 #1
 357
 151
 Homework Statement:
 Given ##f(x,y,z) = f(x,y,z(x,y))## then what are the conditions involving the partial derivatives of $f$
 Relevant Equations:

Apparently they are
##\left(\frac{\partial f}{\partial x} \right)_y =\left(\frac{\partial f}{\partial y} \right)_x = \left(\frac{\partial f}{\partial z} \right)_z = 0##
where the subscript denotes the independent variable being held constant
As far as I know when a function is extremized its partial derivatives are all equal to 0 (provided we aren't dealing with a constraint)
##\left(\frac{\partial f}{\partial x} \right)_{yz} = \left(\frac{\partial f}{\partial y}\right)_{xz} = \left(\frac{\partial f}{\partial z}\right)_{xy} =0##
Let's start off
##\left(\frac{\partial f}{\partial x} \right)_{y} = \left(\frac{\partial f}{\partial x}\right)_{yz} + \left( \frac{\partial f}{\partial z} \right)_{xy} \left( \frac{\partial z}{\partial x} \right)_{y}##
I really don't know where to go from here. I'm not even sure my understanding of implicit functions is up to par to even begin addressing this question.
##\left(\frac{\partial f}{\partial x} \right)_{yz} = \left(\frac{\partial f}{\partial y}\right)_{xz} = \left(\frac{\partial f}{\partial z}\right)_{xy} =0##
Let's start off
##\left(\frac{\partial f}{\partial x} \right)_{y} = \left(\frac{\partial f}{\partial x}\right)_{yz} + \left( \frac{\partial f}{\partial z} \right)_{xy} \left( \frac{\partial z}{\partial x} \right)_{y}##
I really don't know where to go from here. I'm not even sure my understanding of implicit functions is up to par to even begin addressing this question.