Let $f, g \in \mathbb R^n \rightarrow \mathbb R$. Is there some way to make sense of the expression:
$$\frac{\partial f(x_1, x_2, \dots, x_n)}{\partial g(x_1, x_2, \dots, x_n)}$$
I want some way to measure "how much $f$ changes along $g$" --- I'm not sure what a reasonable definition of this. Here is one that came to mind.
Let us define $\frac{\partial f}{\partial g}\big(t \big)$: At each point $t \in \mathbb R^n$, we first compute $g'(t) \in \mathbb R^n$. Now, we compute the directional derivative of $f$ along $g'(t)$:
$$ \frac{\partial f(x_1, x_2, \dots x_n)}{g(x_1, x_2, \dots x_n)} (t) : \mathbb R^n \rightarrow \mathbb R\equiv (\nabla_{g'(t)} f)(t) = \lim_{h \rightarrow 0} \frac{f(t + hg'(t)) - f(t)}{h}$$
- Is this a well-known idea? If so, what is it called?
- My problem with this is that it returns a scalar at each point: what I am actually interested in is to find a new function which tells me "how to move $f$ infinitesimally so we can make it closer to $g$.
The given definition above clearly generalizes to any manifold: all I need is a directional derivative, which I do possess on a manifold: Can we say something interesting about this in a larger setting?