Sympy V2

SymPy - Symbolic Mathematics in Python

Overview

This public intake copy packages plugins/antigravity-awesome-skills/skills/sympy from https://github.com/sickn33/antigravity-awesome-skills into the native Omni Skills editorial shape without hiding its origin.

Use it when the operator needs the upstream workflow, support files, and repository context to stay intact while the public validator and private enhancer continue their normal downstream flow.

This intake keeps the copied upstream files intact and uses the external_source block in metadata.json plus ORIGIN.md as the provenance anchor for review.

SymPy - Symbolic Mathematics in Python

Imported source sections that did not map cleanly to the public headings are still preserved below or in the support files. Notable imported sections: Core Capabilities, Working with SymPy: Best Practices, Common Use Case Patterns, Limitations.

When to Use This Skill

Use this section as the trigger filter. It should make the activation boundary explicit before the operator loads files, runs commands, or opens a pull request.

• 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)

Operating Table

Workflow

This workflow is intentionally editorial and operational at the same time. It keeps the imported source useful to the operator while still satisfying the public intake standards that feed the downstream enhancer flow.

1. With NumPy python import numpy as np from sympy import symbols, lambdify x = symbols('x') expr = x2 + 2x + 1 f = lambdify(x, expr, 'numpy') xarray = np.linspace(-5, 5, 100) yarray = f(xarray) ### 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') xvals = np.linspace(-10, 10, 1000) yvals = f(xvals) plt.plot(xvals, yvals) plt.show() ### With SciPy python from scipy.optimize import fsolve from sympy import symbols, lambdify # Define equation symbolically x = symbols('x') equation = x3 - 2x - 5 # Convert to numerical function f = lambdify(x, equation, 'numpy') # Solve numerically with initial guess solution = fsolve(f, 2)

2. Confirm the user goal, the scope of the imported workflow, and whether this skill is still the right router for the task.
3. Read the overview and provenance files before loading any copied upstream support files.
4. Load only the references, examples, prompts, or scripts that materially change the outcome for the current request.
5. Execute the upstream workflow while keeping provenance and source boundaries explicit in the working notes.
6. Validate the result against the upstream expectations and the evidence you can point to in the copied files.
7. Escalate or hand off to a related skill when the work moves out of this imported workflow's center of gravity.

Imported Workflow Notes

Imported: 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)

Imported: 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.

Imported: 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

Examples

Example 1: Ask for the upstream workflow directly

Use @sympy-v2 to handle <task>. Start from the copied upstream workflow, load only the files that change the outcome, and keep provenance visible in the answer.

Explanation: This is the safest starting point when the operator needs the imported workflow, but not the entire repository.

Example 2: Ask for a provenance-grounded review

Review @sympy-v2 against metadata.json and ORIGIN.md, then explain which copied upstream files you would load first and why.

Explanation: Use this before review or troubleshooting when you need a precise, auditable explanation of origin and file selection.

Example 3: Narrow the copied support files before execution

Use @sympy-v2 for <task>. Load only the copied references, examples, or scripts that change the outcome, and name the files explicitly before proceeding.

Explanation: This keeps the skill aligned with progressive disclosure instead of loading the whole copied package by default.

Example 4: Build a reviewer packet

Review @sympy-v2 using the copied upstream files plus provenance, then summarize any gaps before merge.

Explanation: This is useful when the PR is waiting for human review and you want a repeatable audit packet.

Imported Usage Notes

Imported: 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)
# 2*x*cos(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])

Best Practices

Treat the generated public skill as a reviewable packaging layer around the upstream repository. The goal is to keep provenance explicit and load only the copied source material that materially improves execution.

• Keep the imported skill grounded in the upstream repository; do not invent steps that the source material cannot support.
• Prefer the smallest useful set of support files so the workflow stays auditable and fast to review.
• Keep provenance, source commit, and imported file paths visible in notes and PR descriptions.
• Point directly at the copied upstream files that justify the workflow instead of relying on generic review boilerplate.
• Treat generated examples as scaffolding; adapt them to the concrete task before execution.
• Route to a stronger native skill when architecture, debugging, design, or security concerns become dominant.

Troubleshooting

Problem: The operator skipped the imported context and answered too generically

Symptoms: The result ignores the upstream workflow in plugins/antigravity-awesome-skills/skills/sympy, fails to mention provenance, or does not use any copied source files at all. Solution: Re-open metadata.json, ORIGIN.md, and the most relevant copied upstream files. Check the external_source block first, then restate the provenance before continuing.

Problem: The imported workflow feels incomplete during review

Symptoms: Reviewers can see the generated SKILL.md, but they cannot quickly tell which references, examples, or scripts matter for the current task. Solution: Point at the exact copied references, examples, scripts, or assets that justify the path you took. If the gap is still real, record it in the PR instead of hiding it.

Problem: The task drifted into a different specialization

Symptoms: The imported skill starts in the right place, but the work turns into debugging, architecture, design, security, or release orchestration that a native skill handles better. Solution: Use the related skills section to hand off deliberately. Keep the imported provenance visible so the next skill inherits the right context instead of starting blind.

Imported Troubleshooting Notes

Imported: Troubleshooting Common Issues

1. "NameError: name 'x' is not defined"

   • Solution: Always define symbols using symbols() before use

2. Unexpected numerical results

   • Issue: Using floating-point numbers like 0.5 instead of Rational(1, 2)
   • Solution: Use Rational() or S() for exact arithmetic

3. Slow performance in loops

   • Issue: Using subs() and evalf() repeatedly
   • Solution: Use lambdify() to create a fast numerical function

4. "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

5. 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

Related Skills

• @00-andruia-consultant - Use when the work is better handled by that native specialization after this imported skill establishes context.
• @00-andruia-consultant-v2 - Use when the work is better handled by that native specialization after this imported skill establishes context.
• @10-andruia-skill-smith - Use when the work is better handled by that native specialization after this imported skill establishes context.
• @10-andruia-skill-smith-v2 - Use when the work is better handled by that native specialization after this imported skill establishes context.

Additional Resources

Use this support matrix and the linked files below as the operator packet for this imported skill. They should reflect real copied source material, not generic scaffolding.

Imported Reference Notes

Imported: 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

Imported: 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

Imported: 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

Imported: 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)

Imported: 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()}")

Imported: Limitations

• Use this skill only when the task clearly matches the scope described above.
• Do not treat the output as a substitute for environment-specific validation, testing, or expert review.
• Stop and ask for clarification if required inputs, permissions, safety boundaries, or success criteria are missing.