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Python library for symbolic mathematics — exact algebra, calculus, equation solving, matrices, and LaTeX/code generation instead of numeric approximation. USE WHEN a research task needs exact symbolic results: solving equations, derivatives/integrals, simplification, or deriving closed-form expressions.
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Arcplume runs Grok through the Grok Build CLI's own OAuth-authenticated session (grok login) for image generation, with strict preflight validation, secret-safe handling, and headless CLI-driven execution -- no separate XAI_API_KEY billing. Video falls back to the billed xAI API. USE WHEN a user wants to generate an image via a locally logged-in Grok Build CLI session, e.g. 'generate an image with grok', 'use grok build', or 'use my logged-in grok session'.
Shared reference for the Selemene cluster: the two report surfaces (deterministic Rust reports vs. narrative witness-pipeline readings), the @selemene/bridge CLI contract, the output manifest format, and non-prescriptive witnessing tone. USE WHEN deciding which Selemene surface to invoke or when routing between birth/compatibility/transit reports and solo/dyadic readings.
Route Selemene Engine tasks to the right surface: deterministic reports (birth/compatibility/transit) via selemene-core and the @selemene/bridge CLI, or narrative witness readings via selemene-report. USE WHEN the user wants anything under the Selemene/Noesis umbrella but has not named the exact surface.
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基于 SOC 职业分类
| name | sympy |
| description | Python library for symbolic mathematics — exact algebra, calculus, equation solving, matrices, and LaTeX/code generation instead of numeric approximation. USE WHEN a research task needs exact symbolic results: solving equations, derivatives/integrals, simplification, or deriving closed-form expressions. |
| cluster | research-knowledge |
| version | 1.0.0 |
| origin | antigravity-awesome-skills (MIT) |
| license | https://github.com/sympy/sympy/blob/master/LICENSE |
| metadata | {"skill-author":"K-Dense Inc."} |
| risk | unknown |
| source | https://github.com/sympy/sympy |
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.
Use this skill when:
sqrt(2) not 1.414...)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)
For detailed basics: See references/core-capabilities.md
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
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
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
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
The skill includes comprehensive support for:
For detailed advanced topics: See references/advanced-topics.md
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
from sympy import symbols
x, y, z = symbols('x y z')
# Now x, y, z can be used in expressions
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
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
from sympy import pi, sqrt
result = sqrt(8) + pi
result.evalf() # 5.96371554103586
result.evalf(50) # 50 digits of precision
# 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))
solveset: Algebraic equations (primary)linsolve: Linear systemsnonlinsolve: Nonlinear systemsdsolve: Differential equationssolve: General purpose (legacy, but flexible)This skill uses modular reference files for different capabilities:
core-capabilities.md: Symbols, algebra, calculus, simplification, equation solving
matrices-linear-algebra.md: Matrix operations, eigenvalues, linear systems
physics-mechanics.md: Classical mechanics, quantum mechanics, vectors, units
advanced-topics.md: Geometry, number theory, combinatorics, logic, statistics
code-generation-printing.md: Lambdify, codegen, LaTeX output, printing
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
# 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)
# 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()}")
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)
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()
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)
# 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
from sympy import symbols, solve, sqrt
x = symbols('x')
solution = solve(x**2 - 5*x + 6, x)
# [2, 3]
from sympy import symbols, diff, sin
x = symbols('x')
f = sin(x**2)
df_dx = diff(f, x)
# 2*x*cos(x**2)
from sympy import symbols, integrate, exp
x = symbols('x')
integral = integrate(x * exp(-x**2), (x, 0, oo))
# 1/2
from sympy import Matrix
M = Matrix([[1, 2], [2, 1]])
eigenvals = M.eigenvals()
# {3: 1, -1: 1}
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])
"NameError: name 'x' is not defined"
symbols() before useUnexpected numerical results
0.5 instead of Rational(1, 2)Rational() or S() for exact arithmeticSlow performance in loops
subs() and evalf() repeatedlylambdify() to create a fast numerical function"Can't solve this equation"
solve, solveset, nsolve (numerical)Simplification not working as expected
simplify, factor, expand, trigsimppositive=True)simplify(expr, force=True) for aggressive simplification