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:

core-capabilities.md
: Symbols, algebra, calculus, simplification, equation solving
- Load when: Basic symbolic computation, calculus, or solving equations
matrices-linear-algebra.md
: Matrix operations, eigenvalues, linear systems
- Load when: Working with matrices or linear algebra problems
physics-mechanics.md
: Classical mechanics, quantum mechanics, vectors, units
- Load when: Physics calculations or mechanics problems
advanced-topics.md
: Geometry, number theory, combinatorics, logic, statistics
- Load when: Advanced mathematical topics beyond basic algebra and calculus
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

"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

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

sympy

NPX Install

SKILL.md Content

SymPy - Symbolic Mathematics in Python

Overview

When to Use This Skill

Core Capabilities

1. Symbolic Computation Basics

2. Calculus

3. Equation Solving

4. Matrices and Linear Algebra

5. Physics and Mechanics

6. Advanced Mathematics

7. Code Generation and Output

Working with SymPy: Best Practices

1. Always Define Symbols First

2. Use Assumptions for Better Simplification

3. Use Exact Arithmetic

4. Numerical Evaluation When Needed

5. Convert to NumPy for Performance

6. Use Appropriate Solvers

Reference Files Structure

Common Use Case Patterns

Pattern 1: Solve and Verify

Pattern 2: Symbolic to Numeric Pipeline

Pattern 3: Document Mathematical Results

Integration with Scientific Workflows

With NumPy

With Matplotlib

With SciPy

Quick Reference: Most Common Functions

Getting Started Examples

Example 1: Solve Quadratic Equation

Example 2: Calculate Derivative

Example 3: Evaluate Integral

Example 4: Matrix Eigenvalues

Example 5: Generate Python Function

Troubleshooting Common Issues

Additional Resources