Introduction to Maxima for STACK users
Maxima is a system for the manipulation of symbolic and numerical expressions, including differentiation, integration, Taylor series, Laplace transforms, ordinary differential equations, systems of linear equations, polynomials, sets, lists, vectors, matrices, and tensors.
To write more than very simple questions you will need to use some Maxima commands. This documentation does not provide a detailed tutorial on Maxima. A very good introduction is given in Minimal Maxima, which this document assumes you have read.
STACK then modifies Maxima in a number of ways.
Atoms and subscripts
Everything in Maxima is either an atom or an expression. Atoms are either an integer number, float, string or a name. You can use the predicate atom()
to decide if its argument is an atom. Expressions have an operator and a list of arguments. Note that the underscore symbol is not an operator. Thus a_1
is an atom in maxima. Hence, the atoms a1
and a_1
are not considered to be algebraically equivalent. If you would like to consolidate subscripts in students' input see the documentation on extra options. Also note that since the underscore is not an operator, an expression such as (a_b)_c
is not valid Maxima syntax, but a_b_c
is a valid name for an atom.
You can change the TeX output for an atom with Maxima's texput
command. E.g. texput(blob, "\\diamond")
will display the atom blob
as . If you place texput
commands in the question variables, this affects the display everywhere in the question including the inputs. E.g. if a student types in blob
then the validation feedback should say something like "your last answer was: ".
Display with subscripts is a subtle and potentially confusing issue because subscript notation in mathematics has many different uses. For example,
 Subscripts denote a function of the natural numbers, e.g. when defining terms in a sequence . That is the subscript denotes function application. .
 Subscripts denote differentiation, e.g. is the derivative of with respect to .
 Subscripts denote coordinates in a vector, in .
There are many other possible uses for subscripts, especially in other subjects e.g. in physics or actuarial studies.
Because Maxima considers subscripted expressions to be atoms, the default TeX output of an atom V_alpha
from Maxima is (literally {\it V\_alpha}
) and not as a user might expect. For this reason STACK intercepts and redefines how atoms with the underscore are displayed. In particular STACK (but not core Maxima) takes an atom A_B
, applies the tex()
command to A
and B
separately and concatenates the result using subscripts. For example, if you define
texput(A, "{\\mathcal A}");
texput(B, "\\diamond");
then A_B
is now displayed as .
Below are some examples.
Maxima code  Maxima's tex() command (raw output and displayed) 
STACK (raw output and displayed) 

A_B 
{\it A\_B}

{{A}_{B}}

A[1] 
A_{1}

{A_{1}}

A1 
A_{1}

{A_{1}}

A_1 
A_{1}

{A_{1}}

A_x1 
{\it A\_x}_{1}

{{A}_{x_{1}}}

A_BC 
{\it A\_BC}

{{A}_{{\it BC}}}

A_alpha 
{\it A\_alpha}

{{A}_{\alpha}}

A_B_C 
{\it A\_B\_C}

{{{A}_{B}}_{C}}

x_t(1) 
{\it x\_t}\left(1\right)

{{\it x\_t}\left(1\right)}

A[1,2] 
A_{1,2}

{A_{1,2}}

Notes
 in the above examples three different expressions (atoms
A1
,A_1
and the expressionA[1]
) generate the same tex codeA_{1}
, and so are indistinguishable at the display level.  The expression
x_t(1)
refers to the functionx_t
which is not an atom, and hence STACK's logic for displaying atoms with underscores does not apply (by design). If you want to display a function name including a subscript you can explicitly use, e.g.texput(x_t, "x_t");
to control the display, this is just not done automatically.
One situation where this design is not satisfactory is when you want to use both of the atoms F
and F_1
but with different display. For example F
should display as but F_1
should display as . Such a situation is not hard to imagine, as it is often considered good style to have things like . The above design always splits up the atom F_1
into F
and 1
, so that the atom F_1
will display as . (This is actually what you normally expect, especially with the Greek letter subscripts.) To avoid this problem the logic which splits up atoms containing an underscore checks the texput properties list. If an entry has been made for a specific atom then STACK's display logic uses the entry, and does not split an atom over the underscore. In the above example, use the following texput commands.
texput(F, "{\\mathcal F}");
texput(F_1, "{F_1}");
With this code F
displays as , the atom F_1
displays as , and every subscript will display with calligraphic, e.g. F_alpha
displays as . There is no way to code the reverse logic, i.e. define a special display only for the unique atom F
.
Note that the scientific units code redefines and then assumes that symbols represent units. E.g. F
is assumed to represent Farad, and all units are typeset in Roman type, e.g. rather than the normal . This is typically the right thing to do, but it does restrict the number of letters which can be used for variable names in a particular question. To overcome this problem you will have to redefine some atoms with texput. For example,
stack_unit_si_declare(true);
texput(F_a, "F_a");
will display the atom F_a
as , i.e. not in Roman. If you texput(F, "F")
the symbol F
is no longer written in Roman, as you would expect from units. This may be sensible if Farad could not possibly appear in context, but students might type a range of subscripted atoms involving F
.
The use of texput is global to a question. There is no way to display a particular atom differently in different places (except perhaps in the feedback variables, which is currently untested: assume texput is global).
How would you generate the tex like ? STACK's sequence
command (see below) does output its arguments separated by a comma, so sequence(1,2)
is displayed as , however the Maxima command A_sequence(1,2)
refers to the function A_sequence
, (since the underscore is not an operator). Hence STACK's logic for splitting up atoms containing the underscore does not apply. (In any case, even if the display logic did split up function names we would still have the issue of binding power to sort out, i.e. do we have the atom with parts A
and sequence(1,2)
or the function named A
and sequence
?) To create an output like you have no option but to work at the level of display. Teachers can create an inert function which displays using subscripts.
texsub(a,b)
is typeset as i.e. {a}_{b}
in LaTeX. For example,
texsub(A, sequence(1,2))
will display as , with simplification off,
texsub(F,12)
will be displayed as .
Note that the process of converting theta_07
into the intermediate texsub
form internally results in the texsub(theta,7)
which removes the leading zero. This is a known issue, for which a work around is to directly use texput(theta_07, "{{\\theta}_{07}}")
or texsub(theta,"07")
. The second option does not produce optimal LaTeX, since it uses TeX mbox
, e.g. {{\theta}_{\mbox{07}}}
.
Types of object
Maxima is a very weakly typed language. However, in STACK we need the following "types" of expression:
 equations, i.e. an expression in which the top operation is an equality sign;
 inequalities, for example ;
 sets, for example, ;
 lists, for example, . In Maxima ordered lists are entered using square brackets, for example as
p:[1,1,2,x^2]
. An element is accessed using the syntaxp[1]
.  matrices. The basic syntax for a matrix is
p:matrix([1,2],[3,4])
. Each row is a list. Elements are accessed asp[1,2]
, etc.  logical expression. This is a tree of other expressions connected by the logical
and
andor
. This is useful for expressing solutions to equations, such asx=1 or x=2
. Note, the support for these expressions is unique to STACK.  expressions.
Expressions come last, since they are just counted as being not the others! STACK defines predicate functions to test for each of these types.
Numbers
Numbers are important in assessment, and there is more specific and detailed documentation on how numbers are treated: Numbers in STACK.
Alias
STACK defines the following function alias names
simplify := fullratsimp
int := integrate
The absolute value function in Maxima is entered as abs()
. STACK also permits you to enter using 
symbols, i.e.x
. This is an alias for abs
. Note that abs(x)
will be displayed by STACK as .
STACK also redefined a small number of functions
 The plot command
plot2d
is not used in STACK questions. Useplot
instead, which is documented here. This ensures your image files are available on the server.  The random number command
random
is not used in STACK questions. Use the commandrand
instead, which is documented here. This ensures pseudorandom numbers are generated and a student gets the same version each time they login.
Parts of Maxima expressions
op(x)
 the top operator
It is often very useful to take apart a Maxima expression. To help with this Maxima has a number of commands, including op(ex)
, args(ex)
and part(ex,n)
. Maxima has specific documentation on this.
In particular, op(ex)
returns the main operator of the expression ex
. This command has some problems for STACK.
 calling
op(ex)
on an atom (see Maxima's documentation on the predicateatom(ex)
) such as numbers or variable names, causeop(ex)
to throw an error. op(ex)
sometimes returns a string, sometimes not. the unary minus causes problems. E.g. in
1/(1+x)
the operation is not "/", as you might expect, but it is "" instead!
To overcome these problems STACK has a command
safe_op(ex)
This always returns a string. For an atom this is empty, i.e. ""
. It also sorts out some unary minus problems.
get_ops(ex)
 all operators
This function returns a set of all operators in an expression. Useful if you want to find if multiplication is used anywhere in an expression.
Maxima commands defined by STACK
It is very useful when authoring questions to be able to test out Maxima code in the same environment which STACK uses Maxima. That is to say, with the settings and STACK specific functions loaded. To do this see STACKMaxima sandbox.
STACK creates a range of additional functions and restricts those available, many of which are described within this documentation. See also Predicate functions.
Command  Description 

factorlist(ex) 
Returns a list of factors of ex without multiplicities. 
zip_with(f,a,b) 
This function applies the binary function to two lists and returning a list. An example is given in adding matrices to show working. 
coeff_list(ex,v) 
This function takes an expression ex and returns a list of coefficients of v . 
coeff_list_nz(ex,v) 
This function takes an expression ex and returns a list of nonzero coefficients of v . 
divthru(ex) 
Takes an algebraic fraction, e.g. and divides through by the denominator, to leave a polynomial and a proper fraction. Useful in feedback, or steps of a calculation. 
stack_strip_percent(ex,var) 
Removes any variable beginning with the % character from ex and replace them with variables from var . Useful for use with solve, ode2 etc. Solve and ode2. 
exdowncase(ex) 
Takes the expression ex and substitutes all variables for their lower case version (cf sdowncase(ex) in Maxima). This is very useful if you don't care if a student uses the wrong case, just apply this function to their answer before using an answer test. Note, of course, that exdowncase(X)x=0. 
stack_reset_vars 
Resets constants, e.g. , as abstract symbols, see Numbers. 
safe_op(ex) 
Returns the operation of the expression as a string. Atoms return an empty string (rather than throwing an error as does op ). 
comp_square(ex,v) 
Returns a quadratic ex in the variable v in completed square form. 
degree(ex,v) 
Returns the degree of the expanded form of ex in the variable v . See also Maxima's hipow command. 
unary_minus_sort(ex) 
Tidies up the way unary minus is represented within expressions when simp:false . See also simplification. 
texboldatoms(ex) 
Displays all nonnumeric atoms in bold. Useful for vector questions. 
Assignment
In Maxima the assignment of a value to a variable is very unusual.
Input  Result 

a:1 
Assignment of the value to . 
a=1 
An equation, yet to be solved. 
f(x):=x^2 
Definition of a function. 
In STACK simple assignments are of the more conventional form key : value
, for example,
n : rand(3)+2;
p : (x1)^n;
Of course, these assignments can make use of Maxima's functions to manipulate expressions.
p : expand( (x3)*(x4) );
Another common task is that of substitution. This can be performed with Maxima's subst
command. This is quite useful, for example if we define as follows, in the then we can use this in response processing to determine if the student's answer is odd.
p : ans1 + subst(x,x,ans1);
All sorts of properties can be checked for in this way. For example, interpolates. Another example is a stationary point of at , which can be checked for using
p : subst(a,x,diff(ans1,x));
Here we have assumed a
is some point given to the student, ans1
is the answer and that will be used in the response processing tree.
You can use Maxima's looping structures within Question variables. For example
n : 1;
for a:3 thru 26 step 7 do n:n+a;
The result will be . It is also possible to define functions within the question variables for use within a question.
f(x) := x^2;
n : f(4);
Logarithms
STACK loads the contributed Maxima package log10
. This defines logarithms to base automatically. STACK also creates two aliases
ln
is an alias for , which are natural logarithmslg
is an alias for , which are logarithms to base . It is not possible to redefine the commandlog
to be to the base .
Sets, lists, sequences, ntuples
It is very useful to be able to display expressions such as comma separated lists, and ntuples Maxima has inbuilt functions for lists, which are displayed with square brackets , and sets with curly braces . Maxima has no default functions for ntuples or for sequences.
STACK provides an inert function sequence
. All this does is display its arguments without brackets. For example sequence(1,2,3,4)
is displayed . STACK provides convenience functions.
sequenceify
, creates a sequence from the arguments of the expression. This turns lists, sets etc. into a sequence.sequencep
is a predicate to decide if the expression is a sequence. The atom
dotdotdot
is displayed using the tex\ldots
which looks like . This atom cannot be entered by students.
STACK provides an inert function ntuple
. All this does is display its arguments with round brackets. For example ntuple(1,2,3,4)
is displayed . ntupleify
and ntuplep
construct and test for ntuples. In strict Maxima syntax (a,b,c)
is equivalent to block(a,b,c)
. If students type in (a,b,c)
using a STACK input it is filtered to ntuple(a,b,c)
. Teachers must use the ntuple
function explicitly to construct question variables, teacher's answers, test cases and so on. The ntuple
is useful for students to type in coordinates.
If you want to use these functions, then you can create question variables as follows
L1:[a,b,c,d];
D1:apply(ntuple, L1);
L2:args(D1);
D2:sequenceify(L2);
Then L1
is a list and is displayed with square brackets as normal. D1
has operator ntuple
and so is displayed with round brackets. L2
has operator list
and is displayed with square brackets. Lastly, D2 is an sequence
and is displayed without brackets.
You can, of course, apply these functions directly.
T1:ntuple(a,b,c);
S1:sequence(a,b,c,dotdotdot);
If you want to use sequence
or ntuple
in a PRT comparison, you probably want to turn them back into lists. E.g. ntuple(1,2,3)
is not algebraically equivalent to [1,2,3]
. To do this use the args
function. We may, in the future, give more active meaning to the data types of sequence
and ntuple
.
Currently, students can enter expressions with "implied ntuples" E.g
 Student input of
(1,2,3)
is interpreted asntuple(1,2,3)
.  Student input of
{(1,2,3),(4,5,6)}
is interpreted as{ntuple(1,2,3),ntuple(4,5,6)}
.  Since no operations are defined on ntuples, students cannot currenlty enter things like
(1,2,3)+s*(1,0,0)
. There is nothing to stop a teacher defining the expression treentuple(1,2,3)+s*ntuple(1,0,0)
, but the operations+
and*
are not defined for ntuples and so nothing will happen! If you want a student to enter the equation of a line/plane they should probably use the matrix syntax for vectors. (This may change in the future).
Matrices have options to control the display of the braces. Matrices are displayed without commas.
If you are interacting with javascript do not use sequenceify
. If you are interacting with javascript, such ss JSXGraph, then you may want to output a list of values without all the LaTeX and without Maxima's normal bracket symbols. You can use
stack_disp_comma_separate([a,b,sin(pi)]);
This function turns a list into a string representation of its arguments, without braces.
Internally, it applies string
to the list of values (not TeX!). However, you might still get things like %pi
in the output.
You can use this with mathematical input: {@stack_disp_comma_separate([a,b,sin(pi)])@}
and you will get the result a, b, sin(%pi/7)
(without the string quotes) because when a Maxima variable is a string we strip off the outside quotes and don't typeset this in maths mode.
Functions
It is sometimes useful for the teacher to define functions as part of a STACK question. This can be done in the normal way in Maxima using the notation.
f(x):=x^2;
Using Maxima's define()
command is forbidden. An alternative is to define f
as an "unnamed function" using the lambda
command.
f:lambda([x],x^2);
Here we are giving a name to an "unnamed function" which seems perverse. Unnamed functions are extremely useful in many situations.
For example, a piecewise function can be defined by either of these two commands
f(x):=if (x<0) then 6*x2 else 2*exp(3*x);
f:lambda([x],if (x<0) then 6*x2 else 2*exp(3*x));
You can then plot this using
{@plot(f(x),[x,1,1])@}
Maxima "gotcha"s!
 See the section above on assignment.
 Maxima does not have a
degree
command for polynomials. We define one via thehipow
command.  Matrix multiplication is the dot, e.g.
A.B
. The starA*B
gives elementwise multiplication.  The atoms
a1
anda_1
are not considered to be algebraically equivalent.