Independent / Dependent variables for implicit functions

[phrankle]

This probably has a really simple answer.

Forr u=x^2-y^2 and v=x^2+y^2

x and y are apparently the dependent variables. But the independent variable is the input while the dependent variable is the output, so since u=f1(x,y) and v=f2(x,y) shouldn't they (u and v) be the dependent variables?

 

[mathwonk]

a function has a domain and a range, by convention. an equation can often be solved for some vabls if others are given. doing this allows one to view the solved for vabls as depending on the given ones, and hence produces a function.

but which variables you choose to give, and yiou choose to solve for is up to you. i.e. when you write down an equation there is no way of telling which vbls are dependent and which are independent.

having said that, we are lazy beings, and in the equation v = x^2 + y^2, it is natural nit to want to do any work, and hence natural view v as solved for and x,y as given, because the equation is already solved for v.

so many people would think that here v depends on x,y although the equation can also be solved for either x or y, at least under certain restrictions that square roots are permissible.

having said all this, it still appears to me that you have matters exactly backwards, as in your equations, being lazy, i would have said x,y are apparently INdependent, not dependent vbls.

 

[phrankle]

Thanks for the answer. I understand that it could go either way, but, "being lazy", why would you choose x and y as independent?

"In mathematics, an independent variable is any of the arguments, i.e. "inputs", to a function. These are contrasted with the dependent variable, which is the value, i.e. the "output", of the function. Thus if we have a function f(x), then x is an independent variable, and f(x) is a dependent variable. The dependent variable depends on the independent variables; hence the names."
according to wikipedia's dependent and independent variable page (can't link to it because I haven't made 15 posts yet).

Doesn't this correspond to u=f1(x,y)=x^2-y^2 and v=f2(x,y)=x^2+y^2
with (u,v) being dependent variables?

 

[HallsofIvy]

Where are you told that x and y are the "dependent variables"? There is, in general, no "mathematical" definition of "independent" and "dependent" variables- which you choose to be "independent variables" and which "dependent variables" depends upon the particular problem you are trying to solve.