Expressions and Substitutions¶

Symbolic math variables can be combined into symbolic math expressions. Once in an expression, symbolic math variables can be exchanged with substituion.

Expressions¶

A symbolic math expression is a combination of symbolic math variables with numbers and mathematical operators such as +, -, / and *. The standard Python rules for calculating numbers apply in SymPy symbolic math expressions.

After the symbols x and y are created, a symbolic math expression using x and y can be defined.

from sympy import symbols

x, y = symbols('x y')
expr = 2*x + y

Substitution¶

Use SymPy’s .subs() method to insert a numerical value into a symbolic math expression. The first argument of the .subs() method is the mathematical symbol and the second argument is the numerical value. In the expression below:

\[ 2x + y \]

If we substitute:

\[ x = 2 \]

The resulting expression is:

\[ 2(2) + y \]

\[ 4 + y \]

We can code the substitution above using SymPy’s .subs() method as shown below.

expr.subs(x, 2)

\[\displaystyle y + 4\]

The .subs() method does not replace variables in place, .subs() only completes a one-time substitution. If expr is called after the .subs() method is applied, the original expr expression is returned.

expr

\[\displaystyle 2 x + y\]

To make the substitution permanent, a new expression object needs to be assigned to the output of the .subs() method.

expr = 2*x + y
expr2 = expr.subs(x, 2)
expr2

\[\displaystyle y + 4\]

SymPy variables can also be substituted into SymPy expressions. In the code section below, the symbol \(z\) is substituted for the symbol \(x\) (\(z\) replaces \(x\)).

x, y, z = symbols('x y z')
expr = 2*x + y
expr2 = expr.subs(x, z)
expr2

\[\displaystyle y + 2 z\]

Expressions can also be substituted into other expressions. Consider the following:

\[ y + 2x^2 + z^{-3} \]

substitute in

\[ y = 2x \]

results in

\[ 2x + 2x^2 + z^{-3} \]

x, y, z = symbols('x y z')
expr = y + 2*x**2 + z**(-3)
expr2 = expr.subs(y, 2*x)
expr2

\[\displaystyle 2 x^{2} + 2 x + \frac{1}{z^{3}}\]

A practical example involving symbolic math variables, expressions and substitutions can include a complex expression and several replacements.

\[ n_0e^{-Q_v/RT} \]

\[ n_0 = 3.48 \times 10^{-6} \]

\[ Q_v = 12,700 \]

\[ R = 8.31 \]

\[ T = 1000 + 273 \]

We can create four symbolic math variables and combine the variables into an expression with the code below.

from sympy import symbols, exp
n0, Qv, R, T = symbols('n0 Qv R T')
expr = n0*exp(-Qv/(R*T))

Multiple SymPy subs() methods can be chained together to substitute multiple variables in one line of code.

expr.subs(n0, 3.48e-6).subs(Qv,12700).subs(R, 8031).subs(T, 1000+273)

\[\displaystyle \frac{3.48 \cdot 10^{-6}}{e^{\frac{12700}{10223463}}}\]

To evaluate an expression as a floating point number, use SymPy’s .evalf() method.

expr2 = expr.subs(n0, 3.48e-6).subs(Qv,12700).subs(R, 8031).subs(T, 1000+273)

expr2.evalf()

\[\displaystyle 3.47567968697765 \cdot 10^{-6}\]

You can control the number of digits the .evalf() method outputs by passing a number as an argument.

expr2.evalf(4)

\[\displaystyle 3.476 \cdot 10^{-6}\]

Summary¶

The SymPy functions and methods used in this section are summarized in the table below.

SymPy function or method	Description	Example
`symbols()`	create symbolic math variables	`x, y = symbols('x y')`
`.subs()`	substitute a value into a symbolic math expression	`expr.subs(x,2)`
`.evalf()`	evaluate a symbolic math expression as a floating point number	`expr.evalf()`