# LaTeX Formula Rules

## Strict Formatting Requirements

### Inline Formulas
- **Format**: `$formula$`
- **Rule**: NO spaces inside delimiters
- **Examples**:
  - ✅ `$x + y$`
  - ✅ `$\sum_{i=1}^{n} x_i$`
  - ❌ `$ x + y $`
  - ❌ `$x+y$` (missing spaces around operators is ok, but be consistent)

### Display Formulas
- **Format**: `$$formula$$`
- **Rule**: NO spaces inside delimiters
- **Examples**:
  - ✅ `$$\nabla f(x)$$`
  - ✅ `$$\mathbb{E}_{x \sim p}[f(x)]$$`
  - ❌ `$$ \nabla f(x) $$`

### Common Symbols

| Symbol | LaTeX | Meaning |
|--------|-------|---------|
| Σ | `\sum` | Summation (for loop) |
| ∏ | `\prod` | Product |
| ∫ | `\int` | Integral |
| ∇ | `\nabla` | Gradient (.grad) |
| ∂ | `\partial` | Partial derivative |
| ∈ | `\in` | Element of |
| ⊂ | `\subset` | Subset |
| ∀ | `\forall` | For all |
| ∃ | `\exists` | Exists |
| ℝ | `\mathbb{R}` | Real numbers |
| 𝔼 | `\mathbb{E}` | Expectation |
| θ | `\theta` | Parameters |
| ∇_θ | `\nabla_\theta` | Gradient w.r.t. θ |

### Greek Letters (Common in ML)

```
α \alpha    β \beta    γ \gamma    δ \delta
ε \epsilon  ζ \zeta    η \eta      θ \theta
λ \lambda   μ \mu      ν \nu       ξ \xi
π \pi       ρ \rho     σ \sigma    τ \tau
φ \phi      χ \chi     ψ \psi      ω \omega
```

### Best Practices

1. **Always escape special characters**: `#`, `$`, `%`, `&`, `_`, `{`, `}`
2. **Use `\left(` and `\right)` for auto-sizing parentheses**
3. **Use `\cdot` for multiplication dots**: `$x \cdot y$`
4. **Use `\frac{num}{den}` for fractions**: `$\frac{1}{n}$`
5. **Use `\sqrt{}` for square roots**: `$\sqrt{x^2 + y^2}$`