Core Capabilities

Symbolic computation basics, calculus, equation solving, matrices and linear algebra, physics and mechanics, advanced mathematics, and code generation and output. Per-topic detail is in the other reference files in this directory.

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, Operator, Commutator
A, B = Operator('A'), Operator('B')
psi = Ket('psi')
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