nabla
Let $f:{\mathbb{R}}^{n}\to \mathbb{R}$ be a ${C}^{1}({\mathbb{R}}^{n})$ function, that is, a partially differentiable function in all its coordinates. The symbol $\nabla $, named nabla, represents the gradient operator, whose action on $f({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})$ is given by
$\nabla f$  $=$  $({f}_{{x}_{1}},{f}_{{x}_{2}},\mathrm{\dots},{f}_{{x}_{n}})$  
$=$  $({\displaystyle \frac{\partial f}{\partial {x}_{1}}},{\displaystyle \frac{\partial f}{\partial {x}_{2}}},\mathrm{\dots},{\displaystyle \frac{\partial f}{\partial {x}_{n}}})$ 
Properties

1.
If $f,g$ are functions, then
$$\nabla (fg)=(\nabla f)g+f\nabla g.$$ 
2.
For any scalars $\alpha $ and $\beta $ and functions $f$ and $g$,
$$\nabla (\alpha f+\beta g)=\alpha \nabla f+\beta \nabla g.$$
The $\nabla $ symbolism
Using the $\nabla $ formalism, the divergence operator can be expressed as $\nabla \cdot $, the curl operator as $\nabla \times $, and the Laplacian operator as ${\nabla}^{2}$. To wit, for a given vector field
$$\mathbf{A}={A}_{x}\mathbf{i}+{A}_{y}\mathbf{j}+{A}_{z}\mathbf{k},$$ 
and a given function $f$ we have
$\nabla \cdot \mathbf{A}$  $={\displaystyle \frac{\partial {A}_{x}}{\partial x}}+{\displaystyle \frac{\partial {A}_{y}}{\partial y}}+{\displaystyle \frac{\partial {A}_{z}}{\partial z}}$  
$\nabla \times \mathbf{A}$  $=\left({\displaystyle \frac{\partial {A}_{z}}{\partial y}}{\displaystyle \frac{\partial {A}_{y}}{\partial z}}\right)\mathbf{i}+\left({\displaystyle \frac{\partial {A}_{x}}{\partial z}}{\displaystyle \frac{\partial {A}_{z}}{\partial x}}\right)\mathbf{j}+\left({\displaystyle \frac{\partial {A}_{y}}{\partial x}}{\displaystyle \frac{\partial {A}_{x}}{\partial y}}\right)\mathbf{k}$  
${\nabla}^{2}f$  $={\displaystyle \frac{{\partial}^{2}f}{\partial {x}^{2}}}+{\displaystyle \frac{{\partial}^{2}f}{\partial {y}^{2}}}+{\displaystyle \frac{{\partial}^{2}f}{\partial {z}^{2}}}.$ 
Title  nabla 

Canonical name  Nabla 
Date of creation  20130322 14:00:20 
Last modified on  20130322 14:00:20 
Owner  stevecheng (10074) 
Last modified by  stevecheng (10074) 
Numerical id  7 
Author  stevecheng (10074) 
Entry type  Definition 
Classification  msc 26A06 
Related topic  gradient 
Related topic  NablaActingOnProducts 
Related topic  Gradient 
Related topic  AlternateCharacterizationOfCurl 
Defines  $\nabla $ 