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Multivariate Chain Rule

  1. Jul 21, 2010 #1
    1. The problem statement, all variables and given/known data
    Find
    [tex] \frac{\partial z}{\partial y} [/itex]
    where z=F(u,v,y), u=f(v,x), v=g(x,y).

    3. The attempt at a solution
    If I remember multivariate calculus at all, this should be (please forgive the abuse of notation)

    [tex] \frac{\partial z}{\partial y} = \frac{\partial z}{\partial u}\frac{\partial u}{\partial v}\frac{\partial v}{\partial y} + \frac{\partial z}{\partial v}\frac{\partial v}{\partial y} + \frac{\partial z}{\partial y} [/tex]

    However, I have a book in front of me that says it should be

    [tex] \frac{\partial z}{\partial y} = \frac{\partial z}{\partial v}\frac{\partial v}{\partial u}\frac{\partial u}{\partial y} + \frac{\partial z}{\partial u}\frac{\partial u}{\partial y} + \frac{\partial z}{\partial y} [/tex]

    I think I'm correct and that they must have confused u and v. If someone could verify this for me it would be much appreciated.
     
  2. jcsd
  3. Jul 21, 2010 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    I agree. It think they mixed up u and v.
     
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