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Math Help - constrained partial derivative identity

  1. #1
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    constrained partial derivative identity

    if f(x,y,z)=0 then,
    (\frac{\partial x}{\partial y})_z (\frac{\partial y}{\partial z})_x (\frac{\partial z}{\partial x})_y = -1

    I tried expressing \frac{\partial f}{\partial x} with z constant but this didnt work

    Im lost. Help much appreciated
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  2. #2
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    It should! With z constant and x a function of y, we can think of f as a function of y only:
    \frac{df}{dy}= \frac{\partial f}{\partial x}\frac{\partial x}{\partial y}+ \frac{\partial f}{\partial y}= 0
    so
    \left(\frac{\partial x}{\partial y}\right)_z= -\frac{\frac{\partial f}{\partial y}}{\frac{\partial f}{\partial x}}

    Now do the same for the other two
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  3. #3
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    Why have you written \frac{df}{dy} should it not be \frac{\partial f}{\partial y} as z=g(x,y)

    This is where my confusion arises
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