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\usepackage{listings}

\lstset{language=Python, basicstyle=\ttfamily\bfseries,

commentstyle=\color{red}\itshape, stringstyle=\color{darkgreen},

showstringspaces=false, keywordstyle=\color{blue}\bfseries}

\providecommand{\alert}[1]{\textbf{#1}}

\title{Getting started with symbolics}

\author{FOSSEE}

\date{}

\usetheme{Warsaw}\usecolortheme{default}\useoutertheme{infolines}\setbeamercovered{transparent}

\begin{document}

\maketitle

\begin{frame}

\frametitle{Outline}

\label{sec-1}

\begin{itemize}

\item Defining symbolic expressions in sage.

\item Using built-in constants and functions.

\item Performing Integration, differentiation using sage.

\item Defining matrices.

\item Defining Symbolic functions.

\item Simplifying and solving symbolic expressions and functions.

\end{itemize}

\end{frame}

\begin{frame}

\frametitle{Question 1}

\label{sec-2}

\begin{itemize}

\item Define the following expression as symbolic

    expression in sage.

\begin{itemize}

\item x$^2$+y$^2$

\item y$^2$-4ax

\end{itemize}

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Solution 1}

\label{sec-3}

\lstset{language=Python}

\begin{lstlisting}

var('x,y')

x^2+y^2

var('a,x,y')

y^2-4*a*x

\end{lstlisting}

\end{frame}

\begin{frame}

\frametitle{Question 2}

\label{sec-4}

\begin{itemize}

\item Find the values of the following constants upto 6 digits  precision

\begin{itemize}

\item pi$^2$

\item euler$_{\mathrm{gamma}}$$^2$

\end{itemize}

\end{itemize}

\begin{itemize}

\item Find the value of the following.

\begin{itemize}

\item sin(pi/4)

\item ln(23)

\end{itemize}

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Solution 2}

\label{sec-5}

\lstset{language=Python}

\begin{lstlisting}

n(pi^2,digits=6)

n(sin(pi/4))

n(log(23,e))

\end{lstlisting}

\end{frame}

\begin{frame}

\frametitle{Question 3}

\label{sec-6}

\begin{itemize}

\item Define the piecewise function. 

   f(x)=3x+2 

   when x is in the closed interval 0 to 4.

   f(x)=4x$^2$

   between 4 to 6.

\item Sum  of 1/(n$^2$-1) where n ranges from 1 to infinity.

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Solution 3}

\label{sec-7}

\lstset{language=Python}

\begin{lstlisting}

var('x') 

h(x)=3*x+2 

g(x)= 4*x^2

f=Piecewise([[(0,4),h(x)],[(4,6),g(x)]],x)

f

\end{lstlisting}

\lstset{language=Python}

\begin{lstlisting}

var('n')

f=1/(n^2-1) 

sum(f(n), n, 1, oo)

\end{lstlisting}

\end{frame}

\begin{frame}

\frametitle{Question 4}

\label{sec-8}

\begin{itemize}

\item Differentiate the following.

\begin{itemize}

\item sin(x$^3$)+log(3x), to the second order

\item x$^5$*log(x$^7$), to the fourth order

\end{itemize}

\item Integrate the given expression

\begin{itemize}

\item x*sin(x$^2$)

\end{itemize}

\item Find x

\begin{itemize}

\item cos(x$^2$)-log(x)=0

\item Does the equation have a root between 1,2.

\end{itemize}

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Solution 4}

\label{sec-9}

\lstset{language=Python}

\begin{lstlisting}

var('x')

f(x)= x^5*log(x^7) 

diff(f(x),x,5)

var('x')

integral(x*sin(x^2),x) 

var('x')

f=cos(x^2)-log(x)

find_root(f(x)==0,1,2)

\end{lstlisting}

\end{frame}

\begin{frame}

\frametitle{Question 5}

\label{sec-10}

\begin{itemize}

\item Find the determinant and inverse of :

      A=[[x,0,1][y,1,0][z,0,y]]

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Solution 5}

\label{sec-11}

\lstset{language=Python}

\begin{lstlisting}

var('x,y,z')

A=matrix([[x,0,1],[y,1,0],[z,0,y]])

A.det()

A.inverse()

\end{lstlisting}

\end{frame}

\begin{frame}

\frametitle{Summary}

\label{sec-12}

\begin{itemize}

\item We learnt about defining symbolic expression and functions.

\item Using built-in constants and functions.

\item Using <Tab> to see the documentation of a function.

\item Simple calculus operations .

\item Substituting values in expression using substitute function.

\item Creating symbolic matrices and performing operation on them .

\end{itemize}

\end{frame}

\begin{frame}

\frametitle{Thank you!}

\label{sec-13}

  \begin{block}{}

  \begin{center}

  This spoken tutorial has been produced by the

  \textcolor{blue}{FOSSEE} team, which is funded by the 

  \end{center}

  \begin{center}

    \textcolor{blue}{National Mission on Education through \\

      Information \& Communication Technology \\ 

      MHRD, Govt. of India}.

  \end{center}  

  \end{block}

\end{frame}

\end{document}