I recently asked a question and believe a new forum is necessary. The question was in regard to partial derivatives and how we can treat the variables x, y, z in function F(x, y, z) where z = f(x, y), as functions of x and y.
Someone confirmed that we can treat x for example, as a function of x and y. However, I cannot make sense of such a function. To me, it would be something like x = x^2 + y^3 + xy.
This does not make sense to me as it is not a function of x and y and at most is a function of either x or y but not both. Also, why can we make the assumption that x is a function of x and y?
I really need clarification. Thank you in advance
$x$ can be a function of $y$ (and of any further variable) but not of itself.
However the equation you gave ($x = x^2 + y^3 + xy$) could be read as an implicite function. For instace you might transfer all variables onto a single side (then equating to zero), and then solve for $x$ by means of the "midnight formula", resulting then in some $x=g(y)$.
Wrt. your original question within $F(x,y,z)$ the $z$ could well be given as a further function $f(x,y)$, thus providing thereby an additional restriction. That one reduces your original function $F(x,y,f(x,y))$ into some other function $h(x,y)$ only.
--- rk