nabla

Let $f:{ℝ}^n\to ℝ$ be a$C^1({ℝ}^n)$ function, that is, a partially differentiablefunction in all its coordinates. The symbol $\nabla$, namednabla, 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 theLaplacian 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}}.$