physica.
Mathematical Physicsundergraduategraduate

Laplacian

Also known as: Del squared · Nabla squared · ∇² · Laplace operator

The Laplacian compares the value at a point to the average of its neighbors: positive where the point sits in a valley, negative where it caps a hill, zero where it equals the surrounding average (harmonic).

∇2f=∇⋅∇f=∂2f∂x2+∂2f∂y2+∂2f∂z2\nabla^2 f = \nabla\cdot\nabla f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}
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A smooth bump f(x) (cyan) is shown above its Laplacian f''(x) (orange). A scanning marker rides the curve; where f bulges up the Laplacian dips negative, and in the flanks it rises positive — visualizing 'excess over the neighborhood average'.

Equivalent forms

Δf=∇2f\Delta f = \nabla^2 f
∇2f=1r2∂∂r ⁣(r2∂f∂r)+⋯ (spherical)\nabla^2 f = \frac{1}{r^{2}}\frac{\partial}{\partial r}\!\left(r^{2}\frac{\partial f}{\partial r}\right) + \cdots \ (\text{spherical})
A single operator unifies heat, gravity, electrostatics, diffusion, and the kinetic term of the Schrödinger equation — wherever 'smoothing toward the average' happens, ∇² is there.