Let's say I have a vector field $V$ on $\mathbb{R}^2$. Is there a notion of derivative/divergence at a point that's agnostic to $||V(p)||$? In other words, how do I measure if flows are parallel? For instance, for any scaling of a rotational vector field, $V = \alpha \cdot (y \frac{\partial}{\partial x} - x \frac{\partial}{\partial y}), \alpha: \mathbb{R}^2 \rightarrow \mathbb{R}^+$, this derivative should be zero everywhere.
The closest tool I could think of is directional derivative in orthogonal direction, $D_{V^\perp}\frac{V}{||V||}$, but this does not seem justified.