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 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

• Laplacian on Scalar Field is Divergence of Gradient

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

• Results about the Laplacian can be found here.

Sources

• 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {V}$: Further Applications of the Operator $\nabla$: $1$. The Operator $\operatorname {div} \grad$: $(5.1)$, $(5.2)$
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: The Laplacian: $22.32$