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nabla

(Definition)

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 $f(x_1,x_2,\ldots,x_n)$ is given by

$\displaystyle \nabla f$	$\displaystyle =$	$\displaystyle \left(f_{x_1},f_{x_2},\ldots,f_{x_n}\right)$
	$\displaystyle =$	$\displaystyle \left( \frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},\ldots,\frac{\partial f}{\partial x_n} \right)$

Properties

If are functions, then
$\displaystyle \nabla(fg) = (\nabla f) g + f \nabla g.$
For any scalars $\alpha$ and $\beta$ and functions and ,
$\displaystyle \nabla(\alpha f + \beta g) = \alpha \nabla f + \beta \nabla g.$

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

$\displaystyle \mathbf{A}= A_x\, \mathbf{i}+ A_y\, \mathbf{j}+ A_z\, \mathbf{k},$

and a given function

we have

$\displaystyle \nabla\cdot \mathbf{A}$	$\displaystyle = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} +\frac{\partial A_z}{\partial z}$
$\displaystyle \nabla\times \mathbf{A}$	$\displaystyle = \left(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\pa... ...ac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right)\mathbf{k}$
$\displaystyle \nabla^2 f$	$\displaystyle = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} +\frac{\partial^2 f}{\partial z^2}.$