Definition:Laplacian/Scalar Field/Cartesian 3-Space

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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 known as

The Laplacian is also known as the Laplace operator, Laplace's operator or Laplace-Beltrami operator.

The last name is usually used in the context of submanifolds in Euclidean space and on (pseudo-)Riemannian manifolds.

Also see

  • Results about the Laplacian can be found here.