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Zhongwei Li
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skills/sympy/SKILL.md
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skills/sympy/SKILL.md
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name: sympy
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description: 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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---
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# SymPy - Symbolic Mathematics in Python
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## Overview
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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.
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## When to Use This Skill
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Use this skill when:
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- Solving equations symbolically (algebraic, differential, systems of equations)
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- Performing calculus operations (derivatives, integrals, limits, series)
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- Manipulating and simplifying algebraic expressions
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- Working with matrices and linear algebra symbolically
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- Doing physics calculations (mechanics, quantum mechanics, vector analysis)
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- Number theory computations (primes, factorization, modular arithmetic)
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- Geometric calculations (2D/3D geometry, analytic geometry)
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- Converting mathematical expressions to executable code (Python, C, Fortran)
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- Generating LaTeX or other formatted mathematical output
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- Needing exact mathematical results (e.g., `sqrt(2)` not `1.414...`)
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## Core Capabilities
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### 1. Symbolic Computation Basics
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**Creating symbols and expressions:**
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```python
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from sympy import symbols, Symbol
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x, y, z = symbols('x y z')
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expr = x**2 + 2*x + 1
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# With assumptions
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x = symbols('x', real=True, positive=True)
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n = symbols('n', integer=True)
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```
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**Simplification and manipulation:**
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```python
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from sympy import simplify, expand, factor, cancel
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simplify(sin(x)**2 + cos(x)**2) # Returns 1
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expand((x + 1)**3) # x**3 + 3*x**2 + 3*x + 1
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factor(x**2 - 1) # (x - 1)*(x + 1)
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```
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**For detailed basics:** See `references/core-capabilities.md`
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### 2. Calculus
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**Derivatives:**
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```python
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from sympy import diff
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diff(x**2, x) # 2*x
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diff(x**4, x, 3) # 24*x (third derivative)
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diff(x**2*y**3, x, y) # 6*x*y**2 (partial derivatives)
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```
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**Integrals:**
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```python
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from sympy import integrate, oo
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integrate(x**2, x) # x**3/3 (indefinite)
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integrate(x**2, (x, 0, 1)) # 1/3 (definite)
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integrate(exp(-x), (x, 0, oo)) # 1 (improper)
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```
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**Limits and Series:**
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```python
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from sympy import limit, series
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limit(sin(x)/x, x, 0) # 1
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series(exp(x), x, 0, 6) # 1 + x + x**2/2 + x**3/6 + x**4/24 + x**5/120 + O(x**6)
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```
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**For detailed calculus operations:** See `references/core-capabilities.md`
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### 3. Equation Solving
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**Algebraic equations:**
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```python
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from sympy import solveset, solve, Eq
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solveset(x**2 - 4, x) # {-2, 2}
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solve(Eq(x**2, 4), x) # [-2, 2]
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```
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**Systems of equations:**
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```python
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from sympy import linsolve, nonlinsolve
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linsolve([x + y - 2, x - y], x, y) # {(1, 1)} (linear)
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nonlinsolve([x**2 + y - 2, x + y**2 - 3], x, y) # (nonlinear)
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```
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**Differential equations:**
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```python
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from sympy import Function, dsolve, Derivative
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f = symbols('f', cls=Function)
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dsolve(Derivative(f(x), x) - f(x), f(x)) # Eq(f(x), C1*exp(x))
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```
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**For detailed solving methods:** See `references/core-capabilities.md`
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### 4. Matrices and Linear Algebra
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**Matrix creation and operations:**
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```python
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from sympy import Matrix, eye, zeros
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M = Matrix([[1, 2], [3, 4]])
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M_inv = M**-1 # Inverse
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M.det() # Determinant
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M.T # Transpose
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```
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**Eigenvalues and eigenvectors:**
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```python
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eigenvals = M.eigenvals() # {eigenvalue: multiplicity}
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eigenvects = M.eigenvects() # [(eigenval, mult, [eigenvectors])]
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P, D = M.diagonalize() # M = P*D*P^-1
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```
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**Solving linear systems:**
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```python
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A = Matrix([[1, 2], [3, 4]])
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b = Matrix([5, 6])
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x = A.solve(b) # Solve Ax = b
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```
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**For comprehensive linear algebra:** See `references/matrices-linear-algebra.md`
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### 5. Physics and Mechanics
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**Classical mechanics:**
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```python
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from sympy.physics.mechanics import dynamicsymbols, LagrangesMethod
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from sympy import symbols
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# Define system
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q = dynamicsymbols('q')
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m, g, l = symbols('m g l')
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# Lagrangian (T - V)
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L = m*(l*q.diff())**2/2 - m*g*l*(1 - cos(q))
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# Apply Lagrange's method
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LM = LagrangesMethod(L, [q])
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```
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**Vector analysis:**
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```python
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from sympy.physics.vector import ReferenceFrame, dot, cross
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N = ReferenceFrame('N')
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v1 = 3*N.x + 4*N.y
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v2 = 1*N.x + 2*N.z
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dot(v1, v2) # Dot product
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cross(v1, v2) # Cross product
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```
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**Quantum mechanics:**
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```python
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from sympy.physics.quantum import Ket, Bra, Commutator
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psi = Ket('psi')
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A = Operator('A')
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comm = Commutator(A, B).doit()
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```
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**For detailed physics capabilities:** See `references/physics-mechanics.md`
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### 6. Advanced Mathematics
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The skill includes comprehensive support for:
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- **Geometry:** 2D/3D analytic geometry, points, lines, circles, polygons, transformations
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- **Number Theory:** Primes, factorization, GCD/LCM, modular arithmetic, Diophantine equations
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- **Combinatorics:** Permutations, combinations, partitions, group theory
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- **Logic and Sets:** Boolean logic, set theory, finite and infinite sets
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- **Statistics:** Probability distributions, random variables, expectation, variance
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- **Special Functions:** Gamma, Bessel, orthogonal polynomials, hypergeometric functions
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- **Polynomials:** Polynomial algebra, roots, factorization, Groebner bases
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**For detailed advanced topics:** See `references/advanced-topics.md`
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### 7. Code Generation and Output
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**Convert to executable functions:**
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```python
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from sympy import lambdify
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import numpy as np
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expr = x**2 + 2*x + 1
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f = lambdify(x, expr, 'numpy') # Create NumPy function
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x_vals = np.linspace(0, 10, 100)
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y_vals = f(x_vals) # Fast numerical evaluation
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```
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**Generate C/Fortran code:**
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```python
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from sympy.utilities.codegen import codegen
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[(c_name, c_code), (h_name, h_header)] = codegen(
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('my_func', expr), 'C'
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)
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```
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**LaTeX output:**
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```python
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from sympy import latex
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latex_str = latex(expr) # Convert to LaTeX for documents
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```
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**For comprehensive code generation:** See `references/code-generation-printing.md`
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## Working with SymPy: Best Practices
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### 1. Always Define Symbols First
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```python
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from sympy import symbols
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x, y, z = symbols('x y z')
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# Now x, y, z can be used in expressions
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```
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### 2. Use Assumptions for Better Simplification
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```python
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x = symbols('x', positive=True, real=True)
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sqrt(x**2) # Returns x (not Abs(x)) due to positive assumption
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```
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Common assumptions: `real`, `positive`, `negative`, `integer`, `rational`, `complex`, `even`, `odd`
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### 3. Use Exact Arithmetic
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```python
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from sympy import Rational, S
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# Correct (exact):
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expr = Rational(1, 2) * x
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expr = S(1)/2 * x
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# Incorrect (floating-point):
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expr = 0.5 * x # Creates approximate value
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```
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### 4. Numerical Evaluation When Needed
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```python
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from sympy import pi, sqrt
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result = sqrt(8) + pi
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result.evalf() # 5.96371554103586
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result.evalf(50) # 50 digits of precision
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```
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### 5. Convert to NumPy for Performance
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```python
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# Slow for many evaluations:
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for x_val in range(1000):
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result = expr.subs(x, x_val).evalf()
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# Fast:
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f = lambdify(x, expr, 'numpy')
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results = f(np.arange(1000))
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```
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### 6. Use Appropriate Solvers
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- `solveset`: Algebraic equations (primary)
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- `linsolve`: Linear systems
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- `nonlinsolve`: Nonlinear systems
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- `dsolve`: Differential equations
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- `solve`: General purpose (legacy, but flexible)
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## Reference Files Structure
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This skill uses modular reference files for different capabilities:
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1. **`core-capabilities.md`**: Symbols, algebra, calculus, simplification, equation solving
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- Load when: Basic symbolic computation, calculus, or solving equations
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2. **`matrices-linear-algebra.md`**: Matrix operations, eigenvalues, linear systems
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- Load when: Working with matrices or linear algebra problems
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3. **`physics-mechanics.md`**: Classical mechanics, quantum mechanics, vectors, units
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- Load when: Physics calculations or mechanics problems
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4. **`advanced-topics.md`**: Geometry, number theory, combinatorics, logic, statistics
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- Load when: Advanced mathematical topics beyond basic algebra and calculus
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5. **`code-generation-printing.md`**: Lambdify, codegen, LaTeX output, printing
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- Load when: Converting expressions to code or generating formatted output
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## Common Use Case Patterns
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### Pattern 1: Solve and Verify
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```python
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from sympy import symbols, solve, simplify
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x = symbols('x')
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# Solve equation
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equation = x**2 - 5*x + 6
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solutions = solve(equation, x) # [2, 3]
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# Verify solutions
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for sol in solutions:
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result = simplify(equation.subs(x, sol))
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assert result == 0
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```
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### Pattern 2: Symbolic to Numeric Pipeline
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```python
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# 1. Define symbolic problem
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x, y = symbols('x y')
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expr = sin(x) + cos(y)
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# 2. Manipulate symbolically
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simplified = simplify(expr)
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derivative = diff(simplified, x)
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# 3. Convert to numerical function
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f = lambdify((x, y), derivative, 'numpy')
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# 4. Evaluate numerically
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results = f(x_data, y_data)
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```
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### Pattern 3: Document Mathematical Results
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```python
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# Compute result symbolically
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integral_expr = Integral(x**2, (x, 0, 1))
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result = integral_expr.doit()
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# Generate documentation
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print(f"LaTeX: {latex(integral_expr)} = {latex(result)}")
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print(f"Pretty: {pretty(integral_expr)} = {pretty(result)}")
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print(f"Numerical: {result.evalf()}")
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```
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## Integration with Scientific Workflows
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### With NumPy
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```python
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import numpy as np
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from sympy import symbols, lambdify
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x = symbols('x')
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expr = x**2 + 2*x + 1
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f = lambdify(x, expr, 'numpy')
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x_array = np.linspace(-5, 5, 100)
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y_array = f(x_array)
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```
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### With Matplotlib
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```python
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import matplotlib.pyplot as plt
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import numpy as np
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from sympy import symbols, lambdify, sin
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x = symbols('x')
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expr = sin(x) / x
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f = lambdify(x, expr, 'numpy')
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x_vals = np.linspace(-10, 10, 1000)
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y_vals = f(x_vals)
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plt.plot(x_vals, y_vals)
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plt.show()
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```
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### With SciPy
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```python
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from scipy.optimize import fsolve
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from sympy import symbols, lambdify
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# Define equation symbolically
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x = symbols('x')
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equation = x**3 - 2*x - 5
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# Convert to numerical function
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f = lambdify(x, equation, 'numpy')
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# Solve numerically with initial guess
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solution = fsolve(f, 2)
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```
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## Quick Reference: Most Common Functions
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```python
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# Symbols
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from sympy import symbols, Symbol
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x, y = symbols('x y')
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# Basic operations
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from sympy import simplify, expand, factor, collect, cancel
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from sympy import sqrt, exp, log, sin, cos, tan, pi, E, I, oo
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# Calculus
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from sympy import diff, integrate, limit, series, Derivative, Integral
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# Solving
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from sympy import solve, solveset, linsolve, nonlinsolve, dsolve
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# Matrices
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from sympy import Matrix, eye, zeros, ones, diag
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# Logic and sets
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from sympy import And, Or, Not, Implies, FiniteSet, Interval, Union
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# Output
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from sympy import latex, pprint, lambdify, init_printing
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# Utilities
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from sympy import evalf, N, nsimplify
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```
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## Getting Started Examples
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### Example 1: Solve Quadratic Equation
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```python
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from sympy import symbols, solve, sqrt
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x = symbols('x')
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solution = solve(x**2 - 5*x + 6, x)
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# [2, 3]
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```
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### Example 2: Calculate Derivative
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```python
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from sympy import symbols, diff, sin
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x = symbols('x')
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f = sin(x**2)
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df_dx = diff(f, x)
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# 2*x*cos(x**2)
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```
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### Example 3: Evaluate Integral
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```python
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from sympy import symbols, integrate, exp
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x = symbols('x')
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integral = integrate(x * exp(-x**2), (x, 0, oo))
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# 1/2
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```
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### Example 4: Matrix Eigenvalues
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```python
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from sympy import Matrix
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M = Matrix([[1, 2], [2, 1]])
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eigenvals = M.eigenvals()
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# {3: 1, -1: 1}
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```
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### Example 5: Generate Python Function
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```python
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from sympy import symbols, lambdify
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import numpy as np
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x = symbols('x')
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expr = x**2 + 2*x + 1
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f = lambdify(x, expr, 'numpy')
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f(np.array([1, 2, 3]))
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# array([ 4, 9, 16])
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```
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## Troubleshooting Common Issues
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1. **"NameError: name 'x' is not defined"**
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- Solution: Always define symbols using `symbols()` before use
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2. **Unexpected numerical results**
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- Issue: Using floating-point numbers like `0.5` instead of `Rational(1, 2)`
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- Solution: Use `Rational()` or `S()` for exact arithmetic
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3. **Slow performance in loops**
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- Issue: Using `subs()` and `evalf()` repeatedly
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- Solution: Use `lambdify()` to create a fast numerical function
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4. **"Can't solve this equation"**
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- Try different solvers: `solve`, `solveset`, `nsolve` (numerical)
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- Check if the equation is solvable algebraically
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- Use numerical methods if no closed-form solution exists
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5. **Simplification not working as expected**
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- Try different simplification functions: `simplify`, `factor`, `expand`, `trigsimp`
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- Add assumptions to symbols (e.g., `positive=True`)
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- Use `simplify(expr, force=True)` for aggressive simplification
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## Additional Resources
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- Official Documentation: https://docs.sympy.org/
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- Tutorial: https://docs.sympy.org/latest/tutorials/intro-tutorial/index.html
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- API Reference: https://docs.sympy.org/latest/reference/index.html
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- Examples: https://github.com/sympy/sympy/tree/master/examples
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