mathsoftware

A course about LaTeX and SageMath
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Homework1.tex (4536B)


      1 \documentclass[12pt,a4paper]{article}
      2 \usepackage[utf8]{inputenc}
      3 \usepackage{amsmath}
      4 \usepackage{amsfonts}
      5 \usepackage{amssymb}
      6 \usepackage{amsthm}
      7 \usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
      8 
      9 \title{Mathematical software - homework 1}
     10 \author{Sebastiano Tronto}
     11 
     12 \newtheorem{thm}{Theorem}
     13 \newtheorem{prop}[thm]{Proposition}
     14 
     15 \theoremstyle{definition}
     16 \newtheorem{ex}{Exercise}
     17 
     18 \theoremstyle{definition}
     19 \newtheorem*{remark}{Remark}
     20 
     21 \begin{document}
     22 
     23 \noindent\hrulefill
     24 
     25 \begin{center}
     26 \Huge{\textbf{Mathematical Software - Homework 1}}
     27 \end{center}
     28 
     29 \noindent\hrulefill
     30 \begin{center}
     31 \begin{tabular}{lcr}
     32 \texttt{sebastiano.tronto@uni.lu} & \qquad \qquad \qquad \qquad &
     33 \textbf{Deadline: Sunday, March 28th}
     34 \end{tabular}
     35 \end{center}
     36 
     37 \vspace{1cm}
     38 
     39 \begin{center}
     40   \emph{\large
     41     For each of the following exercises submit a .tex and a .pdf file.
     42   }
     43 \end{center}
     44 
     45 \vspace{1cm}
     46 
     47 \begin{ex}
     48   Write a short Latex document that contains the following theorem-like
     49   environments using the \texttt{\textbackslash newtheorem} command of the
     50   \texttt{amsthm} package (the box around the text is not needed):
     51   \begin{center}
     52     \fbox{\parbox{0.95\textwidth}{
     53       \begin{prop}[Fundamental Theorem of Algebra]
     54         \label{prop:fta}
     55         Let \(p(x)\) be a non-constant polynomial with coefficients in
     56         $\mathbb C$. Then there is \(z\in\mathbb C\) such that $p(z)=0$.
     57       \end{prop}
     58 
     59       \begin{remark}
     60         Proposition \ref{prop:fta} is not true for polynomials with
     61         coefficients in $\mathbb R$. For example
     62         \begin{align}
     63           p(x) = x^2+1
     64         \end{align}
     65         does not have real roots.
     66       \end{remark}
     67 
     68       \begin{thm}
     69         If $X$ and $Y$ are $\sigma$-finite measure spaces and $f:X\times Y\to
     70         \mathbb R$ is measurable and such that
     71         \begin{align*}
     72           \int_{X\times Y}|f(x,y)|\mathrm d(x,y) < \infty
     73         \end{align*}
     74         then
     75         \begin{align}
     76           \label{eq:fubini}
     77           \int_X\left(\int_Yf(x,y)\mathrm d y\right)\mathrm d x =
     78           \int_Y\left(\int_Xf(x,y)\mathrm d x\right)\mathrm d y =
     79           \int_{X\times Y} f(x,y)\mathrm d(x,y)\,.
     80         \end{align}
     81       \end{thm}
     82 
     83       \begin{remark}
     84         In practice, equation \eqref{eq:fubini} means that we can switch the
     85         order of integration in a double integral.
     86       \end{remark}
     87     }}
     88   \end{center}
     89   Notice that Propositions, Theorems and some of the equations are numbered,
     90   and some of them are referred to in the Remarks. This numbering should change
     91   accordingly if more numbered Theorems and equations are added before this
     92   part of the text.
     93 \end{ex}
     94 
     95 \vspace{0.8cm}
     96 
     97 \begin{ex}
     98 [This is basically a ``do whatever you want, just do something'' exercise]
     99 Write a Mathematical document in Latex, where ``Mathematical'' means any text
    100 that contains both inline and displaystyle symbols and formulas, and where most
    101 of the sentences contain a Mathematical symbol.
    102 For example, you can submit (part of) your homework or notes for another course
    103 that you are following at the moment.
    104 
    105 You must follow the following rules:
    106 \begin{itemize}
    107   \item Your document must be at least 2 pages long.
    108   \item Use the following in your preamble:
    109     \begin{verbatim}
    110 \documentaclass[12pt,a4paper]{article}
    111 \usepackage[left=2.5cm,right=2.5cm,top=2.5cm,bottom=2.5cm]{geometry}
    112     \end{verbatim}
    113 \end{itemize}
    114 The document itself can be about any area of Mathematics, Physics or similar.
    115 The correctness of the Mathematics itself will not be judged for this course.
    116 \end{ex}
    117 
    118 \section*{Grading}
    119 
    120 This homework assignment is worth 25\% of your final grade.
    121 
    122 \vspace{0.3cm}
    123 \textbf{Exercise 1 (10 points).}
    124 \begin{itemize}
    125   \item A correct use of the \texttt{\textbackslash newtheorem} command is
    126         worth 4 out of 10 points.
    127   \item A correct use of the labelling and reference system is worth 4 points.
    128   \item Reproducing correctly the mathematical formulas is worth 2 points.
    129 \end{itemize}
    130 
    131 \textbf{Exercise 2 (10 points).}
    132 \begin{itemize}
    133   \item A resulting pdf file that looks readable and correct is worth 6 out of
    134         10 points.
    135   \item The correct use of latex commands and environments is worth 4 out of
    136         10 points. \emph{Examples of mistakes in this sense: adjusting manually
    137         the size of parenthesis instead of using \texttt{\textbackslash left}
    138         and \texttt{\textbackslash right}; using \texttt{\textbackslash Sigma}
    139         when you actually want to write a sum symbol.}
    140 \end{itemize}
    141 
    142 
    143 \end{document}