sympy

Use this skill when working with symbolic mathematics in Python. This skill should be used for symbolic computation tasks including solving equations algebra...

Security Scan

The skill is an instructional wrapper for the SymPy Python library and its requested resources and instructions are consistent with that purpose, but the runtime guidance includes file I/O, pickling, code generation/compilation, and eval-able representations that carry normal-but-real risks if used with untrusted input.

Purpose & Capability

The name, description, and the SKILL.md content consistently document SymPy usage (symbolic algebra, calculus, matrices, physics, code generation). No requested binaries, env vars, or install steps contradict the declared purpose.

Instruction Scope

The instructions stay within SymPy's domain (examples for symbolic math, code generation, printing, and Jupyter integration). However, the docs include examples that perform file writes (output.tex, output.py), pickle.dump/load, autowrap/compilation (cython/f2py), and mention srepr() which can be eval()'ed to recreate expressions. These are legitimate SymPy capabilities but can be dangerous if applied to untrusted input (code execution, native compilation, arbitrary file creation, or unsafe deserialization). The SKILL.md does not direct the agent to read unrelated system files or external endpoints, nor does it request credentials.

Install Mechanism

No install spec is present (instruction-only skill). Nothing will be written to disk by the skill itself during install; risk depends on how the agent executes examples at runtime.

Credentials

The skill requires no environment variables, credentials, or config paths. The requests (none) are proportionate to an instructional SymPy skill.

Persistence & Privilege

always is false and the skill does not request persistent or elevated platform privileges. It does show patterns that could create files or compile code at runtime if the agent follows the examples, but that is a capability of SymPy examples, not a privileged installation behavior.

Scan Findings in Context

[no_code_files_to_scan] expected: The regex-based scanner had nothing to analyze because this is an instruction-only skill containing only documentation and examples; that is expected for a docs-style SymPy skill.

Assessment

This skill is coherent with its stated purpose (teaching/using SymPy). Before installing or allowing the agent to run it: (1) be aware examples demonstrate writing files, pickling, compiling native code (autowrap), and reconstructing expressions via eval()-able srepr() — avoid running these on untrusted input; (2) if you plan to let an agent execute examples autonomously, ensure the execution environment is isolated (no sensitive files accessible) and you trust any inputs that may be deserialized or eval()'ed; (3) autowrap/codegen examples can invoke a C/Fortran build toolchain and produce native code — confirm you want that capability enabled. If you only need read-only guidance, you can use the skill without running any code snippets.

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:

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, 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:

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:

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"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

1. "NameError: name 'x' is not defined"

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

2. Unexpected numerical results

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

3. Slow performance in loops

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

4. "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

5. 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

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