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+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%Tutorial slides on Python.
+%
+% Author: FOSSEE 
+% Copyright (c) 2009, FOSSEE, IIT Bombay
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\documentclass[14pt,compress]{beamer}
+%\documentclass[draft]{beamer}
+%\documentclass[compress,handout]{beamer}
+%\usepackage{pgfpages} 
+%\pgfpagesuselayout{2 on 1}[a4paper,border shrink=5mm]
+
+% Modified from: generic-ornate-15min-45min.de.tex
+\mode<presentation>
+{
+  \usetheme{Warsaw}
+  \useoutertheme{split}
+  \setbeamercovered{transparent}
+}
+
+\usepackage[english]{babel}
+\usepackage[latin1]{inputenc}
+%\usepackage{times}
+\usepackage[T1]{fontenc}
+\usepackage{amsmath}
+
+% Taken from Fernando's slides.
+\usepackage{ae,aecompl}
+\usepackage{mathpazo,courier,euler}
+\usepackage[scaled=.95]{helvet}
+
+\definecolor{darkgreen}{rgb}{0,0.5,0}
+
+\usepackage{listings}
+\lstset{language=Python,
+    basicstyle=\ttfamily\bfseries,
+    commentstyle=\color{red}\itshape,
+  stringstyle=\color{darkgreen},
+  showstringspaces=false,
+  keywordstyle=\color{blue}\bfseries}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Macros
+\setbeamercolor{emphbar}{bg=blue!20, fg=black}
+\newcommand{\emphbar}[1]
+{\begin{beamercolorbox}[rounded=true]{emphbar} 
+      {#1}
+ \end{beamercolorbox}
+}
+\newcounter{time}
+\setcounter{time}{0}
+\newcommand{\inctime}[1]{\addtocounter{time}{#1}{\tiny \thetime\ m}}
+
+\newcommand{\typ}[1]{\lstinline{#1}}
+
+\newcommand{\kwrd}[1]{ \texttt{\textbf{\color{blue}{#1}}}  }
+
+%%% This is from Fernando's setup.
+% \usepackage{color}
+% \definecolor{orange}{cmyk}{0,0.4,0.8,0.2}
+% % Use and configure listings package for nicely formatted code
+% \usepackage{listings}
+% \lstset{
+%    language=Python,
+%    basicstyle=\small\ttfamily,
+%    commentstyle=\ttfamily\color{blue},
+%    stringstyle=\ttfamily\color{orange},
+%    showstringspaces=false,
+%    breaklines=true,
+%    postbreak = \space\dots
+% }
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Title page
+\title[Basic Python]{Interpolation, Differentiation and Integration}
+
+\author[FOSSEE] {FOSSEE}
+
+\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}
+\date[] {31, October 2009\\Day 1, Session 4}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%\pgfdeclareimage[height=0.75cm]{iitmlogo}{iitmlogo}
+%\logo{\pgfuseimage{iitmlogo}}
+
+
+%% Delete this, if you do not want the table of contents to pop up at
+%% the beginning of each subsection:
+\AtBeginSubsection[]
+{
+  \begin{frame}<beamer>
+    \frametitle{Outline}
+    \tableofcontents[currentsection,currentsubsection]
+  \end{frame}
+}
+
+%%\AtBeginSection[]
+%%{
+  %%\begin{frame}<beamer>
+%%    \frametitle{Outline}
+  %%  \tableofcontents[currentsection,currentsubsection]
+  %%\end{frame}
+%%}
+
+% If you wish to uncover everything in a step-wise fashion, uncomment
+% the following command: 
+%\beamerdefaultoverlayspecification{<+->}
+
+%\includeonlyframes{current,current1,current2,current3,current4,current5,current6}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% DOCUMENT STARTS
+\begin{document}
+
+\begin{frame}
+  \titlepage
+\end{frame}
+
+\begin{frame}
+  \frametitle{Outline}
+  \tableofcontents
+%  \pausesections
+\end{frame}
+
+\section{Integration}
+
+\subsection{Quadrature}
+
+\begin{frame}[fragile]
+\frametitle{Quadrature}
+\begin{itemize}
+\item We wish to find area under a curve
+\item Area under $(sin(x) + x^2)$ in $(0,1)$
+\item scipy has functions to do that
+\end{itemize}
+\small{\typ{In []: from scipy.integrate import quad}}
+\begin{itemize}
+\item Inputs - function to integrate, limits
+\end{itemize}
+\begin{lstlisting}
+In []: x = 0
+In []: quad(sin(x)+x**2, 0, 1)
+\end{lstlisting}
+\alert{\typ{error:}}
+\typ{First argument must be a callable function.}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Functions - Definition}
+\begin{lstlisting}
+In []: def f(x):
+           return sin(x)+x**2
+In []: quad(f, 0, 1)
+\end{lstlisting}
+\begin{itemize}
+\item \typ{def}
+\item arguments
+\item \typ{return}
+\end{itemize}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Functions - Calling them}
+\begin{lstlisting}
+In [15]: f()
+---------------------------------------
+\end{lstlisting}
+\alert{\typ{TypeError:}}\typ{f() takes exactly 1 argument}
+\typ{(0 given)}
+\begin{lstlisting}
+In []: f(0)
+Out[]: 0.0
+In []: f(1)
+Out[]: 1.8414709848078965
+\end{lstlisting}
+\end{frame}
+
+
+\begin{frame}[fragile]
+\frametitle{Functions - Default Arguments}
+\begin{lstlisting}
+In []: def f(x=1):
+           return sin(x)+x**2
+In []: f(10)
+Out[]: 99.455978889110625
+In []: f(1)
+Out[]: 1.8414709848078965
+In []: f()
+Out[]: 1.8414709848078965
+\end{lstlisting}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Functions - Keyword Arguments}
+\begin{lstlisting}
+In []: def f(x=1, y=pi):
+           return sin(y)+x**2
+In []: f()
+Out[]: 1.0000000000000002
+In []: f(2)
+Out[]: 4.0
+In []: f(y=2)
+Out[]: 1.9092974268256817
+In []: f(y=pi/2,x=0)
+Out[]: 1.0
+\end{lstlisting}
+\end{frame}
+
+\begin{frame}[fragile]
+  \frametitle{More on functions}
+  \begin{itemize}
+  \item Scope of variables in the function is local
+  \item Mutable items are \alert{passed by reference}
+  \item First line after definition may be a documentation string
+    (\alert{recommended!})
+  \item Function definition and execution defines a name bound to the
+    function
+  \item You \emph{can} assign a variable to a function!
+  \end{itemize}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Quadrature \ldots}
+\begin{lstlisting}
+In []: quad(f, 0, 1)
+\end{lstlisting}
+Returns the integral and an estimate of the absolute error in the result.
+\begin{itemize}
+\item Use \typ{dblquad} for Double integrals
+\item Use \typ{tplquad} for Triple integrals
+\end{itemize}
+\end{frame}
+
+\subsection{ODEs}
+
+\begin{frame}[fragile]
+\frametitle{ODE Integration}
+We shall use the simple ODE of a simple pendulum. 
+\begin{equation*}
+\ddot{\theta} = -\frac{g}{L}sin(\theta)
+\end{equation*}
+\begin{itemize}
+\item This equation can be written as a system of two first order ODEs
+\end{itemize}
+\begin{align}
+\dot{\theta} &= \omega \\
+\dot{\omega} &= -\frac{g}{L}sin(\theta) \\
+ \text{At}\ t &= 0 : \nonumber \\
+ \theta = \theta_0\quad & \&\quad  \omega = 0 \nonumber
+\end{align}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Solving ODEs using SciPy}
+\begin{itemize}
+\item We use the \typ{odeint} function from scipy to do the integration
+\item Define a function as below
+\end{itemize}
+\begin{lstlisting}
+In []: def pend_int(unknown, t, p):
+  ....     theta, omega = unknown
+  ....     g, L = p
+  ....     f=[omega, -(g/L)*sin(theta)]
+  ....     return f
+  ....
+\end{lstlisting}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Solving ODEs using SciPy \ldots}
+\begin{itemize}
+\item \typ{t} is the time variable \\ 
+\item \typ{p} has the constants \\
+\item \typ{initial} has the initial values
+\end{itemize}
+\begin{lstlisting}
+In []: t = linspace(0, 10, 101)
+In []: p=(-9.81, 0.2)
+In []: initial = [10*2*pi/360, 0]
+\end{lstlisting}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Solving ODEs using SciPy \ldots}
+
+\small{\typ{In []: from scipy.integrate import odeint}}
+\begin{lstlisting}
+In []: pend_sol = odeint(pend_int, 
+                         initial,t, 
+                         args=(p,))
+\end{lstlisting}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Things we have learned}
+  \begin{itemize}
+  \item Functions
+    \begin{itemize}
+    \item Definition
+    \item Calling
+    \item Default Arguments
+    \item Keyword Arguments
+    \end{itemize}
+  \item Quadrature
+  \end{itemize}
+\end{frame}
+
+\end{document}
+




