/sympy

name: sympy
description: Performs exact symbolic mathematics with SymPy for algebra, calculus, equation solving, symbolic linear algebra, physics, and lambdify or LaTeX code generation. Use when a task needs symbolic results, explicit assumptions, or exact arithmetic; use NumPy or SciPy for purely numerical workloads.
license: https://github.com/sympy/sympy/blob/master/LICENSE
allowed-tools: Read Write Edit Bash
compatibility: Requires Python 3.9+ and SymPy 1.14.0. Optional NumPy/SciPy/Matplotlib, IPython/ipywidgets, or ANTLR 4.11 parser runtime for relevant examples. Compiled wrappers need a C/Fortran compiler and backend packages; emitting source needs no compiler. Network only for installation/docs.
metadata:
  version: "1.5"
  last-reviewed: "2026-10-01"
  upstream-version: "1.14.0"
  skill-author: K-Dense Inc.

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](references/review.md) 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](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](references/matrices-linear-algebra.md). 5. **Physics and mechanics** — see [references/physics-mechanics.md](references/physics-mechanics.md). 6. **Advanced mathematics** — see [references/advanced-topics.md](references/advanced-topics.md). 7. **Code generation and output** — see [references/code-generation-printing.md](references/code-generation-printing.md).

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

Declare the solution domain: `solveset` defaults to complex numbers, so use `domain=S.Reals` for real-only questions. A returned `ConditionSet` means an unresolved solution condition, not that no solutions exist; distinguish it from `EmptySet`. A numerical `nsolve` result is a local root found from a starting point, not proof that every root was found.

7. Preserve mathematical meaning and input trust

Use assumptions only when justified by the problem. An unconstrained symbol is complex; `sqrt(x**2)` need not equal `x`, and logarithm/power identities depend on branches. Assumption predicates can return `None` (unknown). Keep excluded denominator zeros when cancelling factors, and verify candidate solutions in the original expression and requested domain. `==` compares symbolic structure; use `Eq` to build an equation and targeted simplification to verify an identity.

`parse_expr`, string `sympify`, and `lambdify` can execute code. Accept only trusted