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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)
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)
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 from sympy import solveset, solve, Eq
solveset(x**2 - 4, x) # {-2, 2}
solve(Eq(x**2, 4), x) # [-2, 2]
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)
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
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 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])
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
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, transformationsNumber Theory: Primes, factorization, GCD/LCM, modular arithmetic, Diophantine equationsCombinatorics: Permutations, combinations, partitions, group theoryLogic and Sets: Boolean logic, set theory, finite and infinite setsStatistics: Probability distributions, random variables, expectation, varianceSpecial Functions: Gamma, Bessel, orthogonal polynomials, hypergeometric functionsPolynomials: 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
from sympy.utilities.codegen import codegen
[(c_name, c_code), (h_name, h_header)] = codegen(
('my_func', expr), 'C'
)
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 systemsnonlinsolve: Nonlinear systemsdsolve: Differential equationssolve: General purpose (legacy, but flexible)
Reference Files Structure This skill uses modular reference files for different capabilities:
core-capabilities.md
Load when: Basic symbolic computation, calculus, or solving equations
matrices-linear-algebra.md
Load when: Working with matrices or linear algebra problems
physics-mechanics.md
Load when: Physics calculations or mechanics problems
advanced-topics.md
Load when: Advanced mathematical topics beyond basic algebra and calculus
code-generation-printing.md
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 