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Partial derivatives after a transformation

  1. Nov 25, 2012 #1
    Suppose I have a transformation

    [tex](x'_1,x'_2)=(f(x_1,x_2), g(x_1,x_2))[/tex] and I wish to find [tex]\partial x'_1\over \partial x'_2[/tex] how do I do it?

    If it is difficult to find a general expression for this, what if we suppose [itex]f,g[/itex] are simply linear combinations of [itex]x_1,x_2[/itex] so something like [itex]ax_1+bx_2[/itex] where [itex]a,b[/itex] are constants?
     
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  3. Nov 25, 2012 #2

    arildno

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    Let the following transformation rules hold:
    [tex]x'_1=f(x_1,x_2)[/tex]
    [tex]x'_2=g(x_1,x_2)[/tex]
    [tex]x_1=F(x'_1,x'_2)[/tex]
    [tex]x_2=G(x'_1,x'_2)[/tex]

    Then, we clearly have, for example the identity:
    [tex]x'_1=f(F(x'_1,x'_2),G(x'_1,x'_2))[/tex]
    Since the pair of marked variables (and the unmarked) are independent from each other, we have the requirement on the functional relationship:
    [tex]\frac{\partial{x}_{1}^{'}}{\partial{x}_{2}^{'}}=0=\frac{\partial{f}}{\partial{x}_{1}}\frac{\partial{F}}{\partial{x}_{2}^{'}}+\frac{\partial{f}}{\partial{x}_{2}}\frac{\partial{G}}{\partial{x}_{2}^{'}}[/tex]
     
    Last edited: Nov 25, 2012
  4. Nov 25, 2012 #3

    arildno

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    Suppose you have, say:
    x=u^2-v^2
    y=u+v
    How can you utilize those implied requirements in the previous post to derive the the correct representations of u and v in terms of x and y?

    Note that in this case, it is fairly trivial to find it by algrebrac means; dividing the first relationship by the second, we have x/y=u-v, and thus:
    u=1/2(x/y+y), v=1/2(y-x/y)

    However, using instead the four identities gained from the above requirements, we have
    [tex]\frac{\partial{x}}{\partial{x}}=1=2u\frac{\partial{u}}{\partial{x}}-2v\frac{\partial{v}}{\partial{x}}[/tex]
    [tex]\frac{\partial{x}}{\partial{y}}=0=2u\frac{\partial{u}}{\partial{y}}-2v\frac{\partial{v}}{\partial{y}}[/tex]
    [tex]\frac{\partial{y}}{\partial{y}}=1=\frac{\partial{u}}{\partial{y}}+\frac{\partial{v}}{\partial{y}}[/tex]
    [tex]\frac{\partial{y}}{\partial{x}}=0=\frac{\partial{u}}{\partial{x}}+\frac{\partial{v}}{\partial{x}}[/tex]

    from these, you should be able to derive the above relations.

    Note, for example, that by combining the first and fourth and the known relation y=u+v, you get:
    [tex]\frac{\partial{u}}{\partial{x}}=\frac{1}{2y}[/tex]
    meaning that we must have u(x,y)=x/2y+h(y), where h(y) is some function of y.
    Applying the fourth, you get v(x,y)=b(y)-x/2y, for some b(y)

    By using known and gained information, you will be able to determine h(y) and b(y) from the second and third relationships.
     
    Last edited: Nov 25, 2012
  5. Nov 25, 2012 #4
    Thank you, arildno.
     
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