Editing Del (section)

{{short description|Vector differential operator}}
{{About|the mathematical operator represented by the nabla symbol|the symbol itself|nabla symbol|the operation associated with the symbol ∂, also sometimes referred to as "del"|Partial derivative|other uses}}
{{distinguish|Dell}}
{{No footnotes|date=March 2010}}

[[File:Del.svg|right|100px|thumb|Del operator,<br />represented by<br />the [[nabla symbol]]]]
'''Del''', or '''nabla''', is an [[Operator (mathematics)|operator]] used in mathematics (particularly in [[vector calculus]]) as a [[vector (geometry)|vector]] [[differential operator]], usually represented by the [[nabla symbol]] '''∇'''. When applied to a [[function (mathematics)|function]] defined on a [[dimension (mathematics)|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 operations 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#Interpretation as v·(∇v)|Navier–Stokes equations]]); the [[divergence]] of a vector field; or the [[curl (mathematics)|curl]] (rotation) of a vector field.

Del is a very convenient [[mathematical notation]] for those three operations (gradient, divergence, and curl) that makes many [[equations]] easier to write and remember.  The del symbol (or nabla) can be [[Formal calculation|formally]] defined as a vector operator whose components are the corresponding [[partial derivative]] operators. As a vector operator, it can act on scalar and vector fields in three different ways, giving rise to three different differential operations: first, it can act on scalar fields by a formal scalar multiplication—to give a vector field called the gradient; second, it can act on vector fields by a formal [[dot product]]—to give a scalar field called the divergence; and lastly, it can act on vector fields by a formal [[cross product]]—to give a vector field called the curl. These formal products do not necessarily [[commutative operation|commute]] with other operators or products.  These three uses, detailed below, are summarized as:
* Gradient: <math>\operatorname{grad}f = \nabla f</math>
* Divergence: <math>\operatorname{div}\mathbf v =  \nabla \cdot \mathbf v </math>
* Curl: <math>\operatorname{curl}\mathbf v =  \nabla \times \mathbf v</math>