An equation from terms of operator del to terms of sums

[olgerm]

https://wikimedia.org/api/rest_v1/media/math/render/svg/a7fd3adddbdfb95797d11ef6167ecda4efe3e0b9
https://en.wikipedia.org/wiki/Lorentz_force#Lorentz_force_in_terms_of_potentials
How to write this formula in terms of sums and vector components?

What is $v\cdot\nabla$ ? I think it is some spatial derivative of speed, but since speed is not a field, it can not be that.

I think rest of the equation is $F_x=q\cdot(-\frac{\partial\phi}{\partial x}-\frac{\partial A_x}{\partial t}+\sum(v_i \cdot \frac{\partial A_i}{\partial x})-?)$ .
Is it correct?

 

[pasmith]

The Lorentz force is the electromagnetic force on an object. That object has a position and a velocity, both of which are vector-valued functions of time.

$$((\mathbf{v} \cdot \nabla)\mathbf{A})_x = \sum_{j} v_j \frac{\partial A_x}{\partial x_j}$$

 

[olgerm]

The Lorentz force is the electromagnetic force on an object. That object has a position and a velocity, both of which are vector-valued functions of time.

$$((\mathbf{v} \cdot \nabla)\mathbf{A})_x = \sum_{j} v_j \frac{\partial A_x}{\partial x_j}$$

So the equation on Wikipedia page is
$F_x=q\cdot(-\frac{\partial\phi}{\partial x}-\frac{\partial A_x}{\partial t}+\sum_i(v_i \cdot (\frac{\partial A_i}{\partial x}-\frac{\partial A_x}{\partial x_i})))$ ?

Is it so?