---
name: sympy
description: "SymPy is a Python library for symbolic mathematics that enables exact computation using mathematical symbols rather than numerical approximations."
license: https://github.com/sympy/sympy/blob/master/LICENSE
metadata:
    skill-author: K-Dense Inc.
risk: safe
source: "https://github.com/sympy/sympy"
date_added: "2026-09-04"
---

# SymPy - Symbolic Mathematics in Python

## Detailed Guide

Read [the detailed guide](references/detailed-guide.md) before executing this skill. It retains the complete procedure and reference material. Treat its safety, prerequisites, and validation requirements as mandatory. For focused work, load the relevant sections; for end-to-end work, read the guide completely.

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

### Example 2: Calculate Derivative
```python
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
```python
from sympy import symbols, integrate, exp
x = symbols('x')
integral = integrate(x * exp(-x**2), (x, 0, oo))
# 1/2
```

### Example 4: Matrix Eigenvalues
```python
from sympy import Matrix
M = Matrix([[1, 2], [2, 1]])
eigenvals = M.eigenvals()
# {3: 1, -1: 1}
```

### Example 5: Generate Python Function
```python
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.