I am referring to an important part of the question Relationship between Connection and Material Derivative. Here is a paste and cut of the relevant part.

That is the directional derivative along $\mathbf{u}$ of the function $g$. On vector fields it is defined componentwise, that is, if $\mathbf{v} = (v_1,v_2,v_3)$ then

$$(\mathbf{u}\cdot \nabla)\mathbf{v} = ((\mathbf{u}\cdot \nabla)v_1, (\mathbf{u}\cdot \nabla)v_2, (\mathbf{u}\cdot \nabla)v_3) = (D_{\mathbf{u}}v_1, D_{\mathbf{u}}v_2, D_{\mathbf{u}}v_3).$$

But that latter thing is clearly the Covariant Derivative of $\mathbf{v}$ along $\mathbf{u}$ when we consider the Levi-Civita Connection on $\mathbb{R}^3$ with the usual flat metric tensor, that is

$$(\mathbf{u}\cdot \nabla)\mathbf{v} = \nabla_{\mathbf{u}}\mathbf{v}.$$

Now, is this conclusion right?

... I mean, I don't know that much of connections and how they can be used on Physics, but I know they are usefull. In that case, writing the material derivative in terms of a connection gives some advantage?

The answer and the comments in this cited question do not say anything about the usefulness of the relationship. I am not even sure that the relationship with the covariant derivative as presented in the question (and in the accepted answer) is a standard view on the material derivative. If it is a standard view, where is it used? (I mean a case where the covariant derivative is not the ordinary derivative everywhere, of course.) I know that a similar relationship makes sense and it is very useful when we try to see how the equations of General Relativity relate to the Newtonian equations, but then we are using four-velocity and the covariant derivative on space-time, not only on $\mathbb{R}^3$.


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