- #1
fluidistic
Gold Member
- 3,947
- 263
Homework Statement
Given 4 state variables x, y, z and w such that [itex]F(x,y,z)=0[/itex] and w depends on 2 of the other variables, show the following relations:
1)[itex]\left ( \frac{\partial x }{\partial y } \right ) _z = \frac{1}{\left ( \frac{\partial y }{\partial x } \right ) _z}[/itex]
2)[itex]\left ( \frac{\partial x }{\partial y } \right ) _z \left ( \frac{\partial y }{\partial z } \right ) _x \left ( \frac{\partial z }{\partial x } \right ) _y=-1[/itex]
3)[itex]\left ( \frac{\partial x }{\partial w } \right ) _z=\left ( \frac{\partial x }{\partial y } \right ) _z\left ( \frac{\partial y }{\partial w } \right ) _z[/itex]
4)[itex]\left ( \frac{\partial x }{\partial y } \right ) _z=\left ( \frac{\partial x }{\partial y } \right ) _w+\left ( \frac{\partial x }{\partial w } \right ) _y \left ( \frac{\partial w }{\partial y } \right ) _z[/itex]
Homework Equations
Hints: for 1) and 2) think about x as x(y,z) and then y=y(x,z)
For 3) choose x=x(x,z)
For 4) choose x=(y,w)
The Attempt at a Solution
Stuck on 1). I'd be tempted to consider differentials like numbers and that way 1) would be instantly "proven". However I do not see how to use the tips provided.