Tutorial for Symbolics and Plotting#

This Sage document is one of the tutorials developed for the MAA PREP Workshop “Sage: Using Open-Source Mathematics Software with Undergraduates” (funding provided by NSF DUE 0817071). It is licensed under the Creative Commons Attribution-ShareAlike 3.0 license (CC BY-SA).

This tutorial has the following sections:

It assumes that one is familiar with the absolute basics of functions and evaluation in Sage. We provide a (very) brief refresher.

  1. Make sure the syntax below for defining a function and getting a value makes sense.

  2. Then evaluate the cell by clicking the “evaluate” link, or by pressing Shift + Enter (hold down Shift while pressing the Enter key).

sage: f(x)=x^3+1
sage: f(2)
9

>>> from sage.all import *
>>> __tmp__=var("x"); f = symbolic_expression(x**Integer(3)+Integer(1)).function(x)
>>> f(Integer(2))
9

Symbolic Expressions#

In the first tutorial, we defined functions using notation similar to that one would use in (say) a calculus course.

There is a useful variant on this - defining expressions involving variables. This will give us the opportunity to point out several important, and sometimes subtle, things.

In the cell below, we define an expression \(FV\) which is the future value of an investment of $100, compounded continuously. We then substitute in values for \(r\) and \(t\) which calculate the future value for \(t=5\) years and \(r=5\%\) nominal interest.

sage: var('r,t')
(r, t)
sage: FV=100*e^(r*t)

>>> from sage.all import *
>>> var('r,t')
(r, t)
>>> FV=Integer(100)*e**(r*t)

sage: FV(r=.05,t=5)
128.402541668774

>>> from sage.all import *
>>> FV(r=RealNumber('.05'),t=Integer(5))
128.402541668774

The previous cells point out several things to remember when working with symbolic expressions. Some are fairly standard.

  • An asterisk (*) signifies multiplication. This should be how you always do multiplication.

  • Although it is possible to allow implicit multiplication, this can easily lead to ambiguity.

  • We can access the most important constants; for instance, \(e\) stands for the constant \(2.71828...\). Likewise, pi (or \(\pi\)) and \(I\) (think complex numbers) are also defined.

  • Of course, if you redefine \(e\) to be something else, all bets are off!

However, two others may be unfamiliar, especially if you have not used much mathematical software before.

  • You must tell Sage what the variables are before using them in a symbolic expression.

  • We did that above by typing var('r,t').

  • This is automatically done with the \(f(x)\) notation, but without that it is necessary, so that Sage knows one intends t (for instance) is a symbolic variable and not a number or something else.

  • If you then wish to substitute some values into the expression, you must explicitly tell Sage which variables are being assigned which values.

  • For instance, above we used FV(r=.05,t=5) to indicate the precise values of \(r\) and \(t\).

Notice that when we define a function, we don’t need to specify which variable has which value. In the function defined below, we have already specified an order.

sage: FV2(r,t)=100*e^(r*t)
sage: FV2(.05,5)
128.402541668774

>>> from sage.all import *
>>> __tmp__=var("r,t"); FV2 = symbolic_expression(Integer(100)*e**(r*t)).function(r,t)
>>> FV2(RealNumber('.05'),Integer(5))
128.402541668774

In this case it is clear that \(r\) is first and \(t\) is second.

But with FV=100*e^(r*t), there is no particular reason \(r\) or \(t\) should be first.

sage: FV(r=.05,t=5); FV(t=5,r=.05)
128.402541668774
128.402541668774

>>> from sage.all import *
>>> FV(r=RealNumber('.05'),t=Integer(5)); FV(t=Integer(5),r=RealNumber('.05'))
128.402541668774
128.402541668774

Also remember that when we don’t use function notation, we’ll need to define our variables.

One of the great things we can do with expressions is manipulate them. Let’s make a typical expression.

sage: z = (x+1)^3

>>> from sage.all import *
>>> z = (x+Integer(1))**Integer(3)

In the cells below, you’ll notice something new: the character #. In Sage (and in Python ), anything on a single line after the number/pound sign (the octothorp ) is ignored. We say that # is a comment character. We use it below to mention alternative ways to do the same thing.

sage: expand(z) # or z.expand()
x^3 + 3*x^2 + 3*x + 1

>>> from sage.all import *
>>> expand(z) # or z.expand()
x^3 + 3*x^2 + 3*x + 1

sage: y = expand(z)
sage: y.factor() # or factor(y)
(x + 1)^3

>>> from sage.all import *
>>> y = expand(z)
>>> y.factor() # or factor(y)
(x + 1)^3

In the previous cell, we assigned the expression which is the expansion of \(z\) to the variable \(y\) with the first line. After that, anything we want to do to the expansion of \(z\) can be done by doing it to \(y\).

There are more commands like this as well. Notice that \(z\) will no longer be \((x+1)^3\) after this cell is evaluated, since we’ve assigned \(z\) to a (much more complex) expression.

sage: z = ((x - 1)^(3/2) - (x + 1)*sqrt(x - 1))/sqrt((x - 1)*(x + 1))
sage: z.simplify_full()
-2*sqrt(x - 1)/sqrt(x^2 - 1)

>>> from sage.all import *
>>> z = ((x - Integer(1))**(Integer(3)/Integer(2)) - (x + Integer(1))*sqrt(x - Integer(1)))/sqrt((x - Integer(1))*(x + Integer(1)))
>>> z.simplify_full()
-2*sqrt(x - 1)/sqrt(x^2 - 1)

This is a good place for a few reminders of basic help.

  • You can see various methods for simplifying an expression by using tab completion. Put your cursor at the end of the next cell (after the simplify) and press tab to see lots of different methods.

  • Also remember that you can use the question mark (e.g., z.simplify_rational?) to get help about a particular method.

sage: z.simplify
<built-in method simplify of sage.symbolic.expression.Expression object at ...>

>>> from sage.all import *
>>> z.simplify
<built-in method simplify of sage.symbolic.expression.Expression object at ...>

Finally, recall that you can get nicely typeset versions of the output in several ways.

  • One option is to click the ‘Typeset’ button at the top.

  • Another - which does not require scrolling! - is to use the show command.

sage: show(z.simplify_rational())

>>> from sage.all import *
>>> show(z.simplify_rational())
\[-\frac{2 \, \sqrt{x - 1}}{\sqrt{x^{2} - 1}}\]

Another Sage command that is useful in this context is solve.

Here, we solve the simple equation \(x^2=-1\).

sage: solve(x^2==-1,x) # solve x^2==-1 for x
[x == -I, x == I]

>>> from sage.all import *
>>> solve(x**Integer(2)==-Integer(1),x) # solve x^2==-1 for x
[x == -I, x == I]

  • In the solve command, one types an equals sign in the equation as two equal signs.

  • This is because the single equals sign means assignment to a variable, as we’ve done above, so Sage (along with Python) uses the double equals sign for symbolic equality.

  • We also include the variable we’d like to solve for after the comma.

It’s also possible to solve more than one expression simultaneously.

sage: solve([x^2==1,x^3==1],x)
[[x == 1]]

>>> from sage.all import *
>>> solve([x**Integer(2)==Integer(1),x**Integer(3)==Integer(1)],x)
[[x == 1]]

Basic 2D Plotting#

One of the other basic uses of mathematics software is easy plotting. Here, we include a brief introduction to the sorts of plotting which will prepare us to use Sage in calculus. (There will be a separate tutorial for more advanced plotting techniques.)

Recall that we can generate a plot using fairly simple syntax. Here, we define a function of \(x\) and plot it between \(-1\) and \(1\).

sage: f(x)=x^3+1
sage: plot(f,(x,-1,1))
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> __tmp__=var("x"); f = symbolic_expression(x**Integer(3)+Integer(1)).function(x)
>>> plot(f,(x,-Integer(1),Integer(1)))
Graphics object consisting of 1 graphics primitive

We can give the plot a name, so that if we want to do something with the plot later, we don’t have to type out the entire plot command. Remember, this is called assigning the plot to the name/variable.

In the next cell, we give the plot the name \(P\).

sage: P=plot(f,(x,-1,1))

>>> from sage.all import *
>>> P=plot(f,(x,-Integer(1),Integer(1)))

One plot is nice, but one might want to superimpose plots as well. For instance, the tangent line to \(f\) at \(x=0\) is just the line \(y=1\), and we might want to show this together with the plot.

So let’s plot this line in a different color, and with a different style for the line, but over the same interval.

sage: Q=plot(1,(x,-1,1),color="red", linestyle="--")
sage: Q
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> Q=plot(Integer(1),(x,-Integer(1),Integer(1)),color="red", linestyle="--")
>>> Q
Graphics object consisting of 1 graphics primitive

Because we put \(Q\) in a line by itself at the end, it shows. We were able to use just one cell to define \(Q\) and show it, by putting each command in a separate line in the same input cell.

Now to show the plots superimposed on each other, we simply add them.

sage: P+Q
Graphics object consisting of 2 graphics primitives

>>> from sage.all import *
>>> P+Q
Graphics object consisting of 2 graphics primitives

Suppose we wanted to view a detail of this.

  • We could create another plot with different endpoints.

  • Another way is to keep the currently created plots, but to set the viewing window using the show command, as below.

sage: (P+Q).show(xmin=-.1,xmax=.1,ymin=.99,ymax=1.01)

>>> from sage.all import *
>>> (P+Q).show(xmin=-RealNumber('.1'),xmax=RealNumber('.1'),ymin=RealNumber('.99'),ymax=RealNumber('1.01'))

Since the axes no longer cross in the frame of reference, Sage shows a short gap between the horizontal and vertical axes.

There are many options one can pass in for various purposes.

  • Some require quotes around the values.

  • Such as the color option when we made a "red" line.

  • Some do not.

  • Such as the xmin in the previous plot, where the minimum x value was just \(-.1\)

Usually (though not always) quotes are required for option values which are words or strings of characters, and not required for numerical values.

Two of the most useful of these options help in labeling graphs.

  • The axes_labels option labels the axes.

  • As with the word processor, we can use dollar signs (like in LaTeX) to make the labels typeset nicely.

  • Here we need both quotes and brackets for proper syntax, since there are two axes to label and the labels are not actually numbers.

sage: plot(f,(x,-1,1),axes_labels=['$x$','$y$'],legend_label='$f(x)$',show_legend=True)
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> plot(f,(x,-Integer(1),Integer(1)),axes_labels=['$x$','$y$'],legend_label='$f(x)$',show_legend=True)
Graphics object consisting of 1 graphics primitive

  • The legend_label option is especially useful with multiple plots.

  • LaTeX notation works here too.

  • In the graphic above, we needed to explicitly ask to show the label. With multiple graphs this should not be necessary.

sage: P1 = plot(f,(x,-1,1),axes_labels=['$x$','$y$'],legend_label='$f(x)$')
sage: P2 = plot(sin,(x,-1,1),axes_labels=['$x$','$y$'],legend_label=r'$\sin(x)$',color='red')
sage: P1+P2
Graphics object consisting of 2 graphics primitives

>>> from sage.all import *
>>> P1 = plot(f,(x,-Integer(1),Integer(1)),axes_labels=['$x$','$y$'],legend_label='$f(x)$')
>>> P2 = plot(sin,(x,-Integer(1),Integer(1)),axes_labels=['$x$','$y$'],legend_label=r'$\sin(x)$',color='red')
>>> P1+P2
Graphics object consisting of 2 graphics primitives

One additional useful note is that plots of functions with vertical asymptotes may need their vertical viewing range set manually; otherwise the asymptote may really go to infinity!

sage: plot(1/x^2,(x,-10,10),ymax=10)
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> plot(Integer(1)/x**Integer(2),(x,-Integer(10),Integer(10)),ymax=Integer(10))
Graphics object consisting of 1 graphics primitive

Remember, you can use the command plot? to find out about most of the options demonstrated above.

Below, you can experiment with several of the plotting options.

  • Just evaluate the cell and play with the sliders, buttons, color picker, etc., to change the plot options.

  • You can access low-level options like the initial number of plotted points, or high-level ones like whether axes are shown or not.

  • This uses a feature of Sage called “interacts”, which is a very powerful way to engage students in exploring a problem.

sage: x = var('x')
sage: @interact
sage: def plot_example(f=sin(x^2),r=range_slider(-5,5,step_size=1/4,default=(-3,3)),
....:                  color=color_selector(widget='colorpicker'),
....:                  thickness=(3,(1..10)),
....:                  adaptive_recursion=(5,(0..10)), adaptive_tolerance=(0.01,(0.001,1)),
....:                  plot_points=(20,(1..100)),
....:                  linestyle=['-','--','-.',':'],
....:                  gridlines=False, fill=False,
....:                  frame=False, axes=True
....:                  ):
....:     show(plot(f, (x,r[0],r[1]), color=color, thickness=thickness,
....:                  adaptive_recursion=adaptive_recursion,
....:                  adaptive_tolerance=adaptive_tolerance, plot_points=plot_points,
....:                  linestyle=linestyle, fill=fill if fill else None),
....:                  gridlines=gridlines, frame=frame, axes=axes)

>>> from sage.all import *
>>> x = var('x')
>>> @interact
>>> def plot_example(f=sin(x**Integer(2)),r=range_slider(-Integer(5),Integer(5),step_size=Integer(1)/Integer(4),default=(-Integer(3),Integer(3))),
...                  color=color_selector(widget='colorpicker'),
...                  thickness=(Integer(3),(ellipsis_iter(Integer(1),Ellipsis,Integer(10)))),
...                  adaptive_recursion=(Integer(5),(ellipsis_iter(Integer(0),Ellipsis,Integer(10)))), adaptive_tolerance=(RealNumber('0.01'),(RealNumber('0.001'),Integer(1))),
...                  plot_points=(Integer(20),(ellipsis_iter(Integer(1),Ellipsis,Integer(100)))),
...                  linestyle=['-','--','-.',':'],
...                  gridlines=False, fill=False,
...                  frame=False, axes=True
...                  ):
...     show(plot(f, (x,r[Integer(0)],r[Integer(1)]), color=color, thickness=thickness,
...                  adaptive_recursion=adaptive_recursion,
...                  adaptive_tolerance=adaptive_tolerance, plot_points=plot_points,
...                  linestyle=linestyle, fill=fill if fill else None),
...                  gridlines=gridlines, frame=frame, axes=axes)

Basic 3D Plotting#

There are several mechanisms for viewing three-dimensional plots in Sage, but we will stick to the default option in the notebook interface, which is via javascript applets from the program Jmol/JSmol .

Plotting a 3D plot is similar to plotting a 2D plot, but we need to specify ranges for two variables instead of one.

sage: g(x,y)=sin(x^2+y^2)
sage: plot3d(g,(x,-5,5),(y,-5,5))
Graphics3d Object

>>> from sage.all import *
>>> __tmp__=var("x,y"); g = symbolic_expression(sin(x**Integer(2)+y**Integer(2))).function(x,y)
>>> plot3d(g,(x,-Integer(5),Integer(5)),(y,-Integer(5),Integer(5)))
Graphics3d Object

There is a lot you can do with the 3D plots.

  • Try rotating the plot above by clicking and dragging the mouse inside of the plot.

  • Also, right-click (Control-click if you have only one mouse button) just to the right of the plot to see other options in a menu.

  • If you have a wheel on your mouse or a multi-touch trackpad, you can scroll to zoom.

  • You can also right-click to see other options, such as

  • spinning the plot,

  • changing various colors,

  • and even making the plot suitable for viewing through 3D glasses (under the “style”, then “stereographic” submenus),

When using the plot3d command, the first variable range specified is plotted along the usual “x” axis, while the second range specified is plotted along the usual “y” axis.

The plot above is somewhat crude because the function is not sampled enough times - this is fairly rapidly changing function, after all. We can make the plot smoother by telling Sage to sample the function using a grid of 300 by 300 points. Sage then samples the function at 90,000 points!

sage: plot3d(g,(x,-5,5),(y,-5,5),plot_points=300)
Graphics3d Object

>>> from sage.all import *
>>> plot3d(g,(x,-Integer(5),Integer(5)),(y,-Integer(5),Integer(5)),plot_points=Integer(300))
Graphics3d Object

As with 2D plots, we can superimpose 3D plots by adding them together.

Note that in this one, we do not define the functions, but only use expressions (see the first set of topics in this tutorial), so it is wisest to define the variables ahead of time.

sage: var('x,y')
(x, y)
sage: b = 2.2
sage: P=plot3d(sin(x^2-y^2),(x,-b,b),(y,-b,b), opacity=.7)
sage: Q=plot3d(0, (x,-b,b), (y,-b,b), color='red')
sage: P+Q
Graphics3d Object

>>> from sage.all import *
>>> var('x,y')
(x, y)
>>> b = RealNumber('2.2')
>>> P=plot3d(sin(x**Integer(2)-y**Integer(2)),(x,-b,b),(y,-b,b), opacity=RealNumber('.7'))
>>> Q=plot3d(Integer(0), (x,-b,b), (y,-b,b), color='red')
>>> P+Q
Graphics3d Object

As usual, only the last command shows up in the notebook, though clearly all are evaluated. This also demonstrates that many of the same options work for 3D plots as for 2D plots.

We close this tutorial with a cool plot that we define implicitly as a 3D contour plot.

sage: var('x,y,z')
(x, y, z)
sage: T = golden_ratio
sage: p = 2 - (cos(x + T*y) + cos(x - T*y) + cos(y + T*z) + cos(y - T*z) + cos(z - T*x) + cos(z + T*x))
sage: r = 4.78
sage: implicit_plot3d(p, (x, -r, r), (y, -r, r), (z, -r, r), plot_points=50, color='yellow')
Graphics3d Object

>>> from sage.all import *
>>> var('x,y,z')
(x, y, z)
>>> T = golden_ratio
>>> p = Integer(2) - (cos(x + T*y) + cos(x - T*y) + cos(y + T*z) + cos(y - T*z) + cos(z - T*x) + cos(z + T*x))
>>> r = RealNumber('4.78')
>>> implicit_plot3d(p, (x, -r, r), (y, -r, r), (z, -r, r), plot_points=Integer(50), color='yellow')
Graphics3d Object

The next tutorial will use all that you have learned about Sage basics, symbolics, and plotting in a specific mathematical venue - the calculus sequence!