INTRODUCTION(Click here to return main page.)


1. Introduction and Use of Sage

    

Mathematical tools have long held an important place in the classroom. With the innovation of information and communication technologies(ICT), many tools have appeared and been adapted for educational purposes. Sage is popular mathematical software which was released in 2005. This software has efficient features which can be utilized through the Internet and can handle most mathematical problems, including linear algebra, abstract algebra, combinatorics, number theory, symbolic computation, numerical mathematics and calculus. In this book, we will introduce this powerful software and discuss how it can be used in classes.

Sage is a mathematical Computer Algebra System (CAS) that can be easily used online. The Sage notebook was released in April 2008 at University of Washington, USA. It is free and has powerful capabilities that can be compared with expensive commercial software such as Mathematica, Maple or Matlab. However, it does not require separate installation of the program. It is more like Web-Mathematica, but has some better features. When you connect with any web browser, you can solve almost all calculus problems in the book by using resources that we are offering. You can easily find pre-existing commands to modify for your own problems. The Sage notebook is a platform that allows you to interface with many open-source and commercial programs (if you have a license).

                                     Figure 1. Sage notebook can interface to many programs


Online Sage Servers:

    http://www.sagenb.org   (Sign in)

    http://math1.skku.ac.kr  (ID: skku    Password: math)

      http://math2.skku.ac.kr  (Make your own accounts)

      http://math3.skku.ac.kr  (Mobile Server, No need for login)

      http://sage.skku.edu   (SKKU Sage Cell Server, Mobile, No need for login, Recommended

      https://salv.us  (University of Washington Sage Server)


                                        

                                        Figure 2. Sage Community (Left) http://www.sagemath.org,

                                                       Sage Korean Version (Right) http://math1.skku.ac.kr


Use Firefox or Chrome to connect to one of the above servers. Then register your ID and password to get started.

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2. Development of the Korean Version of Sage

      

There are Quick Reference (simple manual) for Sage in English and Korean. Those Quick References can be downloaded from the web site address below.


http://wiki.sagemath.org/quickref

http://matrix.skku.ac.kr/2010-Album/Sage-QReference-SKKU.pdf


Sage Quick Reference: Basic Math and Calculus

Peter Jipsen, version 1.1 (Basic Math) Latest Version at http://wiki.sagemath.org/quickref   

William Stein (Calculus) Sage Version 3.4  http://wiki.sagemath.org/quickref   

GNU Free Document License, extend for your own use 


Translated to Korean language by Sang-Gu Lee and Jae Hwa Lee (Sungkyunkwan University)

Korean Version at http://matrix.skku.ac.kr/2010-Album/Sage-QReference-SKKU.pdf  

                   

Notebook (and Command line)

Evaluate cell: <shift-enter>

com <tab> tries to complete command

command?<tab> shows documentation

command??<tab> shows source

a.<tab>i shows all methods for object a (more: dir(a))

search_doc('string or regexp') shows links to docs

search_src('string or regexp') shows links to source

lprint() toggle LaTeX output mode

version() print version of Sage

Insert cell: click on blue line between cells

Delete cell: delete content then backspace

                                                                                                 

Numerical types

Sage has many different number systems. ZZ represents set of integers, QQ represents set of rational numbers, RDF represents floating-point real numbers, CDF represents floating-point complex numbers.

Example for Rationals:

sage: a=1/2; b=QQ(4/2); print a+b; b.parent()

output: 5/2

Rational Field


Builtin constants and functions:

Constants: =pi   =e   =I=i   =oo

=oo=infinity    NaN=NaN    log(2)=log2

=golden_ratio   =euler_gamma

...


Builtin functions:

sin  cos  tan  sec  csc  cot  sinh  cosh  tanh  sech  csch  coth  log  ln  exp

=a*b  =a/b  =a^b =sqrt(x) =x^(1/n) =abs(x) =log(x, b)

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

Define functions: e.g.   f(x)=x^2 or f=lambda x: x^2 or def f(x): return x^2

                                 

Operations and equations:

Relations: : f == g, : f != g, : f <= g, : f >= g, : f < g,  : f > g

Solve : solve(f(x)==g(x), x)

Solve : solve([f(x, y)==0, g(x, y)==0], x, y)

Exact roots: (x^3+2*x+1).roots(x)

Real roots: (x^3+2*x+1).roots(x, ring=RR)

Complex roots: (x^3+2*x+1).roots(x, ring=CC)

: var('i'); f(i)=i; sum(f(i), i, 1, 10)

: var('i'); f(i)=i; prod(f(i), i, 1, 10)


Defining symbolic expressions

Create symbolic variables: var('t, u, theta') 


Symbolic functions

Symbolic function (can integrate, differentiate, etc.): f(a, b, theta) = a + b*theta^2

Also, a "formal" function of theta: f = function('f', theta)

Piecewise symbolic functions: Piecewise([[(0, pi/2), sin(1/x)], [(pi/2, pi), x^2+1]])

                                


Python functions

Defining:

def f(a, b, theta=1):

    c = a + b*theta^2

    return c

Inline functions:

    f = lambda a, b, theta = 1: a + b*theta^2

  ...


Factorization

Factored form: (x^3-y^3).factor()

List of (factor, exponent) pairs: (x^3-y^3).factor_list()


Limits

: limit(f(x), x=a)

...


Derivatives

: diff(f(x), x) or f.diff(x)

: diff(f(x, y), x)

   e.g. diff(x*y + sin(x^2) + e^(-x), x)

  

Integrals

: integral(f(x), x) or f.integrate(x)

   e.g. integral(x*cos(x^2), x)

: integral(f(x), x, a, b)

   e.g. integral(x*cos(x^2), x, 0, sqrt(pi))

: numerical_integral(f(x), a, b)[0]

   e.g. numerical_integral(x*cos(x^2), 0, 1)[0]

assume(...): use if integration asks a question

   e.g. assume(x>0)


       

Multivariable calculus

Gradient: f.gradient() or f.gradient(vars)

   e.g. (x^2+y^2).gradient([x, y])

Hessian: f.hessian()

   e.g. (x^2+y^2).hessian()

Jacobian matrix: jacobian(f, vars)

   e.g. jacobian(x^2 - 2*x*y, (x, y))

 

2D graphics

Polynomial: plot(x^2, (x, -2, 5))

Functions: plot(f(), (x, , ), options)

Line: line([(), ()], options) : line([(x_1, y_1), (x_2, y_2)])

Polygon: polygon([(), ..., ()], options)

Circle: circle((), , options)

Parametric functions: parametric_plot((f(), g()), (t, , ), options)

Polar functions: polar_plot(f(), (t, , ), options)

Animate: animate(list of graphics objects, options).show(delay=20)

 

3D graphics

Line: line3d([(), ()], options)

Sphere: sphere((), , options)

Tetrahedron: tetrahedron((), size, options)

Functions: plot3d(f(), (x, ), (y, ), options)

   add option plot_points=[]

Parametric functions: parametric_plot3d((f(), g(), h()), (t, ), options) or

   parametric_plot3d((f(), g(), h(), (u, ), (v, ), options)


Linear algebra

: vector([1, 2])

: A=matrix([[1, 2], [3, 4]]); A

: det(matrix([[1, 2], [3, 4]]))

: A*v

: A^-1 or A.inverse()

: A.transpose()

Other methods: A.<tab>  # to see methods that are available with matrix A


  Sage Problem: Exercise 4 in Section 8.2 (Area of a Surface of Revolution)

★ Find the area of the surface when is a positive even integer.

   .

 Use http://matrix.skku.ac.kr/cal-lab/cal-0-a.html  

k,x=var('k, x');

@interact

def _(y = input_box(sin(k*x), label="y="), kappa=slider(0, 100, 1, default=1, label=' k')):

   html('$$AREA=%s$$'%(

 lim(integral(2*pi*sin(k*x)*sqrt(1+diff(sin(k*x), x)), x, 0, pi), k=kappa) ))

   plot(sin(kappa*x), x, 0, pi, color='purple', fill=true).show(aspect_ratio=1, xmin=0, xmax=pi, ymin=-1, ymax=1)

 


 Answer : 0.


  Sage Problem: Exercise 22 in Section 9.3 (Polar Coordinates)

★ Sketch the curve with the given polar equation.

   

 Use http://matrix.skku.ac.kr/cal-lab/cal-0-b.html

theta=var('theta');

polar_plot(sin(4*theta), (0, 2*pi)).show(aspect_ratio=1, xmin=-1, xmax=1, ymin=-1, ymax=1)

  

                                                     


  Sage Problem: Exercise 16 in Section 11.6 (Cylinders and Quadric Surfaces)

★ Sketch the region bounded by the surfaces and for .

 Use http://matrix.skku.ac.kr/cal-lab/cal-0-c.html

var('x, y, z')

z=(x^2+y^2)^(1/2)

plot3d(z, (x, -2, 2), (y, -2, 2), (z, 2, 4))+implicit_plot3d(x^2+y^2==4,(x, -2, 2), (y, -2, 2), (z, 2, 4), opacity=0.5)

                                        


All of the above Sage commands and many more in the published section of each Sage server (e.g. http://math1.skku.ac.kr/pub/ ) can be copied and pasted, so you can modify them. Sungkyunkwan University has made more than 3,000 Sage commands. We hope everybody can take full advantage of Sage in learning and teaching Calculus.


* http://matrix.skku.ac.kr/cal-lab/Math-CAS.htm

Sage-Calculus--Grapher : http://matrix.skku.ac.kr/cal-lab/sage-grapher.html   

Sage- Parametric Equation Grapher : http://matrix.skku.ac.kr/cal-lab/sage-grapher-para.html  

Sage- Polar Equation Grapher : http://matrix.skku.ac.kr/cal-lab/sage-grapher-polar.html  

Sage- Implicit Function Grapher : http://matrix.skku.ac.kr/cal-lab/sage-grapher-imp.html