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Having problems with the (I think) general chain rule

  1. Mar 7, 2013 #1
    1. The problem statement, all variables and given/known data

    Consider the transformation [itex]\mathbf{x}=G(\mathbf{u}), \text{ where } \mathbf{x}=(x_1,x_2,x_3),\:\mathbf{u}=(u_1,u_2,u_3)[/itex] given by[tex]x_1=u_1+u_3^2[/tex][tex]x_2=u_3-u_1^2[/tex][tex]x_3=u_1+u_2+u_3[/tex]

    I need to compute the derivative of this transformation, and then show that the transformation is locally invertible if [itex]u_1u_3>0[/itex]

    2. Relevant equations

    unsure really.

    3. The attempt at a solution

    well x is a function of [itex]x_1,x_2,x_3[/itex] and all the [itex]x_j's[/itex] are functions of [itex]u_1,u_2,u_3[/itex]. So I am dealing with a map [itex]\mathbb{R}^3[/itex] to [itex]\mathbb{R}^3[/itex] right?

    sorry i'm really lost on what this question is asking. mostly focusing on the firsrt part right now (the derivative)
     
    Last edited: Mar 7, 2013
  2. jcsd
  3. Mar 7, 2013 #2

    LCKurtz

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    I suppose that third line is supposed to read ##x_3=u_1+u_2+u_3##. Is that what is causing your problems? Can you calculate the Jacobian now?
     
  4. Mar 7, 2013 #3
    that's a typo yes, but not what is causing my problem.
     
  5. Mar 7, 2013 #4
    [tex](x_1,x_2,x_3)=G(u_1,u_2,u_3)[/tex]
    [tex]D\mathbf{x}=\left[\begin{matrix} \frac{\partial x_1}{\partial u_1} & \frac{\partial x_1}{\partial u_2} & \frac{\partial x_1}{\partial u_3} \\ \frac{\partial x_2}{\partial u_1} & \frac{\partial x_2}{\partial u_2} & \frac{\partial x_2}{\partial x_3} \\ \frac{\partial x_3}{\partial u_1} & \frac{\partial x_3}{\partial u_2} & \frac{\partial x_3}{\partial u_3}\end{matrix}\right]
    [/tex]
     
  6. Mar 7, 2013 #5
    is that the jacobian?
     
  7. Mar 7, 2013 #6

    LCKurtz

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    Yes. You can read about the Jacobian and its properties here:

    http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant
     
  8. Mar 7, 2013 #7
    ok cool, thanks a lot!

    I'm sure i'll be able to hack through the rest of it. I'm quite frustrated with my textbook right now (folland). It seems to be more of a reference text than something that I can actually learn from. Still looking for a companion text, or even better something like khan academy.
     
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