Symbols:Nabla/Laplacian

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Laplacian

$\nabla^2$


Scalar Field

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

Let $\map U {x_1, x_2, \ldots, x_n}$ be a scalar field over $\R^n$.

Let the partial derivatives of $U$ exist throughout $\R^n$.


The Laplacian of $U$ is defined as:

$\ds \nabla^2 U := \sum_{k \mathop = 1}^n \dfrac {\partial^2 U} {\partial {x_k}^2}$


Vector Field

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:

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


Riemannian Manifold

Let $\struct {M, g}$ be a Riemannian manifold.

Let $f \in \map {\CC^\infty} M : M \to \R$ be a smooth mapping on $M$.

Let $\grad$ be the gradient operator.

Let $\operatorname {div}$ be the divergence operator.


The Laplacian of $f$ is defined as:

$\Delta f := \map {\operatorname {div} } {\grad f}$


The $\LaTeX$ code for \(\nabla^2\) is \nabla^2 .


Sources