1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Finding an equation of Partial Derivatives

  1. Oct 27, 2012 #1
    1. The problem statement, all variables and given/known data

    If f(x,y,z) = 0, then you can think of z as a function of x and y, or z(x,y). y can also be thought of as a function of x and z, or y(z,x)
    Therefore:

    [tex] dz= \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y} dy [/tex]
    and
    [tex] dy= \frac{\partial y}{\partial x}dx + \frac{\partial y}{\partial z} dz [/tex]

    Show that
    [tex] 1= \frac{\partial z}{\partial y} \frac{\partial y}{\partial z} [/tex]
    and then
    [tex] -1= \frac{\partial x}{\partial z} \frac{\partial y}{\partial x}\frac{\partial z}{\partial y} [/tex]
    2. Relevant equations

    3. The attempt at a solution
    Substituting [itex] dy [/itex] into the [itex] dz [/itex] equation you get
    [tex] dz = \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y} \frac{\partial y}{\partial x}dx + \frac{\partial y}{\partial z}\frac{\partial z}{\partial y}dz [/tex]

    This can be rearranged to show
    [tex] dz (1-\frac{\partial z}{\partial y}\frac{\partial y}{\partial z}) = dx(\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial x}) [/tex]
    and then
    [tex](1-\frac{\partial z}{\partial y}\frac{\partial y}{\partial z}) = \frac{dx}{dz}(\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial x}) [/tex]

    In order to show that [itex]\frac{\partial z}{\partial y}\frac{\partial y}{\partial z} = 1 [/itex], I only need to show that [itex] (\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial x})=0 [/itex], but I'm not sure how to do that. As for the second equation, I'm not sure where to get a [itex] \frac{\partial x}{\partial z} [/itex] into the equation in the first place.
     
  2. jcsd
  3. Oct 27, 2012 #2
    Attached
     

    Attached Files:

    • 001.jpg
      001.jpg
      File size:
      16.2 KB
      Views:
      76
  4. Oct 27, 2012 #3
    Thanks for the reply!

    What do [itex] f_y [/itex] and [itex] f_z [/itex] refer to?
     
  5. Oct 27, 2012 #4
    These are the partial derivatives of f with respect to y and z,respectivly.
     
  6. Oct 27, 2012 #5
    Oh okay, thing is, that's the way the instructor did it, while we were asked to derive expressions for the total differential for dz and dy, then substitute the latter into the former, sorry if I wasn't clear. Thanks though!
     
  7. Oct 27, 2012 #6
    Attached
     

    Attached Files:

    • 002.jpg
      002.jpg
      File size:
      17.9 KB
      Views:
      73
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Finding an equation of Partial Derivatives
Loading...