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}