Laplacian operator

The operator

Δ := \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}

is called the Laplacian operator. It appears in Laplace's equation and in Poisson's equation. Also appears in the heat equation.

It is usually written as $Δ = \nabla^{2} = \nabla \cdot \nabla$ . Also observe that $\nabla \cdot$ is the divergence in the 3D case.

In the 2D case, I understand it like a measure of the "force" existing in every point of a kind of "membrane" due to the influence of the surrounding points.