SymPy - Symbolic Mathematics in Python

Overview

SymPy is a Python library for symbolic mathematics that enables exact computation using mathematical symbols rather than numerical approximations. This skill provides comprehensive guidance for performing symbolic algebra, calculus, linear algebra, equation solving, physics calculations, and code generation using SymPy.

Installation

Reviewed against current official documentation and executed with SymPy 1.14.0 on Python 3.13.3 (2026-10-01). Core SymPy requires Python 3.9+; the tested NumPy 2.5.3 / SciPy 1.18.1 stack needs Python 3.12+. SymPy 1.14.0 requires mpmath>=1.1,<1.4; use the compatible 1.3.0, not the newer 1.4.x release. See verification and official sources for coverage.

# Install SymPy using uv
uv pip install "sympy==1.14.0"

# Optional: for lambdify and plotting examples
uv pip install numpy scipy matplotlib

Check your version:

import sympy
print(sympy.__version__)

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...)

Core Capabilities

Seven capability areas are documented in references/core_capabilities.md:

1. Symbolic computation basics — symbols, expressions, simplification, substitution.
2. Calculus — differentiation, integration, limits, series.
3. Equation solving — solve, solveset, linear and nonlinear systems, ODEs.
4. Matrices and linear algebra — see references/matrices-linear-algebra.md.
5. Physics and mechanics — see references/physics-mechanics.md.
6. Advanced mathematics — see references/advanced-topics.md.
7. Code generation and output — see references/code-generation-printing.md.

Deeper treatment of the first three is in references/core-capabilities.md.

Working with SymPy: Best Practices

1. Always Define Symbols First

from sympy import symbols
x, y, z = symbols('x y z')
# Now x, y, z can be used in expressions

2. Use Assumptions for Better Simplification

from sympy import symbols, sqrt
x = symbols('x', positive=True, real=True)
sqrt(x**2)  # Returns x (not Abs(x)) due to positive assumption

Common assumptions: real, positive, negative, integer, rational, complex, even, odd

3. Use Exact Arithmetic

from sympy import Rational, S
# Correct (exact):
expr = Rational(1, 2) * x
expr = S(1)/2 * x

# Approximate (appropriate for measured/numerical inputs):
expr = 0.5 * x  # Creates approximate value

4. Numerical Evaluation When Needed

from sympy import pi, sqrt
result = sqrt(8) + pi
result.evalf()    # 5.96371554103586
result.evalf(50)  # Request 50 decimal digits; cannot recover precision lost in inputs

5. Convert to NumPy for Performance

from sympy import symbols, lambdify
import numpy as np
x = symbols("x")
expr = x**2 + 1
# Slow for many evaluations:
for x_val in range(1000):
    result = expr.subs(x, x_val).evalf()

# Fast:
f = lambdify(x, expr, 'numpy')
results = f(np.arange(1000))

6. Use Appropriate Solvers

• solveset: Algebraic equations (primary)
• linsolve: Linear systems
• nonlinsolve: Nonlinear systems
• dsolve: Differential equations
• solve: General purpose; supports some problems solveset does not

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