>>12093184

>>12093230

The nabla symbol is an operator because it's a differential operator. You can think of it as a vector filled with differential operators. Therefore, all vector arithmatic still aplies so you can take the dot product and the cross product between it and a vector.

vector arithmatic:

vector times scalar

[math]\vec v \alpha = \begin{pmatrix}\alpha v_1\\ \alpha v_2 \\ \alpha v_3 \end{pmatrix}[/math]

vector dot product:

[math]\vec v \cdot \vec u = v_1 u_1 + v_2 u_2 + v_3 u_3[/math]

vector cross product:

[math]\vec v \times \vec u = \begin{pmatrix}v_2 u_3 - v_3 u_2 \\ v_3 u_1 - v_1 u_3 \\ v_1 u_2 - v_2 u_1 \end{pmatrix}[/math]

given some scalar function [math]f(x,y,z)[/math] and some vector function [math]\vec v(x,y,z)=\begin{pmatrix}v_x(x,y,z)\\ v_y(x,y,z)\\v_z(x,y,z)\\\end{pmatrix}[/math] and using the notation [math]\partial_x = \frac{\partial}{\partial x}[/math] while also using vector components x,y,z instead of 1,2,3 we get:

gradient (nabla times some scalar function of x,y,z)

[math] \nabla f(x, y, z) = \begin{pmatrix}\partial_x f(x,y,z)\\ \partial_y f(x,y,z) \\ \partial_z f(x,y,z) \end{pmatrix} [/math]

divergence (dot product of nabla with some vector function of x,y,z)

[math]\nabla \cdot \vec v(x,y,z) = \partial_x v_x(x,y,z) + \partial_y v_y(x,y,z) + \partial_z v_z(x,y,z)[/math]

curl (cross product of nabla with some vector function of x,y,z)

[math]\begin{pmatrix}\partial_y v_z - \partial_z v_y \\ \partial_z v_x - \partial_x v_z \\ \partial_x v_y - \partial_y v_x \end{pmatrix}[/math]