Confusion in notation of functions.

Question

Let us consider the following notations for $x \in X,y\in Y ,z \in Z$.

$$F(x,y,z)=x^yy^z$$

$$F_x(y,z)=x^yy^z$$

I am clear with former notation , but I saw latter one too , what's the difference between them ?

Is there any valid difference ?

If no difference then why multiple notations ?

Answer 1

$F$ is a function of three variables ($x, y, z$). Whereas $F_{x}$ consists of two variables ($y, z$) and $x$ is fixed.

Answer 2

There are many ways to write the evaluation of a univariate function:

• $f(x)$ is the function $f$ evaluated at $x$
• $fx$ is the function $f$ evaluated at $x$. Or occasionally the function $x$ evaluated at $f$
• $a_n$ is the function $a$ evaluated at $n$. Usually $n$ is drawn from some interval of integers and we call $a$ a "sequence" rather than a "function"
• $a^n$ is the function $a$ evaluated at $n$. We don't usually use superscripts this way, but in some contexts it's very useful.
• ${}_na$ is the function $a$ evaluated at $n$. Again this is unusual but occasionally convenient (usually because we are using notation where the other places around $a$ already have something, so we can't use that spot for notating evaluation)

There are surely others that get used, but this is all I could think of off hand.

Parsing your second example, $F_x$ is the function $F$ evaluated at $x$. But the values of $F$ are functions rather than, say, numbers. Thus $F_x(y,z)$ is the evaluation of the function $F_x$ at $(y,z)$.

Note that $F(x,y,z)$ and $F_x(y,z)$ can't make sense in the same context; the usage of $F$ in the two cases is for two different types of things. (although on some occasions there is utility in using the same letter to mean different things, it usually leads to confusion)