Del

Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus. When applied to a field (a function defined on a multi-dimensional domain), it may denote any one of three operators depending on the way it is applied: the gradient or (locally) steepest slope of a scalar field (or sometimes of a vector field, as in the Navier–Stokes equations); the divergence of a vector field; or the curl (rotation) of a vector field.
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 (or nabla) 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 product with a scalar, a dot product, and a cross product, respectively, of the "del operator" with the field. These formal products do not necessarily commute with other operators or products. These three uses, detailed below, are summarized as:

Gradient:



grad
⁡
f
=
∇
f


{\displaystyle \operatorname {grad} f=\nabla f}

Divergence:



div
⁡



v
→



=
∇
⋅



v
→





{\displaystyle \operatorname {div} {\vec {v}}=\nabla \cdot {\vec {v}}}

Curl:



curl
⁡



v
→



=
∇
×



v
→





{\displaystyle \operatorname {curl} {\vec {v}}=\nabla \times {\vec {v}}}

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