# Can we make sense of $\frac{\partial f(x_1, x_2, \dots, x_n)}{\partial g(x_1, x_2, \dots, x_n)}$?

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?

The construction you described doesn't actually work on manifolds - $$g'(t)$$ (or $$dg$$ as we would normally write) is not a tangent vector, but a cotangent vector; so the directional derivative $$\nabla_{dg} f$$ doesn't make sense without a metric (or some other additional structure) to identify $$TM$$ with $$T^* M$$.

It also doesn't seem to have the right behaviour to be casually called "$$\partial f / \partial g$$" - for example, $$\nabla_{dg} f$$ would double if you doubled $$g$$, which is the opposite scaling behaviour to what the notation would suggest.

Remember that partial derivatives are only defined in terms of a whole coordinate system - if you're given the two coordinate systems $$(x,y)$$ and $$(x,z=y-x),$$ the expression $$\partial f/\partial x$$ will have different values depending on whether you're holding $$y$$ or $$z$$ fixed! Thus I'm unsure how to interpret the intended spirit of "$$\partial f/\partial g$$".

Is this a well-known idea? If so, what is it called?

Not commonly enough to have a name; but it's a simple enough expression that you'll find it cropping up in equations here and there in vector calculus (as $$\nabla g \cdot \nabla f$$) and Riemannian geometry (as $$\nabla_{\operatorname{grad}g}f$$). The interpretation isn't really close to what you're looking for, however - it's just "the rate of change of $$f$$ in the direction $$\nabla g$$ (or vice versa).

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".

I'm unsure how to interpret this - to me, the way to "move" a scalar function is to deform it by another scalar function (e.g. deform $$f$$ to $$f + \epsilon \phi$$ for a parameter $$\epsilon$$), in which case what you're describing would just be the scalar $$g - f$$? If you could formalize what you're asking for here (or at least give more of a geometric description) there might be something more useful to say.

• You're right; I'll make my question more precise and ask again --- thanks a lot for pointing out that it doesn't really behave very well. If you could expand your answer on where on might see $\nabla_{grad~ g} f$, I'd be very interested! – Siddharth Bhat Feb 26 '20 at 9:51

I think $$\frac{\partial f(x_1, x_2, \dots, x_n)}{\partial g(x_1, x_2, \dots, x_n)}$$ should be able to predict small changes in $$f$$, given small changes in $$g$$. Now, if all we're given is a small change $$dg$$, then that's just $$\nabla g\cdot \vec{dr}$$. Even knowing $$\nabla{g}$$ at the point, we can't invert this dot product to get back $$\vec{dr}$$. Without knowing $$\vec{dr}$$, we can't predict the change in $$f$$, which is $$\nabla f\cdot \vec{dr}$$. So in this sense, the derivative is not define-able.