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Partial derivative chain rule?

  1. Sep 8, 2013 #1
    Is [tex]\frac{∂y}{∂x}×\frac{∂x}{∂z}=-\frac{∂y}{∂z}[/tex]?
     
  2. jcsd
  3. Sep 8, 2013 #2

    CompuChip

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    No, the chain rule does not involve such a minus sign.
    Why are you asking?
     
  4. Sep 8, 2013 #3

    arildno

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    It is true that IF the 3 variables x,y,z are related together in a sufficiently nice condition G(x,y,z)=0, THEN, we may solve for, in a region about a solution point, one of the variables in terms of the other two, i.e, functions X(y,z), Y(x,z), Z(x,y) exist, so that within that region we have that G(X(y,z),y,z)=G(x,Y(x,z),z)=G(x,y,Z(x,y)=0 identically.

    In these cases, it is true that we have the counter-intuitive result:
    [tex]\frac{\partial{Y}}{\partial{x}}\frac{\partial{X}}{\partial{z}}\frac{\partial{Z}}{\partial{y}}=-1[/tex]
     
  5. Sep 8, 2013 #4

    arildno

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    To give a simple example.
    Consider the function G(x,y,z)=x+y+z

    Now, the condition G(x,y,z)=0 gives rise to the equation x+y+z=0
    We may now form three separate function definitions:
    X(y,z)=-y-z
    Y(x,z)=-x-z
    Z(x,y)=-x-y

    We have now that:
    G(X(y,z),y,z)=-y-z+y+z=0, i.e, the condition G=0 is satisfied IDENTICALLY, for all choices of y and z.
    Similarly with the other two substitutions.

    We see that in this case, that we have:
    [tex]\frac{\partial{Y}}{\partial{x}}=\frac{\partial{X}}{\partial{z}}=\frac{\partial{Z}}{\partial{y}}=-1[/tex]
    and therefore,
    [tex]\frac{\partial{Y}}{\partial{x}}\frac{\partial{X}}{\partial{z}}\frac{\partial{Z}}{\partial{y}}-1[/tex]
     
  6. Sep 13, 2013 #5
    Try using the simple example z = x + y

    Isn't there a minus sign?
     
  7. Sep 13, 2013 #6

    HallsofIvy

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    In that specific case, the equation is true but it is NOT "the chain rule". Your initial post implied that you were offering this as a general formula derived from the chain rule.
     
  8. Sep 13, 2013 #7
    Is there a general formula for partial derivatives or is it a collection of several formulas based on different conditions?

    Anyway, consider a function w ( x, y, z ). We can then derive z ( x, y, w ).

    In this case, is
    [tex]\frac{∂w}{∂z}×\frac{∂z}{∂x}=-\frac{∂w}{∂x}?[/tex]
     
  9. Sep 13, 2013 #8

    arildno

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    "w ( x, y, z ). We can then derive z ( x, y, w )."
    This is meaningless.
     
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