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)` 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: 1. **`core-capabilities.md`**: Symbols, algebra, calculus, simplification, equation solving - Load when: Basic symbolic computation, calculus, or solving equations 2. **`matrices-linear-algebra.md`**: Matrix operations, eigenvalues, linear systems - Load when: Working with matrices or linear algebra problems 3. **`physics-mechanics.md`**: Classical mechanics, quantum mechanics, vectors, units - Load when: Physics calculations or mechanics problems 4. **`advanced-topics.md`**: Geometry, number theory, combinatorics, logic, statistics - Load when: Advanced mathematical topics beyond basic algebra and calculus 5. **`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