Editing Del (section)

==Definition==
In the [[Cartesian coordinate system]] <math>\mathbb{R}^n</math> with coordinates <math>(x_1, \dots, x_n)</math> and [[standard basis]] <math>\{\mathbf e_1, \dots, \mathbf e_n \}</math>, del is a vector operator whose <math>x_1, \dots, x_n</math> components are the [[partial derivative]] operators <math>{\partial \over \partial x_1}, \dots, {\partial \over \partial x_n}</math>; that is,
:<math> \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)</math>

where the expression in parentheses is a row vector.  In [[three-dimensional]] Cartesian coordinate system <math>\mathbb{R}^3</math> with coordinates <math>(x, y, z)</math> and standard basis or unit vectors of axes <math>\{\mathbf e_x, \mathbf e_y, \mathbf e_z \}</math>, del is written as:
:<math>\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) </math>

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, for any scalar field <math>f</math> and any vector field <math>\mathbf{F}=(F_x, F_y, F_z)</math>, if one ''defines''

:<math>\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</math>
:<math>\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}</math>
:<math>\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)</math>
:<math>\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)</math>
:<math>\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),</math>

then using the above definition of <math>\nabla</math>, one may write

:<math>
\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
</math>
and
:<math>
\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}
</math>
and
:<math>\begin{align}
\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({\partial F_z \over \partial y}-{\partial F_y \over \partial z}\right)\mathbf{e}_x + \left({\partial F_x \over \partial z}-{\partial F_z \over \partial x}\right)\mathbf{e}_y + \left({\partial F_y \over \partial x}-{\partial F_x \over \partial y}\right)\mathbf{e}_z
\end{align}</math>

:'''Example:'''
:<math>f(x, y, z) = x + y + z </math>
:<math>\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) </math>
:

Del can also be expressed in other coordinate systems, see for example [[del in cylindrical and spherical coordinates]].