Laplacian

Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $f \colon U \to \mathbb{R}$ be twice differentiable on $U$. The Laplacian of $f$ is the function $\Delta f \colon U \to \mathbb{R}$, $$\Delta f = \operatorname{div}(\nabla f) = \nabla \cdot \nabla f,$$ the divergence of the gradient of $f$. It is also written $\nabla^2 f$.

Remarks

In Cartesian coordinates, $\Delta f$ is the sum of the unmixed second partial derivatives of $f$. The operator $\Delta$ is linear, and it maps functions of class C^k to functions of class $C^{k-2}$ for $k \ge 2$. Let $f$ be of class $C^2$, and let $\bar{f}_r$ be the average of $f$ on the sphere of radius $r$ about $x$. Then $\Delta f(x)$ is $2n$ times the limit of $(\bar{f}_r - f(x)) / r^2$ as $r \to 0$. So $\Delta f(x) > 0$ means that $f(x)$ is below its nearby averages. The Laplacian appears in the heat equation $\partial_t u = \Delta u$, the wave equation and Laplace's equation $\Delta u = 0$.

Explanations

L1 human

The Laplacian compares the value of a function at a point with its values nearby. If $f$ at the point is lower than the average around it, the Laplacian is positive. If it is higher, the Laplacian is negative.