# What is the mean of rate of change of a function with respect to the domain of some other function, as in: $\frac{dFz}{dy}$

While studying 'curl' I came accross these terms: Here, I don't understand the meaning of $\frac{dFz}{dy}$. Fz is a function of 'z', so what is the meaning of rate of change of Fz with respect to 'y'?

I understand differentiation as the rate of change of value of a function with respect to a change in the value of its domain. But 'y' here, is not the domain of 'Fz'. If it is, then what is the meaning of all this?

## 1 Answer

$F_z$ is not (necessarily) a function of $z$. Notationally if $F$ was a function of z we would write $F(z)$, not $F_z$.

What is meant is that $F$ is a "vector field". That is, $\vec{F}(x,y,z)$ is a function which assigns a vector to every position in 3-space. So $F_x$, $F_y$ and $F_z$ refer to the x, y and z-coordinates of $\vec{F}$ at any location.

Any coordinate of $F$ could be a function of any position coordinate (or of all three).