Does a = v dv/dx hold when moving in a plane?

[Feynman's fan]

Not exactly homework in my case but I guess it fits the level.

We know that a = v dv/dx holds when moving on a line. Does it hold when moving on a plane?

If yes, what would be the exact formulation of the statement. I mean, I read a = v dv/dx as a(x) = v(x) dv/dx(x) (or are they functions of t?) but since on a plane a(x), v(x) and dv/dx(x) are all vectors, how do we define the product?

 

[Ssnow]

In one dimension you can use the rule $a=\frac{dv}{dt}=\frac{dv}{dx}\frac{dx}{dt}=v\frac{dv}{dx}$ but in dimension greater than one you must do attention to use the analogue tools ...