A coordinate independent interpretation of $\vec u\cdot\nabla\vec v$ for vector fields $\vec u,\vec v$?

Question

I'm having trouble conceptualizing what exactly is meant by the term $\mathbf u\cdot \nabla\mathbf v$ for vector fields $\mathbf u,\mathbf v$ on $\mathbb{R}^n$, say. I know that we can describe this literally as $$\mathbf u\cdot\nabla\mathbf v = (u^i\partial_iv^j)_j$$ but this doesn't exactly seem to be coordinate invariant. My difficulty is in coming up with change-of-variables formulae, since these seem fairly complicated.

Answer 1

You are right: it may sound surprising if you haven't run into it before, but whereas the analagous expression for scalar functions $$\mathbf{u} \cdot \nabla f$$ (i.e., the directional derivative of $f$) is coordinate-free, the "directional derivative" of a vector field is not (one intuition is that this directional derivative cannot differentiate the "turning" of $\mathbf{v}$ from the turning of the coordinate system.)

The analgous coordinate-free object is the covariant derivative $\nabla_{\mathbf{u}}\mathbf{v}$ which, as you might imagine from the name, does transform correctly under change of coordinates. Defining a covariant derivative requires more than just differential structure; you have to specify how vectors "translate without turning" (parallel transport) along curves in the space. If you have a Riemann metric, that is enough to pin down a canonical covariant derivative (via the Levi-Civita connection), and you can calculate it in coordinates using Christoffel symbols (see https://en.wikipedia.org/wiki/Covariant_derivative#Coordinate_description) which are indeed fairly unpleasant to work with directly.