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Math Help - implicit Partial differentiation

  1. #1
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    implicit Partial differentiation

    Hi,
    I have a problem that has to do with implicit partial differentiation. I was wondering how I could solve it using Jacobian determinants, if not possible, then a straight forward approach is okay. Here we go!

    Given:
    f(u,v) = 0
    u = lx + my + nz
    v = x^{2} + y^{2} + z^{2}
    Prove:
    (ly - mx) + (ny - mx) \frac{\partial z}{\partial x} + (lz - nx)\frac{\partial z}{\partial y} = 0

    I have a problem finding the two partial terms. This is what I have done:

    \frac{\partial z}{\partial x} = -\frac{\frac{\partial (u,v)}{\partial (x,y)}}{\frac{\partial (u,v)}{\partial (z,y)}}

    \frac{\partial z}{\partial y} = -\frac{\frac{\partial (u,v)}{\partial (y,x)}}{\frac{\partial (u,v)}{\partial (z,x)}}

    but I end up with ly - mx = 0, not 0 = 0

    Any suggestions? Thanks!
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  2. #2
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    Re: implicit Partial differentiation

    Quote Originally Posted by gotFermi View Post
    Hi,
    I have a problem that has to do with implicit partial differentiation. I was wondering how I could solve it using Jacobian determinants, if not possible, then a straight forward approach is okay. Here we go!

    Given:
    f(u,v) = 0
    u = lx + my + nz
    v = x^{2} + y^{2} + z^{2}
    Prove:
    (ly - mx) + (ny - mx) \frac{\partial z}{\partial x} + (lz - nx)\frac{\partial z}{\partial y} = 0

    I have a problem finding the two partial terms. This is what I have done:

    \frac{\partial z}{\partial x} = -\frac{\frac{\partial (u,v)}{\partial (x,y)}}{\frac{\partial (u,v)}{\partial (z,y)}}

    \frac{\partial z}{\partial y} = -\frac{\frac{\partial (u,v)}{\partial (y,x)}}{\frac{\partial (u,v)}{\partial (z,x)}}

    but I end up with ly - mx = 0, not 0 = 0

    Any suggestions? Thanks!
    Your notation doesn't make any sense. I would advise writing each function as z in terms of the other variables, then doing the differentiation.
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  3. #3
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    Re: implicit Partial differentiation

    What terms don't make sense? I can change that, I am just copying how my book sells the ideas.

    I'll give the rewriting a try, thanks.
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