I have recently started reading applied-sciences papers and am getting extremely confused about notation when considering function composition and partial derivatives.
For example, say we have a function $f$ that maps $x \in \mathbb R$, $y \in \mathbb R$ to $f(x,y) \in \mathbb R$. Say we also have a function $g$ that maps $z \in \mathbb R$ to $g(z) \in \mathbb R$, and now define $h(x,z) = f(x,g(z))$.
Chain rule yields $\frac{\partial h}{\partial z} (x,z) = \frac{\partial f}{\partial y}(x,g(z))$ $\frac{\partial g}{\partial z}(z)$, which is fine. But papers in this field typically go for $\frac{\partial}{\partial z} f(x,g(z)) = \frac{\partial}{\partial y} f(x,g(z)) \frac{\partial}{\partial z} g(z)$, without ever defining the composition $h$ in the first place. This can be especially confusing if one overloads $y$, yielding $\frac{\partial}{\partial y} f(x,g(y)) = \frac{\partial}{\partial y} f(x,g(y)) \frac{\partial}{\partial y} g(y)$ which simply doesn't make sense to me.
I know defining a new function every time we consider a composition can be cumbersome, but I don't see how the above is acceptable in papers that appear in academic journals. Am I missing something here? If the above is indeed confusing, then what would be a good notation standard to adopt instead of defining new functions for each composition?
Here is an example from a published paper:
My question is: is this notation ambiguous? When writing my own research papers, should I define new functions for each composition, or is the notation above acceptable?
Sometimes, shortcuts are necessary, and some of them are perfectly acceptable. One of those shortcuts is to remove the explicit dependency on the variables, i.e. to write $g$ instead of $g(y)$ in your example. Thus, one could write $$ \frac{\partial }{\partial y} f(x,g) = \frac{\partial }{\partial g} f(x,g)\frac{\partial g}{\partial y} $$ according to the chain rule. This is possible if the reader was told that $g$ is a function of $y$, and that $x$ doesn't depend on $y$. Even shorter $$ \frac{\partial f}{\partial y} = \frac{\partial f}{\partial g} \frac{\partial g}{\partial y}\, , $$ and still perfectly acceptable, provided that $f$ was defined as a function of $x$ and $g$.