# 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 $f(x_{1},x_{2},\ldots,x_{n})$ 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

1. 1.

If $f,g$ are functions, then

 $\nabla(fg)=(\nabla f)g+f\nabla g.$
2. 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

 $\displaystyle\nabla\cdot\mathbf{A}$ $\displaystyle=\frac{\partial A_{x}}{\partial x}+\frac{\partial A_{y}}{\partial y% }+\frac{\partial A_{z}}{\partial z}$ $\displaystyle\nabla\times\mathbf{A}$ $\displaystyle=\left(\frac{\partial A_{z}}{\partial y}-\frac{\partial A_{y}}{% \partial z}\right)\mathbf{i}+\left(\frac{\partial A_{x}}{\partial z}-\frac{% \partial A_{z}}{\partial x}\right)\mathbf{j}+\left(\frac{\partial A_{y}}{% \partial x}-\frac{\partial A_{x}}{\partial y}\right)\mathbf{k}$ $\displaystyle\nabla^{2}f$ $\displaystyle=\frac{\partial^{2}f}{\partial x^{2}}+\frac{\partial^{2}f}{% \partial y^{2}}+\frac{\partial^{2}f}{\partial z^{2}}.$
Title nabla Nabla 2013-03-22 14:00:20 2013-03-22 14:00:20 stevecheng (10074) stevecheng (10074) 7 stevecheng (10074) Definition msc 26A06 gradient NablaActingOnProducts Gradient AlternateCharacterizationOfCurl $\nabla$