sympy

name sympy 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. 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) 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/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