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

### Example 1: Solve Quadratic Equation
```python
from sympy import symbols, solve, sqrt
x = symbols('x')
solution = solve(x**2 - 5*x + 6, x)
# [2, 3]
```

### Example 2: Calculate Derivative
```python
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
```python
from sympy import symbols, integrate, exp
x = symbols('x')
integral = integrate(x * exp(-x**2), (x, 0, oo))
# 1/2
```

### Example 4: Matrix Eigenvalues
```python
from sympy import Matrix
M = Matrix([[1, 2], [2, 1]])
eigenvals = M.eigenvals()
# {3: 1, -1: 1}
```

### Example 5: Generate Python Function
```python
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])
```

## 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

## 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