# Symbols:Nabla

## Symbol

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

The $\LaTeX$ code for \(\nabla\) is `\nabla`

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## 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`

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## 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.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**nabla** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**nabla**