Suppose we have differentiable functions $h: \mathbb{R}^{3} \to \mathbb{R}$, $g: \mathbb{R}^{3} \to \mathbb{R}^{3}$ and $f: \mathbb{R}^{3} \to \mathbb{R}$ such that $$ h(x,y,z) = (f \circ g)(x,y,z) = f(g_{1}(x,y,z), g_{2}(x,y,z), g_{3}(x,y,z)) $$ where the $g_{i}$ are the components of $g$ in each coordinate direction.
If we use the multivariate chain rule to compute the partial derivative of $h$ w.r.t. $x$, we get $$ \frac{\partial h}{\partial x}(x,y,z) = \sum_{i=1}^{3}\frac{\partial f}{\partial g_{i}}(x,y,z) \cdot \frac{\partial g_{i}}{\partial x} (x,y,z) $$ On a theoretical level, I understand what's going on in the chain rule: my question isn't about the chain rule. I know that we're trying to find $Df(g(a)) \circ Dg(a)$ for some $a \in \mathbb{R}^{3}$ and we get the result above through a computation with the Jacobians. It's the notation I'm confused about. I have a few questions about this:
1) When we take the partial of $f$ w.r.t. $g_{i}$ above, what does it mean for us to use $g_{i}$ as a coordinate direction? I know that $f$ takes in $g_{i}$ as an argument, but how do you view a function as a coordinate direction?
2)In the expression above, what should the arguments of $\frac{\partial f}{\partial g_{i}}$ be? Should it be $(x,y,z)$ or $(x, g_{2}(x,y,z), g_{3}(x,y,z))$ (supposing $i=1$)?
3) In the case of $i=1$, for instance, is $ \frac{\partial f}{\partial g_{i}}(x,g_{2}(x,y,z),g_{3}(x,y,z)) = \frac{\partial f}{\partial x}(g_{i}(x,y,z), g_{2}(x,y,z), g_{3}(x,y,z)) $? Because the "regular" chain rule expression requires us to evaluate the partial derivatives of $f$ at $g(x,y,z$).