Definition:Laplacian

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Definition

Real-Valued Function

Let $\R^n$ denote the real Cartesian space of $n$ dimensions.

Let $f \left({x_1, x_2, \ldots, x_n}\right)$ denote a real-valued function on $\R^n$.

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

Let the partial derivative of $f$ with respect to $x_k$ exist for all $x_k$.


The Laplacian of $f$ is defined as:

\(\displaystyle \nabla^2 f\) \(:=\) \(\displaystyle \nabla \cdot \left({\nabla f}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({\sum_{k \mathop = 1}^n \mathbf e_k \dfrac \partial {\partial x_k} }\right) \cdot \left({\sum_{k \mathop = 1}^n \dfrac {\partial f} {\partial x_k} \mathbf e_k}\right)\) Definition of Del Operator
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 1}^n \dfrac {\partial^2 f} {\partial {x_k}^2}\)


Vector-Valued Function

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

Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis on $\mathbf V$.

Let $\mathbf f = \left({f_1 \left({\mathbf x}\right), f_2 \left({\mathbf x}\right), \ldots, f_n \left({\mathbf x}\right)}\right): \mathbf V \to \mathbf V$ be a vector-valued function on $\mathbf V$.


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


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

\(\displaystyle \nabla^2 \mathbf f\) \(:=\) \(\displaystyle \left({\sum_{k \mathop = 1}^n \dfrac {\partial^2 \mathbf f} {\partial {x_k}^2} }\right)\)


Also known as

The Laplacian is also known as the Laplace operator.


Also see

  • Results about the Laplacian can be found here.


Source of Name

This entry was named for Pierre-Simon de Laplace.


Sources