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Partial derivative problem

  1. Mar 12, 2012 #1

    fluidistic

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    Gold Member

    1. The problem statement, all variables and given/known data
    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]


    2. Relevant 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)

    3. 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.
     
  2. jcsd
  3. Mar 12, 2012 #2
    I assume (dx/dy)_z means derivative of x w.r.t. y when z is fixed.
    since F=0, dF=F_x dx+F_y dy+F_z dz=0. When z is fixed, dz=0, so F_x dx=-F_y dy, etc.
     
  4. Mar 12, 2012 #3

    fluidistic

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    Gold Member

    Thank you very much for this huge tip. Will be working on that problem and post if I'm stuck.
     
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