Laplacian

The Laplacian or Laplace operator $\nabla^2$ computes the vector divergence of the gradient of a function $f(\mathbf{x})$,

given by the sum of 'unmixed' second derivatives, or equivalently the trace of the Hessian matrix $H(f(\mathbf{x}))$.

It is proportional to the difference between the average value of $f$ in a small ball around $\mathbf{x}$, and the value at $\mathbf{x}$ itself. When $\nabla^2 f$ is positive, then $f$ is concave up (smaller than the local average), and when $\nabla^2 f$ is negative then $f$ is concave down (larger than its local average). The Laplacian therefore shows up in diffusion processes (such as the heat equation), where the function value at each point is continuously updated towards the average of its neighbors.