Definition:Laplacian/Vector Field/Cartesian 3-Space

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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 $R$.


Definition 1

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

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


Definition 2

Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.

Let $\mathbf V$ be expressed as vector-valued function:

$\mathbf V := V_x \mathbf i + V_y \mathbf j + V_z \mathbf k$

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

$\nabla^2 \mathbf V = \nabla^2 V_x \mathbf i + \nabla^2 V_x \mathbf j + \nabla^2 V_y \mathbf k$

where $\nabla^2 V_x$ and so on are the laplacians of $V_x$, $V_y$ and $V_z$ as scalar fields.


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.