Doing basic geometry with a vector field in the plane, I encountered what looks like a basic object and wonder if it is a well-known concept, which would help me relate this to known literature.
Given a differentiable vector field $\boldsymbol{f}$ in the plane with coordinates $f_1 (x_1,x_2)$ and $f_2 (x_1,x_2)$, does the vector field $\boldsymbol{g}$ defined as \begin{equation} \boldsymbol{g} (x_1,x_2) = f_1 (x_1,x_2) \nabla f_2 (x_1,x_2) - f_2 (x_1,x_2) \nabla f_1 (x_1,x_2) \end{equation} have a name? Can it be easily related to standard vector operators like the curl?
This object has interesting properties that quantify how "parallel" the field is locally. It is zero when all the vectors point in the same direction, independently of their variations in norm. Conversely, the direction $\boldsymbol{g}$ is pointing at what seems to be the direction of "maximal variation in direction" of $\boldsymbol{f}$.
No, this has no name. But here's why your last paragraph takes place. Notice that — provided $f_1\ne 0$ — $$\vec g = f_1^2\, \nabla\left(\frac{f_2}{f_1}\right).$$ Thus, $\vec g = \vec 0$ when the function $f_2/f_1$ is constant, i.e., when the vector field $\vec f$ has constant slope. Thus, as usual with the gradient, $\vec g$ points in the direction in which the slope of $\vec f$ changes most rapidly, as your last observation suggests.