SymPy - Symbolic Mathematics in Python

Detailed Guide

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When to Use This Skill

Use this skill when:

Solving equations symbolically (algebraic, differential, systems of equations)
Performing calculus operations (derivatives, integrals, limits, series)
Manipulating and simplifying algebraic expressions
Working with matrices and linear algebra symbolically
Doing physics calculations (mechanics, quantum mechanics, vector analysis)
Number theory computations (primes, factorization, modular arithmetic)
Geometric calculations (2D/3D geometry, analytic geometry)
Converting mathematical expressions to executable code (Python, C, Fortran)
Generating LaTeX or other formatted mathematical output
Needing exact mathematical results (e.g., sqrt(2) not 1.414...)

Getting Started Examples

Example 1: Solve Quadratic Equation

from sympy import symbols, solve, sqrt
x = symbols('x')
solution = solve(x**2 - 5*x + 6, x)
# [2, 3]

Example 2: Calculate Derivative

from sympy import symbols, diff, sin
x = symbols('x')
f = sin(x**2)
df_dx = diff(f, x)
# 2*x*cos(x**2)

Example 3: Evaluate Integral

from sympy import symbols, integrate, exp
x = symbols('x')
integral = integrate(x * exp(-x**2), (x, 0, oo))
# 1/2

Example 4: Matrix Eigenvalues

from sympy import Matrix
M = Matrix([[1, 2], [2, 1]])
eigenvals = M.eigenvals()
# {3: 1, -1: 1}

Example 5: Generate Python Function

from sympy import symbols, lambdify
import numpy as np
x = symbols('x')
expr = x**2 + 2*x + 1
f = lambdify(x, expr, 'numpy')
f(np.array([1, 2, 3]))
# array([ 4,  9, 16])

Limitations

Use this skill only when the task clearly matches the scope described above.
Do not treat the output as a substitute for environment-specific validation, testing, or expert review.
Stop and ask for clarification if required inputs, permissions, safety boundaries, or success criteria are missing.