# Symbols:Nabla

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## Symbol

Nabla is the name of the symbol $\nabla$.

The $\LaTeX$ code for $\nabla$ is \nabla .

## Del Operator

$\nabla$

Let $\mathbf V$ be a vector space of $n$ dimensions.

Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis of $\mathbf V$.

The del operator is a unary operator on $\mathbf V$ defined as:

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

where $\mathbf v = \ds \sum_{k \mathop = 0}^n x_k \mathbf e_k$ is an arbitrary vector of $\mathbf V$.

## 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 $f \in \map {C^\infty} M : M \to \R$ 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:

$\nabla^2 f := \map {\operatorname {div} } {\grad f}$

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

## Backward Difference Operator

$\nabla$

The backward difference operator on $f$ is defined as:

$\map {\nabla f} x := \map f x - \map f {x - 1}$

## Also known as

The symbol nabla is also known as del, from its use for the Del operator.

## Linguistic Note

The term nabla derives from the ancient Greek word νάβλα for a Phoenician harp.

This arises from the shape of the nabla symbol: $\nabla$.

The term was originally suggested by the encyclopedist William Robertson Smith to Peter Guthrie Tait.

As a result of this suggestion, the term was used in correspondence between Tait and James Clerk Maxwell, mainly in a jocular context.

The name gained official traction as a result of its adoption by Lord Kelvin in his lectures.