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 are summarized as:
Gradient: grad f = ∇ f {\displaystyle \operatorname {grad} f=\nabla f}
Divergence: div v = ∇ ⋅ v {\displaystyle \operatorname {div} \mathbf {v} =\nabla \cdot \mathbf {v} }
Curl: curl v = ∇ × v {\displaystyle \operatorname {curl} \mathbf {v} =\nabla \times \mathbf {v} }
Definition In the Cartesian coordinate system R n {\displaystyle \mathbb {R} ^{n}} with coordinates ( x 1 , … , x n ) {\displaystyle (x_{1},\dots ,x_{n})} and standard basis { e 1 , … , e n } {\displaystyle \{\mathbf {e} _{1},\dots ,\mathbf {e} _{n}\}} , del is a vector operator whose x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} components are the partial derivative operators ∂ ∂ x 1 , … , ∂ ∂ x n {\displaystyle {\partial \over \partial x_{1}},\dots ,{\partial \over \partial x_{n}}} ; that is,
∇ = ∑ i = 1 n e i ∂ ∂ x i = ( ∂ ∂ x 1 , … , ∂ ∂ x n ) {\displaystyle \nabla =\sum _{i=1}^{n}\mathbf {e} _{i}{\partial \over \partial x_{i}}=\left({\partial \over \partial x_{1}},\ldots ,{\partial \over \partial x_{n}}\right)}
where the expression in parentheses is a row vector. In three-dimensional Cartesian coordinate system R 3 {\displaystyle \mathbb {R} ^{3}} with coordinates ( x , y , z ) {\displaystyle (x,y,z)} and standard basis or unit vectors of axes { e x , e y , e z } {\displaystyle \{\mathbf {e} _{x},\mathbf {e} _{y},\mathbf {e} _{z}\}} , del is written as:
∇ = e x ∂ ∂ x + e y ∂ ∂ y + e z ∂ ∂ z = ( ∂ ∂ x , ∂ ∂ y , ∂ ∂ z ) {\displaystyle \nabla =\mathbf {e} _{x}{\partial \over \partial x}+\mathbf {e} _{y}{\partial \over \partial y}+\mathbf {e} _{z}{\partial \over \partial z}=\left({\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\right)}
As a vector operator, del naturally acts on scalar fields via scalar multiplication, and naturally acts on vector fields via dot products and cross products. More specifically, in three dimensions, for any scalar field f {\displaystyle f} and any vector field F = ( F x , F y , F z ) {\displaystyle \mathbf {F} =(F_{x},F_{y},F_{z})} , if one defines
( e i ∂ ∂ x i ) f := ∂ ∂ x i ( e i f ) = ∂ f ∂ x i e i {\displaystyle \left(\mathbf {e} _{i}{\partial \over \partial x_{i}}\right)f:={\partial \over \partial x_{i}}(\mathbf {e} _{i}f)={\partial f \over \partial x_{i}}\mathbf {e} _{i}}
( e i ∂ ∂ x i ) ⋅ F := ∂ ∂ x i ( e i ⋅ F ) = ∂ F i ∂ x i {\displaystyle \left(\mathbf {e} _{i}{\partial \over \partial x_{i}}\right)\cdot \mathbf {F} :={\partial \over \partial x_{i}}(\mathbf {e} _{i}\cdot \mathbf {F} )={\partial F_{i} \over \partial x_{i}}}
( e x ∂ ∂ x ) × F := ∂ ∂ x ( e x × F ) = ∂ ∂ x ( 0 , − F z , F y ) {\displaystyle \left(\mathbf {e} _{x}{\partial \over \partial x}\right)\times \mathbf {F} :={\partial \over \partial x}(\mathbf {e} _{x}\times \mathbf {F} )={\partial \over \partial x}(0,-F_{z},F_{y})}
( e y ∂ ∂ y ) × F := ∂ ∂ y ( e y × F ) = ∂ ∂ y ( F z , 0 , − F x ) {\displaystyle \left(\mathbf {e} _{y}{\partial \over \partial y}\right)\times \mathbf {F} :={\partial \over \partial y}(\mathbf {e} _{y}\times \mathbf {F} )={\partial \over \partial y}(F_{z},0,-F_{x})}
( e z ∂ ∂ z ) × F := ∂ ∂ z ( e z × F ) = ∂ ∂ z ( − F y , F x , 0 ) , {\displaystyle \left(\mathbf {e} _{z}{\partial \over \partial z}\right)\times \mathbf {F} :={\partial \over \partial z}(\mathbf {e} _{z}\times \mathbf {F} )={\partial \over \partial z}(-F_{y},F_{x},0),}
then using the above definition of ∇ {\displaystyle \nabla } , one may write
∇ f = ( e x ∂ ∂ x ) f + ( e y ∂ ∂ y ) f + ( e z ∂ ∂ z ) f = ∂ f ∂ x e x + ∂ f ∂ y e y + ∂ f ∂ z e z {\displaystyle \nabla f=\left(\mathbf {e} _{x}{\partial \over \partial x}\right)f+\left(\mathbf {e} _{y}{\partial \over \partial y}\right)f+\left(\mathbf {e} _{z}{\partial \over \partial z}\right)f={\partial f \over \partial x}\mathbf {e} _{x}+{\partial f \over \partial y}\mathbf {e} _{y}+{\partial f \over \partial z}\mathbf {e} _{z}}
and
∇ ⋅ F = ( e x ∂ ∂ x ⋅ F ) + ( e y ∂ ∂ y ⋅ F ) + ( e z ∂ ∂ z ⋅ F ) = ∂ F x ∂ x + ∂ F y ∂ y + ∂ F z ∂ z {\displaystyle \nabla \cdot \mathbf {F} =\left(\mathbf {e} _{x}{\partial \over \partial x}\cdot \mathbf {F} \right)+\left(\mathbf {e} _{y}{\partial \over \partial y}\cdot \mathbf {F} \right)+\left(\mathbf {e} _{z}{\partial \over \partial z}\cdot \mathbf {F} \right)={\partial F_{x} \over \partial x}+{\partial F_{y} \over \partial y}+{\partial F_{z} \over \partial z}}
and
∇ × F = ( e x ∂ ∂ x × F ) + ( e y ∂ ∂ y × F ) + ( e z ∂ ∂ z × F ) = ∂ ∂ x ( 0 , − F z , F y ) + ∂ ∂ y ( F z , 0 , − F x ) + ∂ ∂ z ( − F y , F x , 0 ) = ( ∂ F z ∂ y − ∂ F y ∂ z ) e x + ( ∂ F x ∂ z − ∂ F z ∂ x ) e y + ( ∂ F y ∂ x − ∂ F x ∂ y ) e z {\displaystyle {\begin{aligned}\nabla \times \mathbf {F} &=\left(\mathbf {e} _{x}{\partial \over \partial x}\times \mathbf {F} \right)+\left(\mathbf {e} _{y}{\partial \over \partial y}\times \mathbf {F} \right)+\left(\mathbf {e} _{z}{\partial \over \partial z}\times \mathbf {F} \right)\\&={\partial \over \partial x}(0,-F_{z},F_{y})+{\partial \over \partial y}(F_{z},0,-F_{x})+{\partial \over \partial z}(-F_{y},F_{x},0)\\&=\left({\text{ }}{\frac {\partial F_{z}}{\partial y}}-{\frac {\partial F_{y}}{\partial z}}\right)\mathbf {e} _{x}+\left({\text{ }}{\frac {\partial F_{x}}{\partial z}}-{\frac {\partial F_{z}}{\partial x}}\right)\mathbf {e} _{y}+\left({\text{ }}{\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\right)\mathbf {e} _{z}\end{aligned}}}
Example:
f ( x , y , z ) = x + y + z {\displaystyle f(x,y,z)=x+y+z}
∇ f = e x ∂ f ∂ x + e y ∂ f ∂ y + e z ∂ f ∂ z = ( 1 , 1 , 1 ) {\displaystyle \nabla f=\mathbf {e} _{x}{\partial f \over \partial x}+\mathbf {e} _{y}{\partial f \over \partial y}+\mathbf {e} _{z}{\partial f \over \partial z}=\left(1,1,1\right)}
Del can also be expressed in other coordinate systems, see for example del in cylindrical and spherical coordinates.
Notational 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.
Gradient The gradient of a scalar field f {\displaystyle f} is its derivative as a resultant vector field, and can be represented as:
∇ f = ∂ f ∂ x x ^ + ∂ f ∂ y y ^ + ∂ f ∂ z z ^ = grad f {\displaystyle \nabla f={\partial f \over \partial x}{\hat {\mathbf {x} }}+{\partial f \over \partial y}{\hat {\mathbf {y} }}+{\partial f \over \partial z}{\hat {\mathbf {z} }}=\operatorname {grad} 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 derivative. In particular, if a hill is defined as a height function over a plane h ( x , y ) {\displaystyle h(x,y)} , the gradient at a given location will be a vector in the xy-plane (visualizable as an arrow on a map) pointing along the steepest direction. The magnitude of the gradient is the value of this steepest slope. In particular, this notation is powerful because the gradient product rule looks very similar to the 1d-derivative case:
∇ ( f g ) = f ∇ g + g ∇ f {\displaystyle \nabla (fg)=f\nabla g+g\nabla f}
However, the rules for dot products do not turn out to be simple, as illustrated by:
∇ ( u ⋅ v ) = ( u ⋅ ∇ ) v + ( v ⋅ ∇ ) u + u × ( ∇ × v ) + v × ( ∇ × u ) {\displaystyle \nabla (\mathbf {u} \cdot \mathbf {v} )=(\mathbf {u} \cdot \nabla )\mathbf {v} +(\mathbf {v} \cdot \nabla )\mathbf {u} +\mathbf {u} \times (\nabla \times \mathbf {v} )+\mathbf {v} \times (\nabla \times \mathbf {u} )}
Divergence The divergence of a vector field
v ( x , y , z ) = v x x ^ + v y y ^ + v z z ^ {\displaystyle \mathbf {v} (x,y,z)=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}+v_{z}{\hat {\mathbf {z} }}} is a scalar field, and can be represented as:
div v = ∂ v x ∂ x + ∂ v y ∂ y + ∂ v z ∂ z = ∇ ⋅ v {\displaystyle \operatorname {div} \mathbf {v} ={\partial v_{x} \over \partial x}+{\partial v_{y} \over \partial y}+{\partial v_{z} \over \partial z}=\nabla \cdot \mathbf {v} }
The divergence is roughly a measure of a vector field's increase in the direction it points; but more accurately, it is a measure of that field's tendency to converge toward or diverge from a point. The power of the del notation is shown by the following product rule:
∇ ⋅ ( f v ) = ( ∇ f ) ⋅ v + f ( ∇ ⋅ v ) {\displaystyle \nabla \cdot (f\mathbf {v} )=(\nabla f)\cdot \mathbf {v} +f(\nabla \cdot \mathbf {v} )}
The formula for the vector product is slightly less intuitive, because this product is not commutative:
∇ ⋅ ( u × v ) = ( ∇ × u ) ⋅ v − u ⋅ ( ∇ × v ) {\displaystyle \nabla \cdot (\mathbf {u} \times \mathbf {v} )=(\nabla \times \mathbf {u} )\cdot \mathbf {v} -\mathbf {u} \cdot (\nabla \times \mathbf {v} )}
Curl The curl of a vector field v ( x , y , z ) = v x x ^ + v y y ^ + v z z ^ {\displaystyle \mathbf {v} (x,y,z)=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}+v_{z}{\hat {\mathbf {z} }}} is a vector function, and can be represented as:
curl v = ( ∂ v z ∂ y − ∂ v y ∂ z ) x ^ + ( ∂ v x ∂ z − ∂ v z ∂ x ) y ^ + ( ∂ v y ∂ x − ∂ v x ∂ y ) z ^ = ∇ × v {\displaystyle \operatorname {curl} \mathbf {v} =\left({\partial v_{z} \over \partial y}-{\partial v_{y} \over \partial z}\right){\hat {\mathbf {x} }}+\left({\partial v_{x} \over \partial z}-{\partial v_{z} \over \partial x}\right){