/sympy

name: sympy
description: Use when you need exact symbolic math in Python — algebra, calculus, equation solving, symbolic linear algebra, or code generation via lambdify/LaTeX. Prefer NumPy or SciPy when floating-point approximations are sufficient.
license: https://github.com/sympy/sympy/blob/master/LICENSE
allowed-tools: Read Write Edit Bash
compatibility: Requires Python 3.9+ and SymPy 1.14+. Optional NumPy/SciPy/Matplotlib for lambdify examples; C/Fortran compiler for autowrap/codegen.
metadata:
  version: "1.3"
  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

Tested against **SymPy 1.14.0** (stable; April 2025). Requires **Python 3.9+**.

# Install SymPy using uv
uv pip install "sympy>=1.14"

# 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

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

# Incorrect (floating-point):
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)  # 50 digits of precision

5. Convert to NumPy for Performance

# 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 (legacy, but flexible)

Reference Files Structure

This skill uses modular reference files for different capabilities:

1. **`core-capabilities.md`**: Symbols, algebra, calculus, simplification, equation solving

• Load when: Basic symbolic computation, calculus, or solving equations

2. **`matrices-linear-algebra.md`**: Matrix operations, eigenvalues, linear systems

• Load when: Working with matrices or linear algebra problems

3. **`physics-mechanics.md`**: Classical mechanics, quantum mechanics, vectors, units

• Load when: Physics calculations or mechanics problems

4. **`advanced-topics.md`**: Geometry, number theory, combinatorics, logic, statistics

• Load when: Advanced mathematical topics beyond basic algebra and calculus

5. **`code-generation-printing.md`**: Lambdify, codegen, LaTeX output, printing

• Load when: Converting expressions to code or generating formatted output

Common Use Case Patterns

Pattern 1: Solve and Verify

from sympy import symbols, solve, simplify
x = symbols('x')

# Solve equation
equation = x**2 - 5*x + 6
solutions = solve(equation, x)  # [2, 3]

# Verify solutions
for sol in solutions:
    result = simplify(equation.subs(x, sol))
    assert result == 0

Pattern 2: Symbolic to Numeric Pipeline

# 1. Define symbolic problem
x, y = symbols('x y')
expr = sin(x) + cos(y)

# 2. Manipulate symbolically
simplified = simplify(expr)
derivative = diff(simplified, x)

# 3. Convert to numerical function
f = lambdify((x, y), derivative, 'numpy')

# 4. Evaluate numerically
results = f(x_data, y_data)

Pattern 3: Document Mathematical Results

# Compute result symbolically
integral_expr = Integral(x**2, (x, 0, 1))
result = integral_expr.doit()

# Generate documentation
print(f"