{"id":"a6752754-a16f-4a15-be6e-fd5a757a821a","shortId":"58kHXm","kind":"skill","title":"sympy","tagline":"SymPy is a Python library for symbolic mathematics that enables exact computation using mathematical symbols rather than numerical approximations.","description":"# SymPy - Symbolic Mathematics in Python\n\n## Overview\n\nSymPy 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.\n\n## When to Use This Skill\n\nUse this skill when:\n- Solving equations symbolically (algebraic, differential, systems of equations)\n- Performing calculus operations (derivatives, integrals, limits, series)\n- Manipulating and simplifying algebraic expressions\n- Working with matrices and linear algebra symbolically\n- Doing physics calculations (mechanics, quantum mechanics, vector analysis)\n- Number theory computations (primes, factorization, modular arithmetic)\n- Geometric calculations (2D/3D geometry, analytic geometry)\n- Converting mathematical expressions to executable code (Python, C, Fortran)\n- Generating LaTeX or other formatted mathematical output\n- Needing exact mathematical results (e.g., `sqrt(2)` not `1.414...`)\n\n## Core Capabilities\n\n### 1. Symbolic Computation Basics\n\n**Creating symbols and expressions:**\n```python\nfrom sympy import symbols, Symbol\nx, y, z = symbols('x y z')\nexpr = x**2 + 2*x + 1\n\n# With assumptions\nx = symbols('x', real=True, positive=True)\nn = symbols('n', integer=True)\n```\n\n**Simplification and manipulation:**\n```python\nfrom sympy import simplify, expand, factor, cancel\nsimplify(sin(x)**2 + cos(x)**2)  # Returns 1\nexpand((x + 1)**3)  # x**3 + 3*x**2 + 3*x + 1\nfactor(x**2 - 1)    # (x - 1)*(x + 1)\n```\n\n**For detailed basics:** See `references/core-capabilities.md`\n\n### 2. Calculus\n\n**Derivatives:**\n```python\nfrom sympy import diff\ndiff(x**2, x)        # 2*x\ndiff(x**4, x, 3)     # 24*x (third derivative)\ndiff(x**2*y**3, x, y)  # 6*x*y**2 (partial derivatives)\n```\n\n**Integrals:**\n```python\nfrom sympy import integrate, oo\nintegrate(x**2, x)              # x**3/3 (indefinite)\nintegrate(x**2, (x, 0, 1))      # 1/3 (definite)\nintegrate(exp(-x), (x, 0, oo))  # 1 (improper)\n```\n\n**Limits and Series:**\n```python\nfrom sympy import limit, series\nlimit(sin(x)/x, x, 0)  # 1\nseries(exp(x), x, 0, 6)  # 1 + x + x**2/2 + x**3/6 + x**4/24 + x**5/120 + O(x**6)\n```\n\n**For detailed calculus operations:** See `references/core-capabilities.md`\n\n### 3. Equation Solving\n\n**Algebraic equations:**\n```python\nfrom sympy import solveset, solve, Eq\nsolveset(x**2 - 4, x)  # {-2, 2}\nsolve(Eq(x**2, 4), x)  # [-2, 2]\n```\n\n**Systems of equations:**\n```python\nfrom sympy import linsolve, nonlinsolve\nlinsolve([x + y - 2, x - y], x, y)  # {(1, 1)} (linear)\nnonlinsolve([x**2 + y - 2, x + y**2 - 3], x, y)  # (nonlinear)\n```\n\n**Differential equations:**\n```python\nfrom sympy import Function, dsolve, Derivative\nf = symbols('f', cls=Function)\ndsolve(Derivative(f(x), x) - f(x), f(x))  # Eq(f(x), C1*exp(x))\n```\n\n**For detailed solving methods:** See `references/core-capabilities.md`\n\n### 4. Matrices and Linear Algebra\n\n**Matrix creation and operations:**\n```python\nfrom sympy import Matrix, eye, zeros\nM = Matrix([[1, 2], [3, 4]])\nM_inv = M**-1  # Inverse\nM.det()        # Determinant\nM.T            # Transpose\n```\n\n**Eigenvalues and eigenvectors:**\n```python\neigenvals = M.eigenvals()  # {eigenvalue: multiplicity}\neigenvects = M.eigenvects()  # [(eigenval, mult, [eigenvectors])]\nP, D = M.diagonalize()  # M = P*D*P^-1\n```\n\n**Solving linear systems:**\n```python\nA = Matrix([[1, 2], [3, 4]])\nb = Matrix([5, 6])\nx = A.solve(b)  # Solve Ax = b\n```\n\n**For comprehensive linear algebra:** See `references/matrices-linear-algebra.md`\n\n### 5. Physics and Mechanics\n\n**Classical mechanics:**\n```python\nfrom sympy.physics.mechanics import dynamicsymbols, LagrangesMethod\nfrom sympy import symbols\n\n# Define system\nq = dynamicsymbols('q')\nm, g, l = symbols('m g l')\n\n# Lagrangian (T - V)\nL = m*(l*q.diff())**2/2 - m*g*l*(1 - cos(q))\n\n# Apply Lagrange's method\nLM = LagrangesMethod(L, [q])\n```\n\n**Vector analysis:**\n```python\nfrom sympy.physics.vector import ReferenceFrame, dot, cross\nN = ReferenceFrame('N')\nv1 = 3*N.x + 4*N.y\nv2 = 1*N.x + 2*N.z\ndot(v1, v2)  # Dot product\ncross(v1, v2)  # Cross product\n```\n\n**Quantum mechanics:**\n```python\nfrom sympy.physics.quantum import Ket, Bra, Commutator\npsi = Ket('psi')\nA = Operator('A')\ncomm = Commutator(A, B).doit()\n```\n\n**For detailed physics capabilities:** See `references/physics-mechanics.md`\n\n### 6. Advanced Mathematics\n\nThe skill includes comprehensive support for:\n\n- **Geometry:** 2D/3D analytic geometry, points, lines, circles, polygons, transformations\n- **Number Theory:** Primes, factorization, GCD/LCM, modular arithmetic, Diophantine equations\n- **Combinatorics:** Permutations, combinations, partitions, group theory\n- **Logic and Sets:** Boolean logic, set theory, finite and infinite sets\n- **Statistics:** Probability distributions, random variables, expectation, variance\n- **Special Functions:** Gamma, Bessel, orthogonal polynomials, hypergeometric functions\n- **Polynomials:** Polynomial algebra, roots, factorization, Groebner bases\n\n**For detailed advanced topics:** See `references/advanced-topics.md`\n\n### 7. Code Generation and Output\n\n**Convert to executable functions:**\n```python\nfrom sympy import lambdify\nimport numpy as np\n\nexpr = x**2 + 2*x + 1\nf = lambdify(x, expr, 'numpy')  # Create NumPy function\nx_vals = np.linspace(0, 10, 100)\ny_vals = f(x_vals)  # Fast numerical evaluation\n```\n\n**Generate C/Fortran code:**\n```python\nfrom sympy.utilities.codegen import codegen\n[(c_name, c_code), (h_name, h_header)] = codegen(\n    ('my_func', expr), 'C'\n)\n```\n\n**LaTeX output:**\n```python\nfrom sympy import latex\nlatex_str = latex(expr)  # Convert to LaTeX for documents\n```\n\n**For comprehensive code generation:** See `references/code-generation-printing.md`\n\n## Working with SymPy: Best Practices\n\n### 1. Always Define Symbols First\n\n```python\nfrom sympy import symbols\nx, y, z = symbols('x y z')\n# Now x, y, z can be used in expressions\n```\n\n### 2. Use Assumptions for Better Simplification\n\n```python\nx = symbols('x', positive=True, real=True)\nsqrt(x**2)  # Returns x (not Abs(x)) due to positive assumption\n```\n\nCommon assumptions: `real`, `positive`, `negative`, `integer`, `rational`, `complex`, `even`, `odd`\n\n### 3. Use Exact Arithmetic\n\n```python\nfrom sympy import Rational, S\n# Correct (exact):\nexpr = Rational(1, 2) * x\nexpr = S(1)/2 * x\n\n# Incorrect (floating-point):\nexpr = 0.5 * x  # Creates approximate value\n```\n\n### 4. Numerical Evaluation When Needed\n\n```python\nfrom sympy import pi, sqrt\nresult = sqrt(8) + pi\nresult.evalf()    # 5.96371554103586\nresult.evalf(50)  # 50 digits of precision\n```\n\n### 5. Convert to NumPy for Performance\n\n```python\n# Slow for many evaluations:\nfor x_val in range(1000):\n    result = expr.subs(x, x_val).evalf()\n\n# Fast:\nf = lambdify(x, expr, 'numpy')\nresults = f(np.arange(1000))\n```\n\n### 6. Use Appropriate Solvers\n\n- `solveset`: Algebraic equations (primary)\n- `linsolve`: Linear systems\n- `nonlinsolve`: Nonlinear systems\n- `dsolve`: Differential equations\n- `solve`: General purpose (legacy, but flexible)\n\n## Reference Files Structure\n\nThis skill uses modular reference files for different capabilities:\n\n1. **`core-capabilities.md`**: Symbols, algebra, calculus, simplification, equation solving\n   - Load when: Basic symbolic computation, calculus, or solving equations\n\n2. **`matrices-linear-algebra.md`**: Matrix operations, eigenvalues, linear systems\n   - Load when: Working with matrices or linear algebra problems\n\n3. **`physics-mechanics.md`**: Classical mechanics, quantum mechanics, vectors, units\n   - Load when: Physics calculations or mechanics problems\n\n4. **`advanced-topics.md`**: Geometry, number theory, combinatorics, logic, statistics\n   - Load when: Advanced mathematical topics beyond basic algebra and calculus\n\n5. **`code-generation-printing.md`**: Lambdify, codegen, LaTeX output, printing\n   - Load when: Converting expressions to code or generating formatted output\n\n## Common Use Case Patterns\n\n### Pattern 1: Solve and Verify\n\n```python\nfrom sympy import symbols, solve, simplify\nx = symbols('x')\n\n# Solve equation\nequation = x**2 - 5*x + 6\nsolutions = solve(equation, x)  # [2, 3]\n\n# Verify solutions\nfor sol in solutions:\n    result = simplify(equation.subs(x, sol))\n    assert result == 0\n```\n\n### Pattern 2: Symbolic to Numeric Pipeline\n\n```python\n# 1. Define symbolic problem\nx, y = symbols('x y')\nexpr = sin(x) + cos(y)\n\n# 2. Manipulate symbolically\nsimplified = simplify(expr)\nderivative = diff(simplified, x)\n\n# 3. Convert to numerical function\nf = lambdify((x, y), derivative, 'numpy')\n\n# 4. Evaluate numerically\nresults = f(x_data, y_data)\n```\n\n### Pattern 3: Document Mathematical Results\n\n```python\n# Compute result symbolically\nintegral_expr = Integral(x**2, (x, 0, 1))\nresult = integral_expr.doit()\n\n# Generate documentation\nprint(f\"LaTeX: {latex(integral_expr)} = {latex(result)}\")\nprint(f\"Pretty: {pretty(integral_expr)} = {pretty(result)}\")\nprint(f\"Numerical: {result.evalf()}\")\n```\n\n## Integration with Scientific Workflows\n\n### With NumPy\n\n```python\nimport numpy as np\nfrom sympy import symbols, lambdify\n\nx = symbols('x')\nexpr = x**2 + 2*x + 1\n\nf = lambdify(x, expr, 'numpy')\nx_array = np.linspace(-5, 5, 100)\ny_array = f(x_array)\n```\n\n### With Matplotlib\n\n```python\nimport matplotlib.pyplot as plt\nimport numpy as np\nfrom sympy import symbols, lambdify, sin\n\nx = symbols('x')\nexpr = sin(x) / x\n\nf = lambdify(x, expr, 'numpy')\nx_vals = np.linspace(-10, 10, 1000)\ny_vals = f(x_vals)\n\nplt.plot(x_vals, y_vals)\nplt.show()\n```\n\n### With SciPy\n\n```python\nfrom scipy.optimize import fsolve\nfrom sympy import symbols, lambdify\n\n# Define equation symbolically\nx = symbols('x')\nequation = x**3 - 2*x - 5\n\n# Convert to numerical function\nf = lambdify(x, equation, 'numpy')\n\n# Solve numerically with initial guess\nsolution = fsolve(f, 2)\n```\n\n## Quick Reference: Most Common Functions\n\n```python\n# Symbols\nfrom sympy import symbols, Symbol\nx, y = symbols('x y')\n\n# Basic operations\nfrom sympy import simplify, expand, factor, collect, cancel\nfrom sympy import sqrt, exp, log, sin, cos, tan, pi, E, I, oo\n\n# Calculus\nfrom sympy import diff, integrate, limit, series, Derivative, Integral\n\n# Solving\nfrom sympy import solve, solveset, linsolve, nonlinsolve, dsolve\n\n# Matrices\nfrom sympy import Matrix, eye, zeros, ones, diag\n\n# Logic and sets\nfrom sympy import And, Or, Not, Implies, FiniteSet, Interval, Union\n\n# Output\nfrom sympy import latex, pprint, lambdify, init_printing\n\n# Utilities\nfrom sympy import evalf, N, nsimplify\n```\n\n## Getting Started Examples\n\n### Example 1: Solve Quadratic Equation\n```python\nfrom sympy import symbols, solve, sqrt\nx = symbols('x')\nsolution = solve(x**2 - 5*x + 6, x)\n# [2, 3]\n```\n\n### Example 2: Calculate Derivative\n```python\nfrom sympy import symbols, diff, sin\nx = symbols('x')\nf = sin(x**2)\ndf_dx = diff(f, x)\n# 2*x*cos(x**2)\n```\n\n### Example 3: Evaluate Integral\n```python\nfrom sympy import symbols, integrate, exp\nx = symbols('x')\nintegral = integrate(x * exp(-x**2), (x, 0, oo))\n# 1/2\n```\n\n### Example 4: Matrix Eigenvalues\n```python\nfrom sympy import Matrix\nM = Matrix([[1, 2], [2, 1]])\neigenvals = M.eigenvals()\n# {3: 1, -1: 1}\n```\n\n### Example 5: Generate Python Function\n```python\nfrom sympy import symbols, lambdify\nimport numpy as np\nx = symbols('x')\nexpr = x**2 + 2*x + 1\nf = lambdify(x, expr, 'numpy')\nf(np.array([1, 2, 3]))\n# array([ 4,  9, 16])\n```\n\n## Troubleshooting Common Issues\n\n1. **\"NameError: name 'x' is not defined\"**\n   - Solution: Always define symbols using `symbols()` before use\n\n2. **Unexpected numerical results**\n   - Issue: Using floating-point numbers like `0.5` instead of `Rational(1, 2)`\n   - Solution: Use `Rational()` or `S()` for exact arithmetic\n\n3. **Slow performance in loops**\n   - Issue: Using `subs()` and `evalf()` repeatedly\n   - Solution: Use `lambdify()` to create a fast numerical function\n\n4. **\"Can't solve this equation\"**\n   - Try different solvers: `solve`, `solveset`, `nsolve` (numerical)\n   - Check if the equation is solvable algebraically\n   - Use numerical methods if no closed-form solution exists\n\n5. **Simplification not working as expected**\n   - Try different simplification functions: `simplify`, `factor`, `expand`, `trigsimp`\n   - Add assumptions to symbols (e.g., `positive=True`)\n   - Use `simplify(expr, force=True)` for aggressive simplification\n\n## Additional Resources\n\n- Official Documentation: https://docs.sympy.org/\n- Tutorial: https://docs.sympy.org/latest/tutorials/intro-tutorial/index.html\n- API Reference: https://docs.sympy.org/latest/reference/index.html\n- Examples: https://github.com/sympy/sympy/tree/master/examples\n\n## Limitations\n- Use this skill only when the task clearly matches the scope described above.\n- Do not treat the output as a substitute for environment-specific validation, testing, or expert review.\n- Stop and ask for clarification if required inputs, permissions, safety boundaries, or success criteria are 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