Variables¶

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description¶

Variables[poly]
    gives the sorted list of independent variables that appear as bases
    of non-numeric subexpressions in poly.
Walks the expression tree and collects symbols and compound forms
(e.g. Sin[x], a[i]) that occur outside numeric arithmetic; duplicates
are removed via canonical order.

Examples¶

All examples below are verified against the current Mathilda build.

In[1]:= Variables[(x + y)^2 + 3 z^2 - y z + 7]
Out[1]= {x, y, z}

In[2]:= Variables[Sin[x] + Cos[x]]
Out[2]= {Cos[x], Sin[x]}

In[3]:= Variables[E^x]
Out[3]= {}

Implementation notes¶

Algorithm. builtin_variables walks the expression with collect_variables, gathering the distinct symbols that occur as polynomial generators (bare symbols and non-numeric bases, excluding numeric atoms and known constants), then sorts the collected Expr* array with qsort under compare_expr_ptrs (the canonical expr_compare order) and wraps the result in a List. The output is the deduplicated, canonically-ordered list of variables on which the input is treated as a polynomial/rational expression.

Protected.
Looks for variables only inside Plus, Times, and Power with rational exponents.
Returns a sorted List of variables.
Symbolic constants like Pi, E, and I are not treated as variables.

Attributes: Protected.

Implementation status¶

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References¶

Geddes, Czapor & Labahn, "Algorithms for Computer Algebra" (1992), Ch. 3 (multivariate polynomial representation and variable sets).
Source: src/poly/poly.c
Specification: docs/spec/builtins/structural-manipulation.md

Notes & additional examples¶

Worked examples¶

In[1]:= Variables[x^2 + y z]
Out[1]= {x, y, z}

In[1]:= Variables[a x^2 + b x + c]
Out[1]= {a, b, c, x}

In[1]:= Variables[Sin[x] + y]
Out[1]= {Sin[x], y}

In[1]:= Variables[x^2 + 3 x + 2]
Out[1]= {x}

Fractional powers are treated as polynomial generators in their own right, so each radical base contributes its variable:

In[1]:= Variables[x^(1/2) + y^(2/3)]
Out[1]= {x, y}

Notes¶

Variables collects the independent generators that appear as bases of non-numeric subexpressions and returns them in canonical sorted order with duplicates removed. Pure numeric coefficients are ignored, so x^2 + 3 x + 2 reports only {x}. Compound non-atomic forms are treated as single generators rather than being broken open: Sin[x] + y yields {Sin[x], y}, keeping Sin[x] whole. Every symbol that occurs outside numeric arithmetic is included, which is why parameters like a, b, c appear alongside the main variable x.