Definition:Laplacian/Vector Field

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Definition

Let $\map {\R^n} {x_1, x_2, \ldots, x_n}$ denote the real Cartesian space of $n$ dimensions.

Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis on $\R^n$.


Let $\mathbf V: \R^n \to \R^n$ be a vector field on $\R^n$:

$\forall \mathbf x \in \R^n: \map {\mathbf V} {\mathbf x} := \ds \sum_{k \mathop = 0}^n \map {V_k} {\mathbf x} \mathbf e_k$

where each of $V_k: \R^n \to \R$ are real-valued functions on $\R^n$.

That is:

$\mathbf V := \tuple {\map {V_1} {\mathbf x}, \map {V_2} {\mathbf x}, \ldots, \map {V_n} {\mathbf x} }$


Let the partial derivative of $\mathbf V$ with respect to $x_k$ exist for all $f_k$.


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

\(\ds \nabla^2 \mathbf V\) \(:=\) \(\ds \sum_{k \mathop = 1}^n \dfrac {\partial^2 \mathbf V} {\partial {x_k}^2}\)


Cartesian $3$-Space

In $3$ dimensions with the standard ordered basis $\tuple {\mathbf i, \mathbf j, \mathbf k}$, this is usually rendered:


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.