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Homework Help: Showing that the complex functions are constant in t. Please Help

  1. Oct 5, 2013 #1
    Showing that the complex functions are constant in t. Please Help!! :)

    1. The problem statement, all variables and given/known data
    The complex position vectors of two parallel interacting equal fluid vortices moving with their axes of rotation always perpendicular to the z-plane are z1 and z2. The location of the vortices are then given by their respective coordinates x1,y 1 and x2, y2. It turns out to be useful to think of these coordinates as specifying a point in the complex plane, in which case the locations of the two fluids at a given time t are given by two complex numbers z1(t)=x1(t)+iy1(t) and z2(t)=x2(t)+iy2(t). The equations governing their motions are:

    (1) Deduce that (a) z1+z2, (b)z1-z2, (c)z12+z22 are all constant in time,(2) and hence descibe the motion geometrically.

    2. Relevant equations
    General z:
    Complex Conjugate:
    Modules of Norm of z:

    A constant function is define by:
    [itex]\frac{df(t)}{dt}[/itex]=0 OR f'(t)=0

    3. The attempt at a solution
    Wow. That was a "wordy" question. So I think I got 1.a but I'm requesting help on the rest of the question.
    (a) z1+z2
    Now I'm not sure if I should be showing whether
    Or if
    Either way for this question I get...

    (b) z1-z2
    This is where I start to get lost. I apply a similar method:
    And I don't know what to do to make it zero.

    (c) I have not attempted yet.

    (2) I don't know how I would represent them geometrically on an x,y plane.

    Thank you for the help, I really appreciate it and the work you guys do on here :)
  2. jcsd
  3. Oct 5, 2013 #2
    First: yes ##\frac {d(z_1 +z_2)}{dt} = \frac {z_1}{dt} + \frac {z_2}{dt}.## d/dt is always a linear operator.
    For part b you are practically there. You didn't apply your conjugate in the last step. (It had to be a minus sign somewhere).

    For the ##z^2## situation, you need to use the chain rule. df/dt = df/dz##\cdot## dz/dt.

    Re geometric representation, unfortunately I am geometry-blind -- I have no idea how things look; but I'm sure someone else will help.
    Last edited: Oct 5, 2013
  4. Oct 5, 2013 #3


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    Science Advisor

    brmath clearly meant [itex]\frac{dz_1}{dt}+ \frac{dz_2}{dt}[/itex].

    For the "geometric representation", if [itex]z_1- z_2[/itex] and [itex]z_1+ z_2[/itex] is constant, then so is [itex](z_1- z_2)(z_1+ z_2)= z_1^2- z_2^2[/itex]. If, in addition, [itex]z_1^2+ z_2^2[/itex] is constant, so is [itex](z_1^2+ z_2^2)+ (z_1^2- z_2^2)= z_1^2[/itex] which, of course, means that [itex]z_2^2[/itex] is also always constant. This set will consist of four points: (a, b), (-a, b), (a, -b), and (-a, -b).
  5. Oct 5, 2013 #4
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