% in Linux systems here at the math department you may put this file in an empty folder and write in a terminal:"latex Handin.text" and it simply produces the pdf-file.

% Everything until the command \begin{document} is preamble. I.e. it sets up the fonts, command, shorthands, language, etc.. If you want to start writing you own Latex document right away you can remove everything between \begin{document} and \end{document} and write you own text and equations.

% I have tried to explain everything in the preamble for those who are interested.

% <- the procentage sign tells the latex command to ignore - which is why I can write anything here and it wont impact the pdf output file.

\documentclass[reqno,english,11pt]{amsart}
% Loads a style file, which determines most of the layout of the document.
\pdfoutput=1                             
% Sets output format as pdf-file

\usepackage[latin1]{inputenc}            
\usepackage[T1]{fontenc}                 
% To get Swedish characters in the output
\usepackage[swedish,english]{babel}      
% Last language is used - exchange order to get Swedish. This has a big impact. Indeed, if not done properly Latex will not hyphenate any words.

\usepackage[all]{xy}                     
% For making diagrams - see:http://en.wikibooks.org/wiki/LaTeX/Xy-pic for help on this
\usepackage{mathtools}
% Some extra tools you might need
\usepackage{euscript}
% Nice script fonts - I used script A for atlasses.
\usepackage{url}
% Use this to write an url in a document as \url{www.math.uu.se}.

\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\newtheorem{example}[definition]{Example}
\newtheorem{exercise}[definition]{Exercise}
\newtheorem{question}[definition]{Question}
\newtheorem{observation}[definition]{Observation}
\newtheorem{theorem}[definition]{Theorem}
\newtheorem{lemma}[definition]{Lemma}
% socalled environments. The fact that they all "linked" with definition means that they are all numbered using the same system so you e.g. get: Example 1.1, then Lemma 1.2, then observation 1.3, then Lemma 1.4 and so on. Removing the definition part of each subsequent command would make each of them numbered independently - remember that LaTeX numbers everything for you.

\DeclareMathOperator{\dist}{dist}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\U}{U}
\DeclareMathOperator{\Or}{O}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\rang}{rang}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\proj}{proj}
\DeclareMathOperator{\adj}{adj}
% Most operators in mathematics like "sin" is written in a different font than variables. Indeed, this makes "sin" different from (s times i times n). "sin" is automatically defined by LaTeX, but the above are not.
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
% the command $\mathbb{R}$ writes the type of R that denotes the real numbers, the above thus defines shorthands for real, natural, complex numbers and integers.
\newcommand{\Ss}{\mathcal{S}}
\newcommand{\A}{\mathcal{A}}
% These provides shorthands for script A and S - the nice fonts we got from the package euscript loaded above - I used the script A for atlasses.

\DeclarePairedDelimiter\absv{\lvert}{\rvert}
\DeclarePairedDelimiter\norm{\lVert}{\rVert}
\DeclarePairedDelimiter\inner{\langle}{\rangle}
\DeclarePairedDelimiter\sqbrac{[}{]}
\DeclarePairedDelimiter\pare{(}{)}
% shorthands for brackets (paranthesis, square brackets, inner product...). This will not work if mathtools package is not loaded as above. E.g. to get the inner product <v,w> looking nice in a math equation write \inner{v,w}. 

\newcommand{\osub}{\mathring{\subset}}
\newcommand{\osup}{\mathring{\supset}}
% defining short hands for open subsets and open supersets. The mathring command puts a small circle over any math symbol.

\def\co{\colon\thinspace}
% a colon in math is spaced wrongly - so if you want to write maps you chouls use this colon. E.g. you may write $f \co X \to Y$ and get f:X->Y in the correct style.


\begin{document}

\section{Hand in Assignment}

The following hand in assignment is due on April 4th. It will be pass or fail, but it will be possible to retry and hand in a revised version later provided the first attempt was a serious one. You may write questions via email, and if necessary I will update the web page with additional hints.

I would suggest writing it using \LaTeX~and emailing me a pdf-file. If you are not familiar with \LaTeX~it may be a good opportunity to learn. I will be happy to answer any questions you have if you choose to learn it and I will also put the latex version of this file on the course web page \url{http://www2.math.uu.se/~tkragh/1MA259-2013/Kurs.html}. I have written in comments in the pre-amble of the file to help you understand each part of it. I would suggest opening it in a text editor which color codes latex files - if not it will look completely confusing. I usually use Emacs (GNU Emacs 23). How precisely you turn the tex-file into a pdf-file depends on your operating system and setup, but it should be installed and work easily on the math departments computers. Of course you are also welcome to use other formats, and even hand write it. However, if you hand in a hard copy and not via email you should make sure it arrives on April 3rd at my box here at the math department.

\begin{exercise} [Example 11.12 from notes]
  Let $M$
  % Here is the first use of "$" to indicate to latex that M is a math symbol, so it writes it in the correct font.
  be the Möbious strip (as a manifold) and let $S^1 \subset M$ be the inclusion described in example 11.11 and 11.12 (or equivalently as the zero section in the Möbious band - thought of as a vector bundle over $S^1$). Taking the product with the open interval $(0,1)$ we get an inclusion $S^1\times (0,1) \subset M \times (0,1)$.

  Prove that a global chart of $S^1\times (0,1)$ is not the restriction of a chart of a neighborhood in $M\times (0,1)$.

  Hint: tangent bundles.
\end{exercise}


\begin{exercise}[Collar neighborhood theorem]
  Prove that the boundary $\partial M \subset M$ of a compact manifold w.b. has a neighborhood diffeomorphic to $\partial M \times [0,1)$.

  Hint: try and construct the map from a neighborhood to $\partial M \times [0,1)$ (each of the two components at a time). There are several ways of solving this exercise, and this hint just refers to one.
\end{exercise}

The following is somewhat different, but I found several blogs online praising this type of assignments - even in math. So I am curious how it turns out.

\begin{exercise}
  Write a so called reflective journal about the course (so far). That is, write a short text (max 1 page - can be shorter) addressing the general topic of the following questions (those you find relevant).
  \begin{itemize}
  \item What in the course do you believe is most important?
  \item How do parts of this course relate to other courses you have had or things you have learned? E.g. have you now seen proofs of theorems/lemmas that you have used elsewhere, but never seen proved before.
  \item Are there any parts of the course you had not heard of before and/or seems unrelated to anything else you have learned?
  \item Are there any parts of the course that seems contradictory to things you have learned in other courses? (this is interesting even if you know why the difference is there - e.g. if a small variation of a definition changes some subtle points or even the entire landscape of the subject).
  \end{itemize}
  I would suggest that after you read these questions you go through the curriculum considering all of them (and questions similar to them), and then write the short text.
\end{exercise}


\end{document}