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) # x3 + 3*x2 + 3*x + 1 factor(x2 - 1) # (x - 1)*(x + 1)

For detailed basics: See references/core-capabilities.md

2. Calculus

Derivatives:

from sympy import diff diff(x2, x) # 2*x diff(x4, x, 3) # 24x (third derivative) diff(x**2y3, x, y) # 6xy2 (partial derivatives)

Integrals:

from sympy import integrate, oo integrate(x2, x) # x3/3 (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 + x2/2 + x3/6 + x4/24 + x5/120 + O(x**6)

For detailed calculus operations: See references/core-capabilities.md

3. Equation Solving

Algebraic equations:

from sympy import solveset, solve, Eq solveset(x2 - 4, x) # {-2, 2} solve(Eq(x2, 4), x) # [-2, 2]

Systems of equations:

from sympy import linsolve, nonlinsolve linsolve([x + y - 2, x - y], x, y) # {(1, 1)} (linear) nonlinsolve([x2 + y - 2, x + y2 - 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 = PDP^-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*(lq.diff())**2/2 - mgl(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 = 3N.x + 4N.y v2 = 1N.x + 2N.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:

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

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)

2xcos(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

• 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