Del

Del, or nabla, is an operator used in mathematics, in particular 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 its standard derivative as defined in calculus. When applied to a field (a function defined on a multi-dimensional domain), it may denote the gradient (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, 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 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}}} 2Definition In the Cartesian coordinate system Rn with coordinates ( x 1 , … , x n ) {\displaystyle (x_{1},\dots ,x_{n})} and standard basis { e → 1 , … , e → n } {\displaystyle \{{\vec {e}}_{1},\dots ,{\vec {e}}_{n}\}} , del is defined in terms of partial derivative operators as ∇ = ∑ i = 1 n e → i ∂ ∂ x i = ( ∂ ∂ x 1 , … , ∂ ∂ x n ) {\displaystyle \nabla =\sum _{i=1}^{n}{\vec {e}}_{i}{\partial \over \partial x_{i}}=\left({\partial \over \partial x_{1}},\ldots ,{\partial \over \partial x_{n}}\right)} In three-dimensional Cartesian coordinate system R3 with coordinates ( x , y , z ) {\displaystyle (x,y,z)} and standard basis or unit vectors of axes { e → x , e → y , e → z } {\displaystyle \{{\vec {e}}_{x},{\vec {e}}_{y},{\vec {e}}_{z}\}} , del is written as ∇ = e → x ∂ ∂ x + e → y ∂ ∂ y + e → z ∂ ∂ z = ( ∂ ∂ x , ∂ ∂ y , ∂ ∂ z ) {\displaystyle \nabla ={\vec {e}}_{x}{\partial \over \partial x}+{\vec {e}}_{y}{\partial \over \partial y}+{\vec {e}}_{z}{\partial \over \partial z}=\left({\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\right)} Del can also be expressed in other coordinate systems, see for example del in cylindrical and spherical coordinates. 2Notational uses Del is used as a shorthand form to simplify many long mathematical expressions. It is most commonly used to simplify expressions for the gradient, divergence, curl, directional derivative, and Laplacian. 3Gradient The vector derivative of a scalar field f {\displaystyle f} is called the gradient, and it can be represented as: grad ⁡ f = ∂ f ∂ x e → x + ∂ f ∂ y e → y + ∂ f ∂ z e → z = ∇ f {\displaystyle \operatorname {grad} f={\partial f \over \partial x}{\vec {e}}_{x}+{\partial f \over \partial y}{\vec {e}}_{y}+{\partial f \over \partial z}{\vec {e}}_{z}=\nabla f} It always points in the direction of greatest increase of f {\displaystyle f} , and it has a magnitude equal to the maximum rate of increase at the point—just like a standard derivativ