# Symbols:Nabla

## Contents

## 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:

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

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

## Laplacian

- $\nabla^2$

### 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:

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

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