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| -rw-r--r-- | getting-started-with-symbolics/script.rst | 340 | ||||
| -rw-r--r-- | getting-started-with-symbolics/slides.org | 160 | ||||
| -rw-r--r-- | getting-started-with-symbolics/slides.tex | 252 |
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diff --git a/getting-started-with-symbolics/quickref.tex b/getting-started-with-symbolics/quickref.tex deleted file mode 100644 index b26d168..0000000 --- a/getting-started-with-symbolics/quickref.tex +++ /dev/null @@ -1,8 +0,0 @@ -Creating a linear array:\\ -{\ex \lstinline| x = linspace(0, 2*pi, 50)|} - -Plotting two variables:\\ -{\ex \lstinline| plot(x, sin(x))|} - -Plotting two lists of equal length x, y:\\ -{\ex \lstinline| plot(x, y)|} diff --git a/getting-started-with-symbolics/script.rst b/getting-started-with-symbolics/script.rst deleted file mode 100644 index 4f714d4..0000000 --- a/getting-started-with-symbolics/script.rst +++ /dev/null @@ -1,340 +0,0 @@ -.. Objectives -.. ---------- - -.. By the end of this tutorial, you will be able to - -.. 1. Defining symbolic expressions in sage. -.. # Using built-in constants and functions. -.. # Performing Integration, differentiation using sage. -.. # Defining matrices. -.. # Defining Symbolic functions. -.. # Simplifying and solving symbolic expressions and functions. - - -.. Prerequisites -.. ------------- - -.. 1. getting started with sage notebook - - -.. Author : Amit - Internal Reviewer : - External Reviewer : - Language Reviewer : Bhanukiran - Checklist OK? : <, if OK> [2010-10-05] - -Symbolics with Sage -------------------- - -Hello friends and welcome to the tutorial on Symbolics with Sage. - -{{{ Show welcome slide }}} - -During the course of the tutorial we will learn - -{{{ Show outline slide }}} - -* Defining symbolic expressions in Sage. -* Using built-in constants and functions. -* Performing Integration, differentiation using Sage. -* Defining matrices. -* Defining symbolic functions. -* Simplifying and solving symbolic expressions and functions. - -In addtion to a lot of other things, Sage can do Symbolic Math and we shall -start with defining symbolic expressions in Sage. - -Have your Sage notebook opened. If not, pause the video and -start you Sage notebook right now. - -On the sage notebook type:: - - sin(y) - -It raises a name error saying that ``y`` is not defined. We need to -declare ``y`` as a symbol. We do it using the ``var`` function. -:: - - var('y') - -Now if you type:: - - sin(y) - -Sage simply returns the expression. - -Sage treats ``sin(y)`` as a symbolic expression. We can use this to do -symbolic math using Sage's built-in constants and expressions. - -Let us try out a few examples. :: - - var('x,alpha,y,beta') - x^2/alpha^2+y^2/beta^2 - -We have defined 4 variables, ``x``, ``y``, ``alpha`` and ``beta`` and -have defined a symbolic expression using them. - -Here is an expression in ``theta`` :: - - var('theta') - sin(theta)*sin(theta)+cos(theta)*cos(theta) - -Now that you know how to define symbolic expressions in Sage, here is -an exercise. - -{{ show slide showing question 1 }} - -%% %% Define following expressions as symbolic expressions in Sage. - - 1. x^2+y^2 - #. y^2-4ax - -Please, pause the video here. Do the exercise and then continue. - -The solution is on your screen. - -{{ show slide showing solution 1 }} - -Sage also provides built-in constants which are commonly used in -mathematics, for instance pi, e, infinity. The function ``n`` gives -the numerical values of all these constants. -:: - n(pi) - n(e) - n(oo) - -If you look into the documentation of function ``n`` by doing - -:: - n(<Tab> - -You will see what all arguments it takes and what it returns. It will -be very helpful if you look at the documentation of all functions -introduced in the course of this script. - -Also we can define the number of digits we wish to have in the -constants. For this we have to pass an argument -- digits. Type - -:: - - n(pi, digits = 10) - -Apart from the constants Sage also has a lot of built-in functions -like ``sin``, ``cos``, ``log``, ``factorial``, ``gamma``, ``exp``, -``arcsin`` etc ... - -Lets try some of them out on the Sage notebook. -:: - - sin(pi/2) - - arctan(oo) - - log(e,e) - -Following are exercises that you must do. - -{{ show slide showing question 2 }} - -%% %% Find the values of the following constants upto 6 digits - precision - - 1. pi^2 - #. euler_gamma^2 - - -%% %% Find the value of the following. - - 1. sin(pi/4) - #. ln(23) - -Please, pause the video here. Do the exercises and then continue. - -The solutions are on your screen - -{{ show slide showing solution 2 }} - -Given that we have defined variables like x, y etc., we can define an -arbitrary function with desired name in the following way.:: - - var('x') - function('f',x) - -Here f is the name of the function and x is the independent variable . -Now we can define f(x) to be :: - - f(x) = x/2 + sin(x) - -Evaluating this function f for the value x=pi returns pi/2.:: - - f(pi) - -We can also define functions that are not continuous but defined -piecewise. Let us define a function which is a parabola between 0 -to 1 and a constant from 1 to 2 . Type the following -:: - - - var('x') - h(x)=x^2 - g(x)=1 - - f=Piecewise([[(0,1),h(x)],[(1,2),g(x)]],x) - f - -We can also define functions convergent series and other series. - -We first define a function f(n) in the way discussed above.:: - - var('n') - function('f', n) - - -To sum the function for a range of discrete values of n, we use the -sage function sum. - -For a convergent series , f(n)=1/n^2 we can say :: - - var('n') - function('f', n) - f(n) = 1/n^2 - sum(f(n), n, 1, oo) - - -Lets us now try another series :: - - - f(n) = (-1)^(n-1)*1/(2*n - 1) - sum(f(n), n, 1, oo) - -This series converges to pi/4. - -Following are exercises that you must do. - -{{ show slide showing question 3 }} - -%% %% 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. - -%% %% Sum of 1/(n^2-1) where n ranges from 1 to infinity. - -Please, pause the video here. Do the exercise(s) and then continue. - -{{ show slide showing solution 3 }} - -Moving on let us see how to perform simple calculus operations using Sage - -For example lets try an expression first :: - - diff(x**2+sin(x),x) - -The diff function differentiates an expression or a function. It's -first argument is expression or function and second argument is the -independent variable. - -We have already tried an expression now lets try a function :: - - f=exp(x^2)+arcsin(x) - diff(f(x),x) - -To get a higher order differential we need to add an extra third argument -for order :: - - diff(f(x),x,3) - -in this case it is 3. - -Just like differentiation of expression you can also integrate them :: - - x = var('x') - s = integral(1/(1 + (tan(x))**2),x) - s - -Many a times we need to find factors of an expression, we can use the -"factor" function - -:: - - y = (x^100 - x^70)*(cos(x)^2 + cos(x)^2*tan(x)^2) - f = factor(y) - -One can simplify complicated expression :: - - f.simplify_full() - -This simplifies the expression fully. We can also do simplification of -just the algebraic part and the trigonometric part :: - - f.simplify_exp() - f.simplify_trig() - -One can also find roots of an equation by using ``find_root`` function:: - - phi = var('phi') - find_root(cos(phi)==sin(phi),0,pi/2) - -Let's substitute this solution into the equation and see we were -correct :: - - var('phi') - f(phi)=cos(phi)-sin(phi) - root=find_root(f(phi)==0,0,pi/2) - f.substitute(phi=root) - -as we can see when we substitute the value the answer is almost = 0 showing -the solution we got was correct. - -Following are a few exercises that you must do. - -%% %% Differentiate the following. - - 1. sin(x^3)+log(3x) , degree=2 - #. x^5*log(x^7) , degree=4 - -%% %% Integrate the given expression - - sin(x^2)+exp(x^3) - -%% %% Find x - cos(x^2)-log(x)=0 - Does the equation have a root between 1,2. - -Please, pause the video here. Do the exercises and then continue. - - -Lets us now try some matrix algebra symbolically :: - - var('a,b,c,d') - A=matrix([[a,1,0],[0,b,0],[0,c,d]]) - A - -Now lets do some of the matrix operations on this matrix -:: - A.det() - A.inverse() - - -Following is an (are) exercise(s) that you must do. - -%% %% Find the determinant and inverse of : - - A=[[x,0,1][y,1,0][z,0,y]] - -Please, pause the video here. Do the exercise(s) and then continue. - - -{{{ Show the summary slide }}} - -That brings us to the end of this tutorial. In this tutorial we learnt -how to - -* define symbolic expression and functions -* use built-in constants and functions -* use <Tab> to see the documentation of a function -* do simple calculus -* substitute values in expressions using ``substitute`` function -* create symbolic matrices and perform operations on them - diff --git a/getting-started-with-symbolics/slides.org b/getting-started-with-symbolics/slides.org deleted file mode 100644 index 5d9391e..0000000 --- a/getting-started-with-symbolics/slides.org +++ /dev/null @@ -1,160 +0,0 @@ -#+LaTeX_CLASS: beamer -#+LaTeX_CLASS_OPTIONS: [presentation] -#+BEAMER_FRAME_LEVEL: 1 - -#+BEAMER_HEADER_EXTRA: \usetheme{Warsaw}\usecolortheme{default}\useoutertheme{infolines}\setbeamercovered{transparent} -#+COLUMNS: %45ITEM %10BEAMER_env(Env) %10BEAMER_envargs(Env Args) %4BEAMER_col(Col) %8BEAMER_extra(Extra) -#+PROPERTY: BEAMER_col_ALL 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 :ETC - -#+LaTeX_CLASS: beamer -#+LaTeX_CLASS_OPTIONS: [presentation] - -#+LaTeX_HEADER: \usepackage[english]{babel} \usepackage{ae,aecompl} -#+LaTeX_HEADER: \usepackage{mathpazo,courier,euler} \usepackage[scaled=.95]{helvet} - -#+LaTeX_HEADER: \usepackage{listings} - -#+LaTeX_HEADER:\lstset{language=Python, basicstyle=\ttfamily\bfseries, -#+LaTeX_HEADER: commentstyle=\color{red}\itshape, stringstyle=\color{darkgreen}, -#+LaTeX_HEADER: showstringspaces=false, keywordstyle=\color{blue}\bfseries} - -#+TITLE: Getting started with symbolics -#+AUTHOR: FOSSEE -#+EMAIL: -#+DATE: - -#+DESCRIPTION: -#+KEYWORDS: -#+LANGUAGE: en -#+OPTIONS: H:3 num:nil toc:nil \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t -#+OPTIONS: TeX:t LaTeX:nil skip:nil d:nil todo:nil pri:nil tags:not-in-toc - -* Outline - - Defining symbolic expressions in sage. - - Using built-in constants and functions. - - Performing Integration, differentiation using sage. - - Defining matrices. - - Defining Symbolic functions. - - Simplifying and solving symbolic expressions and functions. - -* Question 1 - - Define the following expression as symbolic - expression in sage. - - - x^2+y^2 - - y^2-4ax - -* Solution 1 -#+begin_src python - var('x,y') - x^2+y^2 - - var('a,x,y') - y^2-4*a*x -#+end_src python -* Question 2 - - Find the values of the following constants upto 6 digits precision - - - pi^2 - - euler_gamma^2 - - - - Find the value of the following. - - - sin(pi/4) - - ln(23) - -* Solution 2 -#+begin_src python - n(pi^2,digits=6) - n(sin(pi/4)) - n(log(23,e)) -#+end_src python -* Question 3 - - 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. - - - Sum of 1/(n^2-1) where n ranges from 1 to infinity. - -* Solution 3 -#+begin_src python - 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_src python - -#+begin_src python - var('n') - f=1/(n^2-1) - sum(f(n), n, 1, oo) -#+end_src python - -* Question 4 - - Differentiate the following. - - - sin(x^3)+log(3x), to the second order - - x^5*log(x^7), to the fourth order - - - Integrate the given expression - - - x*sin(x^2) - - - Find x - - cos(x^2)-log(x)=0 - - Does the equation have a root between 1,2. - -* Solution 4 -#+begin_src python - 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_src - -* Question 5 - - Find the determinant and inverse of : - - A=[[x,0,1][y,1,0][z,0,y]] - -* Solution 5 -#+begin_src python - var('x,y,z') - A=matrix([[x,0,1],[y,1,0],[z,0,y]]) - A.det() - A.inverse() -#+end_src -* Summary - - We learnt about defining symbolic expression and functions. - - Using built-in constants and functions. - - Using <Tab> to see the documentation of a function. - - Simple calculus operations . - - Substituting values in expression using substitute function. - - Creating symbolic matrices and performing operation on them . -* Thank you! -#+begin_latex - \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_latex - - - diff --git a/getting-started-with-symbolics/slides.tex b/getting-started-with-symbolics/slides.tex deleted file mode 100644 index 51e8997..0000000 --- a/getting-started-with-symbolics/slides.tex +++ /dev/null @@ -1,252 +0,0 @@ -% Created 2010-11-11 Thu 02:03 -\documentclass[presentation]{beamer} -\usepackage[latin1]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{fixltx2e} -\usepackage{graphicx} -\usepackage{longtable} -\usepackage{float} -\usepackage{wrapfig} -\usepackage{soul} -\usepackage{textcomp} -\usepackage{marvosym} -\usepackage{wasysym} -\usepackage{latexsym} -\usepackage{amssymb} -\usepackage{hyperref} -\tolerance=1000 -\usepackage[english]{babel} \usepackage{ae,aecompl} -\usepackage{mathpazo,courier,euler} \usepackage[scaled=.95]{helvet} -\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} |