Recently, I've encountered this question about the chain rule here. The OP asked that
Let $F$ be a function of $x$ and $z$, and $z$ is a function of $x$. Then by the chain rule: $$\frac{dF}{dx} = \frac{\partial{F}}{\partial{x}} + \frac{\partial{F}}{\partial{z}}\frac{\partial{z}}{\partial{x}}$$ I'm not sure how the equation is derived.
In this question and the answer of the above question, the partial derivatives $\frac{\partial{F}}{\partial{x}}$ and $\frac{\partial{F}}{\partial{z}}$ appear. Moreover, the OP did not mention the domain and the codomain of $F$.
My textbook and Wikipedia only define partial derivatives when the domain of our function is $\mathbb R^n$. Is correct that the concept of partial derivatives is defined only for functions' the domain is $\mathbb R^n$?
Is it correct that the notations $\frac{\partial{F}}{\partial{x}}$ and $\frac{\partial{F}}{\partial{z}}$ in the linked question only make sense when the domain of $F$ is $\mathbb R^n$?
Thank you for your help!