nabla

Let $f:{ℝ}^n\to ℝ$ be a $C^1({ℝ}^n)$ 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

$$𝐀=A_x𝐢+A_y𝐣+A_z𝐤,$$

and a given function $f$ we have

$\nabla \cdot 𝐀$ $={\displaystyle \frac{\partial A_x}{\partial x}}+{\displaystyle \frac{\partial A_y}{\partial y}}+{\displaystyle \frac{\partial A_z}{\partial z}}$

$\nabla \times 𝐀$ $=\left({\displaystyle \frac{\partial A_z}{\partial y}}-{\displaystyle \frac{\partial A_y}{\partial z}}\right)𝐢+\left({\displaystyle \frac{\partial A_x}{\partial z}}-{\displaystyle \frac{\partial A_z}{\partial x}}\right)𝐣+\left({\displaystyle \frac{\partial A_y}{\partial x}}-{\displaystyle \frac{\partial A_x}{\partial y}}\right)𝐤$

${\nabla }^{2}f$ $={\displaystyle \frac{{\partial }^{2}f}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}f}{\partial y^2}}+{\displaystyle \frac{{\partial }^{2}f}{\partial z^2}}.$

Title	nabla
Canonical name	Nabla
Date of creation	2013-03-22 14:00:20
Last modified on	2013-03-22 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	$\nabla$