# Definition:Del Operator

## Definition

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 $\mathbf V$.

### Cartesian $3$-Space

Let $\R^3$ be a Cartesian $3$-space.

Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.

The **del operator** is defined in $\R^3$ as:

- $\operatorname {del} = \nabla := \mathbf i \dfrac \partial {\partial x} + \mathbf j \dfrac \partial {\partial y} + \mathbf k \dfrac \partial {\partial z}$

## Also known as

The **del operator** is often seen referred to as **nabla**, but the latter term is technically to be applied to the $\nabla$ symbol itself.

Other earlier writers have referred to it as **atled**, but that idea never really caught on.

Its full name is the **differential operator**, but that term is rarely used.

## Also see

- Results about
**the $\nabla$ operator**can be found here.

## Historical Note

The **del operator** was introduced by William Rowan Hamilton, and initially developed by Peter Guthrie Tait.

The name **del** was introduced by Josiah Willard Gibbs.

## Sources

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