
About
SymPy is a Python library for symbolic mathematics that enables exact computation using mathematical symbols rather than numerical approximations.
name: sympy description: "SymPy is a Python library for symbolic mathematics that enables exact computation using mathematical symbols rather than numerical approximations." license: https://github.com/sympy/sympy/blob/master/LICENSE metadata: skill-author: K-Dense Inc. risk: unknown source: "https://github.com/sympy/sympy"
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)not1.414...)
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/