\vspace{.5cm}

\begin{solution}
Using the overlap map $y = \frac{1}{x}$ we can say:
\begin{equation*}
\frac{\partial}{\partial x} = \frac{\text{d}y}{\text{d}x} \frac{\partial}{\partial y} = -\frac{1}{x^2} \frac{\partial}{\partial y} = -y^2 \frac{\partial}{\partial y}
\end{equation*}
As $x \rightarrow \infty$ the line $[x,1]$ tends to $[1,0]$. Consequently $y \rightarrow 0$ in this direction telling us that:
\begin{equation*}
\frac{\partial}{\partial x} \rightarrow 0 \hspace{.3cm} \text{as} \hspace{.3cm} x \rightarrow \infty
\end{equation*}
Therefore, $\frac{\partial}{\partial x}$ extends to a smooth vector field on $\mathbb{R} \cup \{\infty\}$ that vanishes at $\infty$. 
\end{solution}



\end{document}