\documentclass{article}
\usepackage[landscape]{geometry}
\usepackage{url}
\usepackage{multicol}
\usepackage{amsmath}
\usepackage{amsfonts}
\advance\topmargin-.8in
\advance\textheight2in
\advance\textwidth3in
\advance\oddsidemargin-1.5in
\advance\evensidemargin-1.5in
\parindent0pt
\parskip2pt
\newcommand{\hr}{\centerline{\rule{3.5in}{1pt}}}
\begin{document}
\begin{multicols*}{3}
\begin{center}
\textbf{Sage Quick Reference (Basic Math)}\\
Peter Jipsen, version 1.1\\
latest version at \url{wiki.sagemath.org/quickref}\\
GNU Free Document License, extend for your own use\\
Aim: map standard math notation to Sage commands
\end{center}

\textbf{Notebook (and commandline)}

Evaluate cell: $\langle$shift-enter$\rangle$

\emph{com}$\langle$tab$\rangle$ tries to complete \emph{command}

\emph{command}\verb|?|$\langle$tab$\rangle$ shows documentation

\emph{command}\verb|??|$\langle$tab$\rangle$ shows source

\verb|a.|$\langle$tab$\rangle$ shows all methods for object \verb|a| \quad
(more: \verb|dir(a)|)

\verb|search_doc('|\emph{string or regexp}\verb|')| \quad shows links to docs

\verb|search_src('|\emph{string or regexp}\verb|')| \quad shows links to source

\verb|lprint()| \quad toggle LaTeX output mode

\verb|version()| \quad print version of Sage

Insert cell: click on blue line between cells

Delete cell: delete content then backspace

%*********************************************
\hr\textbf{Numerical types}

Integers: $\mathbb Z=$ \verb|ZZ | e.g. \verb|-2  -1  0  1  10^100|

Rationals: $\mathbb Q=$ \verb|QQ | e.g. \verb|1/2  1/1000  314/100  -42|

Decimals: $\mathbb R\approx$ \verb|RR | e.g. \verb|.5  0.001  3.14  -42.|

Complex: $\mathbb C\approx$ \verb|CC | e.g. \verb|1+i  2.5-3*i|

%*********************************************
\hr\textbf{Basic constants and functions}

Constants: $\pi=$ \verb|pi| \quad $e=$ \verb|e| \quad $i=$ \verb|i| 
\quad $\infty=$ \verb|oo|

Approximate: \verb|pi.n(digits=18)| $=3.14159265358979324$

%Binary operations: \verb|+  -  *  /  ^|

Functions: \verb|sin cos tan sec csc cot sinh cosh tanh| \verb|sech csch coth log ln exp|

$ab=$ \verb|a*b| \quad $\frac a b=$ \verb|a/b| 
\quad 
$a^b=$ \verb|a^b| \quad $\sqrt{x}=$ \verb|sqrt(x)|

$\sqrt[n]{x}=$ \verb|x^(1/n)|
\quad 
$|x|=$ \verb|abs(x)|
\quad 
$\log_b(x)=$ \verb|log(x,b)|

Symbolic variables: e.g. \verb|t,u,v,y,z = var('t u v y z')|

Define function: e.g. $f(x)=x^2$ \qquad \verb| f(x)=x^2|

or \verb| f=lambda x: x^2 | or \verb| def f(x): return x^2|

%*********************************************
\hr\textbf{Operations on expressions}

\verb|factor(...)|\qquad \verb|expand(...)|\qquad \verb|(...).simplify_...|

Symbolic equations: \verb|f(x)==g(x)|

\verb|_| is previous output

\verb|_+a| \quad \verb|_-a| \quad \verb|_*a| \quad \verb|_/a | manipulates equation

Solve $f(x)=g(x)$: \verb| solve(f(x)==g(x),x)|

\verb|solve([f(x,y)==0, g(x,y)==0], x,y)|

\verb|find_root(f(x), a, b)|\quad find $x\in [a,b]$ s.t. $f(x)\approx 0$

$\displaystyle\sum_{i=k}^n f(i)=$ \verb|sum([f(i) for i in [k..n]])|

$\displaystyle\prod_{i=k}^n f(i)=$ \verb|prod([f(i) for i in [k..n]])|

%*********************************************
\hr\textbf{Calculus}

$\displaystyle\lim_{x\to a} f(x)=$ \verb|limit(f(x), x=a)|

$\displaystyle\lim_{x\to a^-} f(x)=$ \verb|limit(f(x), x=a, dir='minus')|

$\displaystyle\lim_{x\to a^+} f(x)=$ \verb|limit(f(x), x=a, dir='plus')|

$\frac{d}{dx}(f(x))=$ \verb|diff(f(x),x)|

$\frac{\partial}{\partial x}(f(x,y))=$ \verb|diff(f(x,y),x)|

\verb|diff| $=$ \verb|differentiate| $=$ \verb|derivative|

$\int f(x)dx=$ \verb|integral(f(x),x)|

\verb|integral| = \verb|integrate|

$\int_a^b f(x)dx=$ \verb|integral(f(x),x,a,b)|

Taylor polynomial, deg $n$ about $a$: \texttt{taylor(f(x),x,$a$,$n$)} 

%*********************************************
\hr\textbf{2d graphics}

\texttt{line([($x_1$,$y_1$),$\ldots$,($x_n$,$y_n$)],$\it options$)}

\texttt{polygon([($x_1$,$y_1$),$\ldots$,($x_n$,$y_n$)],$\it options$)}

\texttt{circle(($x$,$y$),$r$,$\it options$)}

\texttt{text("txt",($x$,$y$),$\it options$)}

\emph{options} as in \verb|plot.options|, 
e.g. \texttt{thickness=$\it pixel$},

\texttt{rgbcolor=($r$,$g$,$b$)},
\quad \texttt{hue=$h$} \quad where $0\le r,b,g,h\le 1$

use option \verb|figsize=[w,h]| to adjust aspect ratio

\texttt{plot(f($x$),$x_{\rm min}$,$x_{\rm max}$,$\it options$)}

\texttt{parametric\_plot((f($t$),g($t$)),$t_{\rm min}$,$t_{\rm max}$,$\it options$)}

\texttt{polar\_plot(f($t$),$t_{\rm min}$,$t_{\rm max}$,$\it options$)}

combine graphs: \verb|circle((1,1),1)+line([(0,0),(2,2)])|

\texttt{animate(}\emph{list of graphics objects, options}\texttt{).show(delay=20)}

%*********************************************
\hr\textbf{3d graphics}

\texttt{line3d([($x_1$,$y_1$,$z_1$),$\ldots$,($x_n$,$y_n$,$z_n$)],$\it options$)}

\texttt{sphere(($x$,$y$,$z$),$r$,$\it options$)}

\texttt{tetrahedron(($x$,$y$,$z$),$size$,$\it options$)}

\texttt{cube(($x$,$y$,$z$),$size$,$\it options$)}

\texttt{octahedron(($x$,$y$,$z$),$size$,$\it options$)}

\texttt{dodecahedron(($x$,$y$,$z$),$size$,$\it options$)}

\texttt{icosahedron(($x$,$y$,$z$),$size$,$\it options$)}

\mbox{\emph{options} e.g. \texttt{aspect\_ratio=$[1,1,1]$ color='red' opacity}}

\texttt{plot3d(f($x,y$),[$x_{\rm b}$,$x_{\rm e}$],[$y_{\rm b}$,$y_{\rm e}$],$\it options$)}

add option \verb|plot_points=[|$m,n$\verb|]| or use \texttt{plot3d\_adaptive}

\texttt{parametric\_plot3d((f($t$),g($t$),h($t$)),[$t_{\rm b}$,$t_{\rm e}$],$\it options$)}

\texttt{parametric\_plot3d((f($u,v$),g($u,v$),h($u,v$)),}

\qquad \qquad \qquad \qquad \qquad \qquad 
\texttt{[$u_{\rm b}$,$u_{\rm e}$],[$v_{\rm b}$,$v_{\rm e}$],$\it options$)}

use + to combine graphics objects

%*********************************************
\hr\textbf{Discrete math}

$\lfloor x\rfloor=$ \verb|floor(x)| 
\quad 
$\lceil x\rceil=$ \verb|ceil(x)|

Remainder of $n$ divided by $k=$ \verb|n%k| \quad\, $k|n$ iff \verb| n%k==0|

$n!=$ \verb|factorial(n)| \qquad
${x\choose m}=$ \verb|binomial(x,m)|

$\phi=$ \verb|golden_ratio| \qquad $\phi(n)=$ \texttt{euler\_phi($n$)}

Strings: e.g. \ \verb|s = 'Hello'| = \verb|"Hello"| = \verb|""+"He"+'llo'|

\texttt{s[0]='H' \quad s[-1]='o' \quad s[1:3]='el' \quad s[3:]='lo'}

Lists: e.g. \ \verb|[1,'Hello',x]| = \verb|[]+[1,'Hello']+[x]|

Tuples: e.g. \ \verb|(1,'Hello',x)| \quad (immutable)

Sets: e.g. \ $\{1,2,1,a\}=$ \verb|Set([1,2,1,'a'])| \ ($=\{1,2,a\}$)

List comprehension $\approx$ set builder notation, e.g.

$\{f(x):x\in X, x>0\}=$ \verb|Set([f(x) for x in X if x>0])|

%*********************************************
\hr\textbf{Linear algebra}

$\begin{pmatrix}1\\2\end{pmatrix}=$ \verb|vector([1,2])|

$\begin{pmatrix}1&2\\3&4\end{pmatrix}=$ \verb|matrix([[1,2],[3,4]])|

$\left|\begin{matrix}1&2\\3&4\end{matrix}\right|=$
\verb|det(matrix([[1,2],[3,4]]))|

$Av=$ \verb|A*v| \quad $A^{-1}=$ \verb|A^-1| \quad $A^t=$ \verb|A.transpose()|

methods: \verb|nrows() ncols() nullity() rank() trace()|...

%*********************************************
\hr\textbf{Sage modules and packages}

\verb|from| \emph{module\_name} \verb|import *| \qquad (many preloaded)

e.g. \texttt{calculus coding combinat crypto functions games
geometry graphs groups logic matrix numerical plot probability 
rings sets stats}

\verb|sage.|\emph{module\_name}\verb|.all.|$\langle$tab$\rangle$ shows exported commands

\mbox{Std packages: Maxima GP/PARI GAP Singular R Shell ...}

Opt packages: Biopython Fricas(Axiom) Gnuplot Kash ...

\%\emph{package\_name} then use package command syntax

\verb|time| \emph{command} \quad to show timing information

\end{multicols*}

\end{document}