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sympy

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.

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Download and install this skill https://openskills.cc/api/download?slug=k-dense-ai-skills-sympy&locale=en&source=copy
name:sympydescription: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/LICENSEallowed-tools:Read Write Edit Bashcompatibility:Requires Python 3.9+ and SymPy 1.14+. Optional NumPy/SciPy/Matplotlib for lambdify examples; C/Fortran compiler for autowrap/codegen.metadata:[object Object]

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

    1. Symbolic Computation Basics

    Creating symbols and expressions:

    from sympy import symbols, Symbol
x, y, z = symbols('x y z')
expr = x**2 + 2*x + 1

# With assumptions
x = symbols('x', real=True, positive=True)
n = symbols('n', integer=True)

    Simplification and manipulation:

    from sympy import simplify, expand, factor, cancel
simplify(sin(x)**2 + cos(x)**2)  # Returns 1
expand((x + 1)**3)  # x**3 + 3*x**2 + 3*x + 1
factor(x**2 - 1)    # (x - 1)*(x + 1)

    For detailed basics: See references/core-capabilities.md

    2. Calculus

    Derivatives:

    from sympy import diff
diff(x**2, x)        # 2*x
diff(x**4, x, 3)     # 24*x (third derivative)
diff(x**2*y**3, x, y)  # 6*x*y**2 (partial derivatives)

    Integrals:

    from sympy import integrate, oo
integrate(x**2, x)              # x**3/3 (indefinite)
integrate(x**2, (x, 0, 1))      # 1/3 (definite)
integrate(exp(-x), (x, 0, oo))  # 1 (improper)

    Limits and Series:

    from sympy import limit, series
limit(sin(x)/x, x, 0)  # 1
series(exp(x), x, 0, 6)  # 1 + x + x**2/2 + x**3/6 + x**4/24 + x**5/120 + O(x**6)

    For detailed calculus operations: See references/core-capabilities.md

    3. Equation Solving

    Algebraic equations:

    from sympy import solveset, solve, Eq
solveset(x**2 - 4, x)  # {-2, 2}
solve(Eq(x**2, 4), x)  # [-2, 2]

    Systems of equations:

    from sympy import linsolve, nonlinsolve
linsolve([x + y - 2, x - y], x, y)  # {(1, 1)} (linear)
nonlinsolve([x**2 + y - 2, x + y**2 - 3], x, y)  # (nonlinear)

    Differential equations:

    from sympy import Function, dsolve, Derivative
f = symbols('f', cls=Function)
dsolve(Derivative(f(x), x) - f(x), f(x))  # Eq(f(x), C1*exp(x))

    For detailed solving methods: See references/core-capabilities.md

    4. Matrices and Linear Algebra

    Matrix creation and operations:

    from sympy import Matrix, eye, zeros
M = Matrix([[1, 2], [3, 4]])
M_inv = M**-1  # Inverse
M.det()        # Determinant
M.T            # Transpose

    Eigenvalues and eigenvectors:

    eigenvals = M.eigenvals()  # {eigenvalue: multiplicity}
eigenvects = M.eigenvects()  # [(eigenval, mult, [eigenvectors])]
P, D = M.diagonalize()  # M = P*D*P^-1

    Solving linear systems:

    A = Matrix([[1, 2], [3, 4]])
b = Matrix([5, 6])
x = A.solve(b)  # Solve Ax = b

    For comprehensive linear algebra: See references/matrices-linear-algebra.md

    5. Physics and Mechanics

    Classical mechanics:

    from sympy.physics.mechanics import dynamicsymbols, LagrangesMethod
from sympy import symbols

# Define system
q = dynamicsymbols('q')
m, g, l = symbols('m g l')

# Lagrangian (T - V)
L = m*(l*q.diff())**2/2 - m*g*l*(1 - cos(q))

# Apply Lagrange's method
LM = LagrangesMethod(L, [q])

    Vector analysis:

    from sympy.physics.vector import ReferenceFrame, dot, cross
N = ReferenceFrame('N')
v1 = 3*N.x + 4*N.y
v2 = 1*N.x + 2*N.z
dot(v1, v2)  # Dot product
cross(v1, v2)  # Cross product

    Quantum mechanics:

    from sympy.physics.quantum import Ket, Bra, Operator, Commutator
A, B = Operator('A'), Operator('B')
psi = Ket('psi')
comm = Commutator(A, B).doit()

    For detailed physics capabilities: See references/physics-mechanics.md

    6. Advanced Mathematics

    The skill includes comprehensive support for:

  • Geometry: 2D/3D analytic geometry, points, lines, circles, polygons, transformations

  • Number Theory: Primes, factorization, GCD/LCM, modular arithmetic, Diophantine equations

  • Combinatorics: Permutations, combinations, partitions, group theory

  • Logic and Sets: Boolean logic, set theory, finite and infinite sets

  • Statistics: Probability distributions, random variables, expectation, variance

  • Special Functions: Gamma, Bessel, orthogonal polynomials, hypergeometric functions

  • Polynomials: Polynomial algebra, roots, factorization, Groebner bases

    • For detailed advanced topics: See references/advanced-topics.md

    7. Code Generation and Output

    Convert to executable functions:

    from sympy import lambdify
import numpy as np

expr = x**2 + 2*x + 1
f = lambdify(x, expr, 'numpy')  # Create NumPy function
x_vals = np.linspace(0, 10, 100)
y_vals = f(x_vals)  # Fast numerical evaluation

    Generate C/Fortran code:

    from sympy.utilities.codegen import codegen
[(c_name, c_code), (h_name, h_header)] = codegen(
    ('my_func', expr), 'C'
)

    LaTeX output:

    from sympy import latex
latex_str = latex(expr)  # Convert to LaTeX for documents

    For comprehensive code generation: See references/code-generation-printing.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:

  • core-capabilities.md: Symbols, algebra, calculus, simplification, equation solving

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

  • matrices-linear-algebra.md: Matrix operations, eigenvalues, linear systems

    • - Load when: Working with matrices or linear algebra problems

  • physics-mechanics.md: Classical mechanics, quantum mechanics, vectors, units

    • - Load when: Physics calculations or mechanics problems

  • advanced-topics.md: Geometry, number theory, combinatorics, logic, statistics

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

  • 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"LaTeX: {latex(integral_expr)} = {latex(result)}")
print(f"Pretty: {pretty(integral_expr)} = {pretty(result)}")
print(f"Numerical: {result.evalf()}")

    Integration with Scientific Workflows

    With NumPy

    import numpy as np
from sympy import symbols, lambdify

x = symbols('x')
expr = x**2 + 2*x + 1

f = lambdify(x, expr, 'numpy')
x_array = np.linspace(-5, 5, 100)
y_array = f(x_array)

    With Matplotlib

    import matplotlib.pyplot as plt
import numpy as np
from sympy import symbols, lambdify, sin

x = symbols('x')
expr = sin(x) / x

f = lambdify(x, expr, 'numpy')
x_vals = np.linspace(-10, 10, 1000)
y_vals = f(x_vals)

plt.plot(x_vals, y_vals)
plt.show()

    With SciPy

    from scipy.optimize import fsolve
from sympy import symbols, lambdify

# Define equation symbolically
x = symbols('x')
equation = x**3 - 2*x - 5

# Convert to numerical function
f = lambdify(x, equation, 'numpy')

# Solve numerically with initial guess
solution = fsolve(f, 2)

    Quick Reference: Most Common Functions

    # Symbols
from sympy import symbols, Symbol
x, y = symbols('x y')

# Basic operations
from sympy import simplify, expand, factor, collect, cancel
from sympy import sqrt, exp, log, sin, cos, tan, pi, E, I, oo

# Calculus
from sympy import diff, integrate, limit, series, Derivative, Integral

# Solving
from sympy import solve, solveset, linsolve, nonlinsolve, dsolve

# Matrices
from sympy import Matrix, eye, zeros, ones, diag

# Logic and sets
from sympy import And, Or, Not, Implies, FiniteSet, Interval, Union

# Output
from sympy import latex, pprint, lambdify, init_printing

# Utilities
from sympy import evalf, N, nsimplify

    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])

    Troubleshooting Common Issues

  • "NameError: name 'x' is not defined"

    • - Solution: Always define symbols using symbols() before use

  • Unexpected numerical results

    • - Issue: Using floating-point numbers like 0.5 instead of Rational(1, 2)
    - Solution: Use Rational() or S() for exact arithmetic

  • Slow performance in loops

    • - Issue: Using subs() and evalf() repeatedly
    - Solution: Use lambdify() to create a fast numerical function

  • "Can't solve this equation"

    • - Try different solvers: solve, solveset, nsolve (numerical)
    - Check if the equation is solvable algebraically
    - Use numerical methods if no closed-form solution exists

  • Simplification not working as expected

    • - Try different simplification functions: simplify, factor, expand, trigsimp
    - Add assumptions to symbols (e.g., positive=True)
    - Use simplify(expr, force=True) for aggressive simplification

    Additional Resources

  • Official Documentation: https://docs.sympy.org/

  • Tutorial: https://docs.sympy.org/latest/tutorials/intro-tutorial/index.html

  • API Reference: https://docs.sympy.org/latest/reference/index.html

  • Examples: https://github.com/sympy/sympy/tree/master/examples