Definition:Laplacian/Scalar Field/Cartesian 3-Space

Definition
Let $R$ be a region of Cartesian $3$-space $\R^3$.

Let $\map U {x, y, z}$ be a scalar field acting over $R$.

The Laplacian of $U$ is defined as:


 * $\nabla^2 U := \dfrac {\partial^2 U} {\partial x^2} + \dfrac {\partial^2 U} {\partial y^2} + \dfrac {\partial^2 U} {\partial z^2}$

where $\nabla$ denotes the del operator.

Also see

 * Laplacian on Scalar Field is Divergence of Gradient


 * Definition:Laplacian on Vector Field on Cartesian 3-Space