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Math Help - Implicit Function Theorem

  1. #1
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    Exclamation Implicit Function Theorem

    6. (20 points) Consider the system of equations
    3
    x2y xz + wz = 3
    xy yz + 2wx = 4atthepointx=1,y=2,z=1,w=2. Canweviewyandwasfunctionsofxand


    z? If so, find ∂w.∂z
    I am having trouble understanding why dw/dz is -1 rather than one, as the Implicit function theorem has a negative sign when it is used. Any quick help would be greatly appreciated. Thanks!
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  2. #2
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    Re: Implicit Function Theorem

    You have:
    F(x,y,z,w)=0
    G(x,y,z,w)=0
    So
    Fxdx+Fydy+Fzdz+Fwdw=0
    Gxdx+Gydy+Gzdz+Gwdw=0

    Solve for dz and dw:
    dz=Adx+Bdy
    dw=Cdx+Ddy

    A=dz/dx, B=dz/dy, C=dw/dx, D=dw/dy

    That’s the principle. The rest is doing the partial derivatives Fx..., and algebra, if that interests you.

    EDIT: Or solve for dy and dw as functions of dx and dz to get partial you want.
    Last edited by Hartlw; April 2nd 2014 at 11:09 AM.
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