Definition:Laplacian/Vector Field/Cartesian 3-Space/Definition 1

Definition

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

Let $\map {\mathbf V} {x, y, z}$ be a vector field acting over $R$.

The Laplacian on $\mathbf V$ is defined as:

$\nabla^2 \mathbf V = \dfrac {\partial^2 \mathbf V} {\partial x^2} + \dfrac {\partial^2 \mathbf V} {\partial y^2} + \dfrac {\partial^2 \mathbf V} {\partial z^2}$

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

• Equivalence of Definitions of Laplacian on Vector Field on Cartesian 3-Space

• Results about the Laplacian can be found here.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: The Laplacian: $22.33$