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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?
  2. jcsd
  3. Nov 25, 2012 #2


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    Let the following transformation rules hold:

    Then, we clearly have, for example the identity:
    Since the pair of marked variables (and the unmarked) are independent from each other, we have the requirement on the functional relationship:
    Last edited: Nov 25, 2012
  4. Nov 25, 2012 #3


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    Suppose you have, say:
    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

    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:
    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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