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diff --git a/info/undergradmath/undergradmath.tex b/info/undergradmath/undergradmath.tex index 5d95b59e04..f041bab041 100644 --- a/info/undergradmath/undergradmath.tex +++ b/info/undergradmath/undergradmath.tex @@ -1,6 +1,7 @@ \RequirePackage{mmap} % make PDF copy and paste-able \documentclass[twocolumn]{article} \usepackage{amsmath, amsthm, amssymb} +\usepackage{mathrsfs} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} % \usepackage{lmodern} % from http://tex.stackexchange.com/a/115089/121234 @@ -23,7 +24,7 @@ \lstdefinestyle{inline}{basicstyle=\ttfamily} \usepackage[dvipsnames]{xcolor} - \definecolor{headcolor}{HTML}{E34234} % vermillion + \definecolor{headcolor}{HTML}{004225} % British racing green {E34234} % vermillion \usepackage{titlesec} % \titleformat{ command }[ shape ]{ format }{ label }{ sep }{ before-code }[ after-code ] % \titlespacing*{ command }{ left }{ before-sep }{ after-sep }[ right-sep ] @@ -42,16 +43,15 @@ \section{Rule One} Any mathematics at all, -even a single character, goes in a mathematical setting. -Thus, for ``the value of \( x \) is \( 7 \)\,'' enter -`\lstinline[style=inline]!the value of \( x \) is \( 7 \)!'. +even a single character, gets a mathematical setting. +Thus, for ``the value of $x$ is $7$\,'' enter +\lstinline[style=inline]!the value of $x$ is $7$!. \section{Template} Your document should contain at least this. \begin{lstlisting} \documentclass{article} -\usepackage{amsmath, amssymb, amsthm} -\usepackage[utf8]{inputenc} +\usepackage{mathtools,amssymb,amsthm} % imports amsmath \begin{document} --document body here-- @@ -62,19 +62,19 @@ Your document should contain at least this. \section{Common constructs} \begin{center} \begin{tabular}{@{} *{1}{l@{\hspace{1.5em}}} l @{}} - \(x^2\)\quad\lstinline[style=inline]!x^2! - &\(\sqrt{2}\), \(\sqrt[n]{3}\)\quad\lstinline[style=inline]!\sqrt{2}!, + $x^2$\quad\lstinline[style=inline]!x^2! + &$\sqrt{2}$, $\sqrt[n]{3}$\quad\lstinline[style=inline]!\sqrt{2}!, \lstinline[style=inline]!\sqrt[n]{3}! \\ - \(x_{i,j}\)\quad\lstinline[style=inline]!x_{i,j}! - &\(\frac{2}{3}\), \(2/3\)\quad\lstinline[style=inline]!\frac{2}{3}!, + $x_{i,j}$\quad\lstinline[style=inline]!x_{i,j}! + &$\frac{2}{3}$, $2/3$\quad\lstinline[style=inline]!\frac{2}{3}!, \lstinline[style=inline]!2/3! \end{tabular} \end{center} \section{Calligraphic letters} -Use as \lstinline[style=inline]!\( \mathcal{A} \)!. +Use as in \lstinline[style=inline]!$\mathcal{A}$!. \begin{center} - \( \mathcal{A}\mathcal{B}\mathcal{C} + $ \mathcal{A}\mathcal{B}\mathcal{C} \mathcal{D}\mathcal{E}\mathcal{F} \mathcal{G}\mathcal{H}\mathcal{I} \mathcal{J}\mathcal{K}\mathcal{L} @@ -82,9 +82,13 @@ Use as \lstinline[style=inline]!\( \mathcal{A} \)!. \mathcal{P}\mathcal{Q}\mathcal{R} \mathcal{S}\mathcal{T}\mathcal{U} \mathcal{V}\mathcal{W}\mathcal{X} - \mathcal{Y}\mathcal{Z} \) + \mathcal{Y}\mathcal{Z} $ \end{center} - +Get script letters, such as $\mathscr{P}$ +from \lstinline[style=inline]!$\mathscr{P}$!, +by putting +\lstinline[style=inline]!\usepackage{mathrsfs}! +in the preamble. \section{Greek} \begin{center} @@ -137,19 +141,19 @@ Use as \lstinline[style=inline]!\( \mathcal{A} \)!. \begin{center} \begin{tabular}{@{}*{2}{l@{\hspace{1em}}}l@{}} $\cup$\quad \lstinline[style=inline]!\cup! - &\(\mathbb{R}\)\quad\lstinline[style=inline]!\mathbb{R}! + &$\mathbb{R}$\quad\lstinline[style=inline]!\mathbb{R}! &$\forall$\quad \lstinline[style=inline]!\forall! \\ $\cap$\quad \lstinline[style=inline]!\cap! - &\(\mathbb{Z}\)\quad\lstinline[style=inline]!\mathbb{Z}! + &$\mathbb{Z}$\quad\lstinline[style=inline]!\mathbb{Z}! &$\exists$\quad \lstinline[style=inline]!\exists! \\ - \(\subset\)\quad\lstinline[style=inline]!\subset! - &\(\mathbb{Q}\)\quad\lstinline[style=inline]!\mathbb{Q}! + $\subset$\quad\lstinline[style=inline]!\subset! + &$\mathbb{Q}$\quad\lstinline[style=inline]!\mathbb{Q}! &$\neg$\quad \lstinline[style=inline]!\neg! \\ $\subseteq$\quad \lstinline[style=inline]!\subseteq! - &\(\mathbb{N}\)\quad\lstinline[style=inline]!\mathbb{N}! + &$\mathbb{N}$\quad\lstinline[style=inline]!\mathbb{N}! &$\vee$\quad \lstinline[style=inline]!\vee! \\ $\supset$\quad \lstinline[style=inline]!\supset! - &\(\mathbb{C}\)\quad\lstinline[style=inline]!\mathbb{C}! + &$\mathbb{C}$\quad\lstinline[style=inline]!\mathbb{C}! &$\wedge$\quad \lstinline[style=inline]!\wedge! \\ $\supseteq$\quad \lstinline[style=inline]!\supseteq! &$\varnothing$\quad \lstinline[style=inline]!\varnothing! @@ -157,24 +161,18 @@ Use as \lstinline[style=inline]!\( \mathcal{A} \)!. $\in$\quad \lstinline[style=inline]!\in! &$\emptyset$\quad \lstinline[style=inline]!\emptyset! &$\models$\quad \lstinline[style=inline]!\models! \\ - $\ni$\quad \lstinline[style=inline]!\ni! + $\notin$\quad \lstinline[style=inline]!\notin! &$\aleph$\quad \lstinline[style=inline]!\aleph! - &$\Rightarrow$\quad \lstinline[style=inline]!\Rightarrow! \\ - $\notin$\quad \lstinline[style=inline]!\notin! - &$\setminus$\quad \lstinline[style=inline]!\setminus! - &$\nRightarrow$\quad \lstinline[style=inline]!\nRightarrow! \\ - $\not\in$\quad \lstinline[style=inline]!\not\in! - &$\equiv$\quad \lstinline[style=inline]!\equiv! - & \\ + &$\setminus$\quad \lstinline[style=inline]!\setminus! \\ + % $\ni$\quad \lstinline[style=inline]!\ni! \end{tabular} \end{center} -Negate an operator, as in \(\not\subset\), +Negate an operator, as in $\not\subset$, with \lstinline[style=inline]!\not\subset!. Get the set complement -$A^{\mathsf{c}}$ with \lstinline[style=inline]!A^{\mathsf{c}}!, -get $A^{\complement}$ with \lstinline[style=inline]!A^{\complement}!, -or -get $\bar{A}$ with \lstinline[style=inline]!\bar{A}!. +$A^{\mathsf{c}}$ with \lstinline[style=inline]!A^{\mathsf{c}}! +(or $A^{\complement}$ with \lstinline[style=inline]!A^{\complement}!, +or $\overline{A}$ with \lstinline[style=inline]!\overline{A}!). @@ -199,6 +197,7 @@ If the decorated letter is $i$ or~$j$ then some decorations need as in \lstinline[style=inline]!\vec{\imath}!. Some authors use boldface for vectors: \lstinline[style=inline]!\boldsymbol{x}!. +% Mention \bm? Entering \lstinline[style=inline]!\overline{x+y}! produces $\overline{x+y}$, @@ -213,8 +212,8 @@ Comment on an expression as here \section{Dots} Use low dots in a list $\{0,1,2,\,\ldots\}$, entered as \lstinline[style=inline]!\{0,1,2,\,\ldots\}!. -(If you use \lstinline[style=inline]!\ldots! in plain text as -\lstinline[style=inline]!London, Paris, \ldots{}\,.! note +(If you use \lstinline[style=inline]!\ldots! in plain text as with +\lstinline[style=inline]!London, Paris, \ldots{}\,.! then note the thinspace \lstinline[style=inline]!\,! before the period.) Use centered dots in a sum or product $1+\cdots+100$, entered as \lstinline[style=inline]!1+\cdots+100!. @@ -315,35 +314,13 @@ These get the same treatment. &$\checkmark$\quad \lstinline[style=inline]!\checkmark! \\ \end{tabular} \end{center} -Enter \lstinline[style=inline]!a|b! for the divides relation \( a|b \). -Use \lstinline[style=inline]!\mid! as in -\lstinline[style=inline]!\{a\in S\mid\text{\(a=0\) or \(a\) is odd}\}! -for the set \( \{a\in S \mid\text{\(a=0\) or \(a\) is odd}\} \). - -\section{Variable-sized operators} -The summation -\(\sum_{j=0}^3j^2\)\quad\lstinline[style=inline]!\sum_{j=0}^3 j^2! -and the integral $\int_{x=0}^3x^2\,dx$ -\lstinline[style=inline]!\int_{x=0}^3 x^2\,dx! -expand when displayed. -\begin{equation*} - \sum_{j=0}^3j^2 - \qquad - \int_{x=0}^3x^2\,dx -\end{equation*} -These do the same. -\begin{center} - \begin{tabular}{@{}*{2}{l@{\hspace{.35in}}}l@{}} - $\int$\quad\lstinline[style=inline]!\int! - &$\iiint$\quad\lstinline[style=inline]!\iiint! - &$\bigcup$\quad\lstinline[style=inline]!\bigcup! \\ - $\iint$\quad\lstinline[style=inline]!\iint! - &$\oint$\quad\lstinline[style=inline]!\oint! - &$\bigcap$\quad\lstinline[style=inline]!\bigcap! \\ - % &$\bigvee$\quad\lstinline[style=inline]!\bigvee! - % &$\bigwedge$\quad\lstinline[style=inline]!\bigwedge! \\ - \end{tabular} -\end{center} +Use \lstinline[style=inline]!a\mid b! for the divides relation, +\mbox{$a\mid b$}, +and \lstinline[style=inline]!a\nmid b! for the negation, \mbox{$a\nmid b$}. +Also use \lstinline[style=inline]!\mid! to get +set builder notation \mbox{$\{a\in S \mid\text{$a$ is odd}\}$}, +with +\lstinline[style=inline]!\{a\in S\mid\text{$a$ is odd}\}!. \section{Arrows} \begin{center} @@ -369,6 +346,31 @@ The right arrows in the first column have matching left arrows, such as \lstinline[style=inline]!\nleftarrow!, and there are some other matches for down arrows, etc. +\section{Variable-sized operators} +The summation +$\sum_{j=0}^3j^2$\quad\lstinline[style=inline]!\sum_{j=0}^3 j^2! +and the integral $\int_{x=0}^3x^2\,dx$ +\lstinline[style=inline]!\int_{x=0}^3 x^2\,dx! +expand when displayed. +\begin{equation*} + \sum_{j=0}^3j^2 + \qquad + \int_{x=0}^3x^2\,dx +\end{equation*} +These do the same. +\begin{center} + \begin{tabular}{@{}*{2}{l@{\hspace{.35in}}}l@{}} + $\int$\quad\lstinline[style=inline]!\int! + &$\iiint$\quad\lstinline[style=inline]!\iiint! + &$\bigcup$\quad\lstinline[style=inline]!\bigcup! \\ + $\iint$\quad\lstinline[style=inline]!\iint! + &$\oint$\quad\lstinline[style=inline]!\oint! + &$\bigcap$\quad\lstinline[style=inline]!\bigcap! \\ + % &$\bigvee$\quad\lstinline[style=inline]!\bigvee! + % &$\bigwedge$\quad\lstinline[style=inline]!\bigwedge! \\ + \end{tabular} +\end{center} + \section{Fences} \begin{center} \begin{tabular}{@{}*{2}{l@{\hspace{1.5em}}}l@{}} @@ -382,8 +384,18 @@ and there are some other matches for down arrows, etc. &$\lceil\;\rceil$\quad\lstinline[style=inline]!\lceil\rceil! \end{tabular} \end{center} -They will grow with the enclosed formula -using \lstinline[style=inline]!\left! and \lstinline[style=inline]!\right!. +Fix the size with +\lstinline[style=inline]!\big!, +\lstinline[style=inline]!\Big!, +\lstinline[style=inline]!\bigg!, or +\lstinline[style=inline]!\Bigg!. +\begin{center} + $\displaystyle\Big[\sum_{k=0}^n e^{k^2}\Big]$\quad + \lstinline[style=inline]!\Big[\sum_{k=0}^n e^{k^2}\Big]! +\end{center} +To have them grow with the enclosed formula, +use \lstinline[style=inline]!\left! and \lstinline[style=inline]!\right! +(although sometimes \lstinline[style=inline]!\big!, etc., are necessary). \begin{center} $\displaystyle\left\langle i,2^{2^i}\right\rangle$\quad \lstinline[style=inline]!\left\langle i,2^{2^i}\right\rangle! @@ -392,22 +404,13 @@ Every \lstinline[style=inline]!\left! must match a \lstinline[style=inline]!\right! and they must end on the same line in the output. -For a one-sided fence put a period +For a one-sided fence, put a \lstinline[style=inline]!\left.! or -\lstinline[style=inline]!\right.! on the other side.% +\lstinline[style=inline]!\right.! on the other side. \begin{center} $\displaystyle\left.\frac{df}{dx}\right|_{x_0}$\quad \lstinline[style=inline]!\left.\frac{df}{dx}\right|_{x_0}! \end{center} -Fix the size with -\lstinline[style=inline]!\big!, -\lstinline[style=inline]!\Big!, -\lstinline[style=inline]!\bigg!, or -\lstinline[style=inline]!\Bigg!. -\begin{center} - $\displaystyle\Big[\sum_{k=0}^n e^{k^2}\Big]$\quad - \lstinline[style=inline]!\Big[\sum_{k=0}^n e^{k^2}\Big]! -\end{center} \section{Arrays, Matrices} @@ -441,7 +444,7 @@ Definition by cases is an array with two columns. \begin{center} $\displaystyle f_n=\begin{cases} - a &\text{if \(n=0\)} \\ + a &\text{if $n=0$} \\ r\cdot f_{n-1} &\text{else} \end{cases} $ @@ -450,14 +453,15 @@ Definition by cases is an array with two columns. \begin{lstlisting}[xleftmargin=-1em] f_n= \begin{cases} - a &\text{if \(n=0\)} \\ + a &\text{if $n=0$} \\ r\cdot f_{n-1} &\text{else} \end{cases} \end{lstlisting} \end{minipage} \end{center} -A matrix is another array variant. -With this abbreviation you need not specify column alignments. +A matrix is an array with fences. +With a \lstinline[style=inline]!pmatrix! environment, +you need not specify column alignments. \begin{center} $\displaystyle \begin{pmatrix} @@ -479,39 +483,49 @@ For the determinant use \lstinline[style=inline]!|A|! inline and \lstinline[style=inline]!vmatrix! in display. \section{Spacing in mathematics} -\begin{center} - \begin{tabular}{@{}*{1}{l@{\hspace{2.25em}}}l@{}} - \(\rightarrow\,\leftarrow\)\quad\lstinline[style=inline]!\,! - &\(\rightarrow\quad\leftarrow\)\quad\lstinline[style=inline]!\quad! \\ - \(\rightarrow\:\leftarrow\)\quad\lstinline[style=inline]!\:! - &\(\rightarrow\qquad\leftarrow\)\quad\lstinline[style=inline]!\qquad! \\ - \(\rightarrow\;\leftarrow\)\quad\lstinline[style=inline]!\;! - &\(\rightarrow\!\leftarrow\)\quad\lstinline[style=inline]+\!+ - \end{tabular} -\end{center} -The left column spaces are in ratio \(3\mathbin{:}4\mathbin{:}5\). -The last in the right column is a negative space, -opposite to \lstinline[style=inline]!\,!. +Improve $\sqrt{2}x$ to $\sqrt{2}\,x$ with a thin space, +as in \lstinline[style=inline]!\sqrt{2}\,x!. +Slightly wider are \lstinline[style=inline]!\:! +and \lstinline[style=inline]!\;! +(the three are in ratio $3\mathbin{:}4\mathbin{:}5$). +Get the improvement of $n/\!\log n$ instead of $n/\log n$ +by using a negative thin space, +as in \lstinline[style=inline]+n/\!\log n+. +Bigger spaces are: \lstinline[style=inline]!\quad! for +$\rightarrow\quad\leftarrow$, and \lstinline[style=inline]!\qquad! for +$\rightarrow\qquad\leftarrow$, +which are useful between parts of a display. +% \begin{center} +% \begin{tabular}{@{}*{1}{l@{\hspace{2.25em}}}l@{}} +% $\rightarrow\mkern3mu\leftarrow$\quad\lstinline[style=inline]!\,! +% &$\rightarrow\quad\leftarrow$\quad\lstinline[style=inline]!\quad! \\ +% $\rightarrow\mkern4mu\leftarrow$\quad\lstinline[style=inline]!\:! +% &$\rightarrow\qquad\leftarrow$\quad\lstinline[style=inline]!\qquad! \\ +% $\rightarrow\mkern5mu\leftarrow$\quad\lstinline[style=inline]!\;! +% &$\rightarrow\!\leftarrow$\quad\lstinline[style=inline]+\!+ +% \end{tabular} +% \end{center} Get arbitrary space as in -% \(\rightarrow\hspace*{0.5cm}\leftarrow\) -\lstinline[style=inline]!\hspace{0.5cm}!. +% $\rightarrow\hspace*{0.5cm}\leftarrow$ +\lstinline[style=inline]!\hspace*{0.5cm}!. \section{Displayed equations} -Put equations on a separate line with the -\lstinline[style=inline]!equation*! environment. +The +\lstinline[style=inline]!equation*! environment +puts an equation on a separate line. \begin{center} \vspace*{-\topsep} \begin{minipage}{0.25\linewidth}\vspace*{-\abovedisplayskip} \begin{equation*} - S=k\log W + S=k\cdot\lg W \end{equation*} \end{minipage} \quad \begin{minipage}{0.65\linewidth} \begin{lstlisting}[xleftmargin=0ex] \begin{equation*} - S=k\log W + S=k\cdot\lg W \end{equation*} \end{lstlisting} \end{minipage} @@ -535,7 +549,7 @@ You can break into multiple lines. \end{lstlisting} \end{minipage} \end{center} -Align using the \lstinline[style=inline]!align*! environment +Align equations using \lstinline[style=inline]!align*! \vspace*{-\topsep} \begin{center} \begin{minipage}[c]{0.25\linewidth}\vspace*{-\abovedisplayskip} @@ -554,7 +568,7 @@ Align using the \lstinline[style=inline]!align*! environment \end{lstlisting} \end{minipage} \end{center} -(you can have an empty left or right side of the alignment). +(the left or right side of an alignment can be empty). For each environment, get a numbered version by dropping the asterisk from the name. @@ -563,19 +577,19 @@ dropping the asterisk from the name. The last three here are display style. \begin{center} \small \def\arraystretch{2} \vspace*{-\topsep}\vspace*{-0.25cm} - \begin{tabular}{@{}r@{\hspace{1em}}l@{}} - \( f\colon\mathbb{R}\to\mathbb{R} \) - &\lstinline[style=inline]!f\colon\mathbb{R}\to\mathbb{R}! \\ - \( 9.8~\text{m}/\text{s}^2 \) - &\lstinline[style=inline]!9.8~\text{m}/\text{s}^2! \\ + \begin{tabular}{@{}l@{}} + $ f\colon\mathbb{R}\to\mathbb{R} $ + \quad\lstinline[style=inline]!f\colon\mathbb{R}\to\mathbb{R}! \\ + $ 9.8~\text{m}/\text{s}^2 $ + \quad\lstinline[style=inline]!9.8~\text{m}/\text{s}^2! \\ $\displaystyle \lim_{h\to 0}\frac{f(x+h)-f(x)}{h} $ - &\lstinline[style=inline]!\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}! \\ + \quad\lstinline[style=inline]!\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}! \\ $\displaystyle \int x^2\,dx=x^3/3+C $ - &\lstinline[style=inline]!\int x^2\,dx=x^3/3+C! \\ + \quad\lstinline[style=inline]!\int x^2\,dx=x^3/3+C! \\ % $\displaystyle % \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx} % $ @@ -585,41 +599,41 @@ The last three here are display style. +\boldsymbol{j}\frac{d}{dy} +\boldsymbol{k}\frac{d}{dz} $ - &\lstinline[style=inline]!\nabla=\boldsymbol{i}\frac{d}{dx}+!\,\(\cdots\)\\ + \quad\lstinline[style=inline]!\nabla=\boldsymbol{i}\frac{d}{dx}+!\,$\cdots$\\ \end{tabular} \end{center} \section{Discrete mathematics examples} There are four modulo forms: -\(m\bmod n\) is from \lstinline[style=inline]!m\bmod n!, -and \(a\equiv b\pmod m\) is from \lstinline[style=inline]!a\equiv b\pmod m!, -and \(a\equiv b\mod m\) is from \lstinline[style=inline]!a\equiv b\mod m!, -and \(a\equiv b\pod m\) is from \lstinline[style=inline]!a\equiv b\pod m!. +$m\bmod n$ is from \lstinline[style=inline]!m\bmod n!, +and $a\equiv b\pmod m$ is from \lstinline[style=inline]!a\equiv b\pmod m!, +and $a\equiv b\mod m$ is from \lstinline[style=inline]!a\equiv b\mod m!, +and $a\equiv b\pod m$ is from \lstinline[style=inline]!a\equiv b\pod m!. For combinations the binomial symbol $\binom{n}{k}$ is from \lstinline[style=inline]!\binom{n}{k}!. This resizes to be bigger in a display (to require the display version use \lstinline[style=inline]!\dbinom{n}{k}! -and for the inline version use \lstinline[style=inline]!\tbinom{n}{k}!). +and require the inline version with \lstinline[style=inline]!\tbinom{n}{k}!). For permutations use -\( n^{\underline{r}} \) from \lstinline[style=inline]!n^{\underline{r}}! -(some authors use \( P(n,r) \), -or \( {}_nP_r \) from \lstinline[style=inline]!{}_nP_r!). +$ n^{\underline{r}} $ from \lstinline[style=inline]!n^{\underline{r}}! +(some authors use $ P(n,r) $, +or $ {}_nP_r $ from \lstinline[style=inline]!{}_nP_r!). \section{Statistics examples} \begin{center} \small\def\arraystretch{1.5} \vspace*{-\topsep}\vspace*{-0.15cm} - \begin{tabular}{@{}r@{\hspace{1em}}l@{}} + \begin{tabular}{@{}l@{}} $ - \sigma^2=\sqrt{\sum (x_i-\mu)^2/N} + \sigma^2=\sqrt{\,\sum (x_i-\mu)^2/N} $ - &\lstinline[style=inline]!\sigma^2=\sqrt{\sum (x_i-\mu)^2/N}! \\ + \quad\lstinline[style=inline]!\sigma^2=\sqrt{\,\sum (x_i-\mu)^2/N}! \\ $ E(X)=\mu_X=\sum (x_i-P(x_i)) $ - &\lstinline[style=inline]! E(X)=\mu_X=\sum (x_i-P(x_i))! \\ + \quad\lstinline[style=inline]! E(X)=\mu_X=\sum (x_i-P(x_i))! \\ \end{tabular} \end{center} The probability density of the normal distribution @@ -644,6 +658,6 @@ and De\TeX ify at \url{detexify.kirelabs.org/classify.html}. \noindent\parbox{\columnwidth}{\small% {\color{headcolor}\hrulefill\\ \vspace*{0ex} -Jim Hef{}feron, Saint Michael's College, VT USA\hfill 2017-Jan-10 +Jim Hef{}feron, Saint Michael's College, VT USA\hfill 2020-Dec-30 }} \end{document} |