Definition:Del Operator

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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.