Laplacian: ∇²f = ∂²f/∂x² + ∂²f/∂y² (+ ∂²f/∂z² in 3D). Measures how f at a point differs from its average nearby. ∇²f = 0: harmonic (Laplace equation). ∇²f = source: Poisson equation. Appears in heat, wave, and Schrödinger equations.
How to Use
Select a scalar field:
- f(x,y): Choose preset
- Point: (x₀, y₀)
- Output: ∇²f at point
Harmonic Functions
∇²f = 0 everywhere. Examples: ln(x²+y²), Re(zⁿ) for complex z. Satisfy the mean value property: value at center = average over any surrounding circle.
In Physics
- Heat: ∂u/∂t = α∇²u
- Wave: ∂²u/∂t² = c²∇²u
- Schrödinger: iℏ∂ψ/∂t = −ℏ²∇²ψ/(2m) + Vψ
Step-by-Step Instructions
- 1Select f(x,y).
- 2Enter point.
- 3View ∇²f.
- 4Check if harmonic.
- 5Compare with zero.