davila7/claude-code-templates/sympy

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Use this skill when working with symbolic mathematics in Python. This skill should be used for symbolic computation tasks including solving equations algebraically, performing calculus operations (derivatives, integrals, limits), manipulating algebraic expressions, working with matrices symbolically, physics calculations, number theory problems, geometry computations, and generating executable code from mathematical expressions. Apply this skill when the user needs exact symbolic results rather than numerical approximations, or when working with mathematical formulas that contain variables and parameters.

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npx vskill@latest install davila7/claude-code-templates/sympy

SKILL.md

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.

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:

python
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:

python
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:

python
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:

python
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:

python
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:

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

Systems of equations:

python
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:

python
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:

python
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:

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

Solving linear systems:

python
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:

python
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:

python
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:

python
from sympy.physics.quantum import Ket, Bra, Commutator
psi = Ket('psi')
A = Operator('A')
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:

python
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:

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

LaTeX output:

python
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
python
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
python
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
python
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
python
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
python
# 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
  1. `matrices-linear-algebra.md`: Matrix operations, eigenvalues, linear systems
  • Load when: Working with matrices or linear algebra problems
  1. `physics-mechanics.md`: Classical mechanics, quantum mechanics, vectors, units
  • Load when: Physics calculations or mechanics problems
  1. `advanced-topics.md`: Geometry, number theory, combinatorics, logic, statistics
  • Load when: Advanced mathematical topics beyond basic algebra and calculus
  1. `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
python
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
python
# 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
python
# 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
python
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
python
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
python
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

python
# 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
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])

Troubleshooting Common Issues

  1. "NameError: name 'x' is not defined"
  • Solution: Always define symbols using symbols() before use
  1. Unexpected numerical results
  • Issue: Using floating-point numbers like 0.5 instead of Rational(1, 2)
  • Solution: Use Rational() or S() for exact arithmetic
  1. Slow performance in loops
  • Issue: Using subs() and evalf() repeatedly
  • Solution: Use lambdify() to create a fast numerical function

This is the start of the file. Read the full SKILL.md on GitHub.

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Popularity

Repository stars
23,956
Repository forks
2,333
Repository npm downloads, 7 days
6,860
7-day trend score
71

Stars and forks are for the whole GitHub repository, not this skill alone. Downloads are for the repository's npm package. Refreshed 1 Apr 2026, 6 months ago.

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