# Del

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In vector calculus, del is a vector differential operator, usually represented by the nabla symbol $\nabla$. When applied to a function defined on a one-dimensional domain, it denotes its standard derivative as defined in calculus. When applied to a field (a function defined on a multi-dimensional domain), del may denote the gradient (locally steepest slope) of a scalar field, the divergence of a vector field, or the curl (rotational) of a vector field, depending on the way it is applied.

Strictly speaking, del is not a specific operator, but rather a convenient mathematical notation for those three operators, that makes many equations easier to write and remember. The del symbol can be interpreted as a vector of partial derivative operators, and its three possible meanings—gradient, divergence, and curl—can be formally viewed as the scalar product, dot product, and cross product, respectively, of the del "operator" with the field. These formal products may not commute with other operators or products.

## Contents

### Definition

In the three-dimensional Cartesian coordinate system R3 with coordinates (x, y, z), del is defined in terms of partial derivative operators as

where $\{\mathbf{\hat{x}}, \mathbf{\hat{y}},\mathbf{\hat{z}} \}$ are the unit vectors in their respective directions. Though this page chiefly treats del in three dimensions, this definition can be generalized to the n-dimensional Euclidean space Rn. In the Cartesian coordinate system with coordinates (x1, x2, ..., xn), del is: