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#+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)
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#+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:
#+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
*
#+begin_latex
\begin{center}
\vspace{12pt}
\textcolor{blue}{\huge Getting started with Symbolics}
\end{center}
\vspace{18pt}
\begin{center}
\vspace{10pt}
\includegraphics[scale=0.95]{../images/fossee-logo.png}\\
\vspace{5pt}
\scriptsize Developed by FOSSEE Team, IIT-Bombay. \\
\scriptsize Funded by National Mission on Education through ICT\\
\scriptsize MHRD,Govt. of India\\
\includegraphics[scale=0.30]{../images/iitb-logo.png}\\
\end{center}
#+end_latex
* Objectives
At the end of this tutorial, you will be able to,
- Define symbolic expressions in sage.
- Use built-in constants and functions.
- Perform Integration, differentiation using sage.
- Define matrices.
- Define Symbolic functions.
- Simplify0and solve symbolic expressions and functions.
* Pre-requisite
Spoken tutorial on -
- Getting started with Sage Notebook.
* Exercise 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
* Exercise 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
* Exercise 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
* Exercise 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
* Exercise 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
In this tutorial, we have learnt to,
- Define symbolic expression and functions using the method ``var``.
- Use built-in constants like pi,e,oo and functions like
sum,sin,cos,log,exp and many more.
- Use <Tab> to see the documentation of a function.
- Do simple calculus using functions like --
- diff()--to find a differential of a function
- integral()--to integrate an expression
- simplify--to simplify complicated expression.
- Substitute values in expressions using ``substitute`` function.
- Create symbolic matrices and perform operations on them like --
- det()--to find out the determinant of a matrix
- inverse()--to find out the inverse of a matrix.
* Evaluation
1. How do you define a name 'y' as a symbol?
2. Get the value of pi upto precision 5 digits using sage?
3. Find third order differential function of
f(x)=sin(x^2)+exp(x^3)
* Solutions
1. var('y')
2. n(pi,5)
3. diff(f(x),x,3)
*
#+begin_latex
\begin{block}{}
\begin{center}
\textcolor{blue}{\Large THANK YOU!}
\end{center}
\end{block}
\begin{block}{}
\begin{center}
For more Information, visit our website\\
\url{http://fossee.in/}
\end{center}
\end{block}
#+end_latex
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