Del

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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 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); the divergence of a vector field; or the 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 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 commute with other operators or products. These three uses, detailed below, are summarized as:

• Gradient: ${\displaystyle \operatorname {grad} f=\nabla f}$
• Divergence: ${\displaystyle \operatorname {div} \mathbf {v} =\nabla \cdot \mathbf {v} }$
• Curl: ${\displaystyle \operatorname {curl} \mathbf {v} =\nabla \times \mathbf {v} }$