Why does $F(x,y,z)= \text{constant}$ imply that the total differentials $dx,dy,dz$ exist?

Question

I'm confused about the assumption on the wikipedia page for exact differentials. Why must it hold that $F(x,y,z)=\text{constant}$ for the total differentials of $x,y,z$ to exist? It seems like they should exist regardless.

Answer 1

Notice the phrase “$x$, $y$, $z$ are bound by...”.

This is a way of saying that the equation $F(x,y,z)=C$ implicitly defines functions $$ x = f(y,z) ,\quad y = g(x,z) ,\quad z = h(x,y) , $$ and it's the differentials $df$ and $dh$ that are computed (under the names of $dx$ and $dz$, since the functions $f$ and $h$ were never given any names).

(Assuming also, of course, that the conditions for the implicit function theorem are fulfilled.)

Answer 2

Just to add to Hans answer which gives a great explanation.

They are not necessarily functions on a global scale, but locally, if $F$ is nice enough. For example $$F(x,y) = x^2+y^2 = 1$$ often has two global $f(y)$ but only one that makes sense locally around some point on the surface.