A clarification of partial derivatives

I have been wondering, why do textbooks say things like "partial derivative with respect to $$x$$", for say a function from $$\mathbb{R}^4$$ to $$\mathbb{R}$$. The function could easily have been written with variables like $$a,b,c,d$$. Shouldn't it really be "partial derivative with respect to the first coordinate"? Is there a textbook that doesn't make this error?

• I don't know your textbook but I'm assuming that the function is written as $f: \mathbb R^4 \to \mathbb R$, $f(x,y,z,t) = ...$ where it would make sense to talk about a partial derivative with respect to $x$. If we have $f(x) = f(x_1, \dots, x_n)$, then it would not make much sense. So it really depends on how the variables are named. – Lukas Mar 18 at 15:56
• Yeah you are right. The first coordinate can be called anything. It's just customary to call it $x$, since authors trust people will understand what they mean. Did you ever have this question about functions $\mathbb R^2 \to \mathbb R$? They always say $x$ and $y$ in that case. – Jackozee Hakkiuz Mar 18 at 17:24

Well, it is customary to define the first coordinate as x. Had they chosen to define the first coordinate as a, then it would say the partial derivative with respect to a. You'll never find a partial derivative with respect to a coordinate where that coordinate has not yet been assigned a variable. Does that make sense? In other words, first we define the $$i^{th}$$ coordinate as some variable, and then we say "the partial derivative with respect to ____". The blank space being the given variable.