Basic Sage Facts
1. Introduction
Sage on open-source alternative to commercial computer algebra systems (CAS) such as Maple, Mathematica, and Matlab. Sage has many capabilities similar to the commercial versions. They include graphing,
solving equations analytically or numerically, and symbolic manipulations such as differentiation and antidifferentiation of functions. The purpose of this handout is to (re)introduce you to Sage and some of its
general features.
2. Workbooks, Folders, & Worksheets
A Sage session begins by using a web browser, such as Firefox or Google, to open the Sage login page
with url
sage.luther.edu
Once logged in, your Sage “workbook” will open. The Sage workbook and has three folders, Active,
Archived, and Trash. A worksheet is made up of a collection of cells. The Active folder’s listing
of worksheets is what you will see upon opening the workbook. If you want to work between multiple
worksheets, you need to open multiple Sage workbooks.
3. Cell Manipulations
A cell within a Sage worksheet is where a command, or sequence of commands, may be carried out. A cell
is represented as a rectangular region in the worksheet, outlined by a thin blue boundary. Activating the
cell
plot(x∧2-2*x-3,(x,-1,3))
generates a plot of the function y = x2 + 1 on the x-interval [−1, 3].
A cell can be added between existed cells by clicking the blue line when it appears as the pointer is dragged
vertically between cells. A new cell may be designated a text/HTML cell by “shift-clicking” the blue
line between cells. A cell is split using “control-;”. Two adjacent cells are joined by “control-backspace”
in the second cell of the sequence. A cell is deleted by deleting all of the cell’s content and then hitting
“backspace.”
4. On-Line Help
Sage has an on-line help system that is invoked by typing the command name followed by a question mark
in a cell. For example
plot?
5. Symbols and Conventions
The following table summarizes some basic Sage symbols and conventions.
Symbol
{
}
[
]
’
’
f(x)
Description
set delimiter (an unordered list)
list delimiter (lists are ordered)
string delimiter
function definition
Example
{a, b, c }
[ a, b, c ]
text(’txt’,(x,y))
f(x)=x^2 + 2*x +1
Table 1: Basic Symbols and Operations
6. Operations, Functions, and Constants
The table below provides information on basic mathematical operations, functions, and constants.
Symbol
+, -, *, /, ∧
sin, cos, tan,
cot, sec, csc
arcsin, arctan, ...
sinh, tanh, ...
exp
ln
log base b
abs
sqrt
factorial
=, <>, <=,
<, >, >=
pi,i,e
oo
Description
add, subtract, multiply, divide, power
trigonometric functions
Example
3*x∧(-4) + x/7
sin(x)*cos(x+1)
inverse trigonometric functions
hyperbolic functions
natural exponential function
natural logarithm
logarithm base b
absolute value
square root
factorial
equal, not equal, less than or equal,
less than,
√ greater than, greater than or equal
π, i = −1, e
∞
tanh(sin(1))
tanh(sin(1))
exp(1/x)
ln(x + 1)
log(5,b)
abs(x+2)
sqrt(1-x)
factorial(6)
x <>7
x >= 9
exp(Pi*I)
limit(f(x), x=oo)
Table 2: Basic Operations, Functions, and Constants
7. Frequently Used Commands
Command
var
plot
parametric plot
Description
define a Sage variable
draw a 2-dimensional plot
2-dimensional parametric plot
implicit plot
find root
limit
diff
draw a 2-dimensional plot of
an implicitly defined curve
draw a 3-dimensional plot
draw a 3-dimensional plot of
an implicitly defined surface
solve a single equation
solve a system of equations
find the root numerically
calculate a limit
find the derivative
integral
integral
find the indefinite integral
find the definite integral
numerical integral
desolve
numerical approximation
analytic solution to an ODE
general solution
particular solution
plot3d
implicit plot3d
solve
Example
x=var(’x’)
plot(sin(x),(x,0,pi)
parametric_plot((cos(t),sin(t)),
(t,0,2*pi))
implicit_plot(x∧2+y∧2==4,(x,-3,3),
(y,-3,3))
plot3d(x*cos(y),(x,0,2),(y,-pi,pi))
implicit_plot3d(x∧2+y∧2+z∧2==8,
(x,-3,3),(y,-3,3),(z,-3,3))
solve(x + 2 == 5, x)
solve([2*x + 3*y == 1,x - 4*y == 2],x,y)
find_root(x + 2 - 5,0,5)
limit(sin(1/x),x=oo)
diff(x*sin(y),x)
NOTE: diff=differentiate=derivative
integral(x*sin(x),x)
integral(x + 1,(x,0,1))
NOTE: integral=integrate
numerical_integral(exp(x^2),0,1)
y=function(’y’,t)
desolve(diff(y,t)==y, y)
desolve(diff(y,t)==y, y, ics=[0,2])
Table 3: Frequently Used Commands