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Examples of closed loop functions

  1. Feb 7, 2013 #1
    Can someone please give me a list of examples of closed loop functions, the only one I know is the equation for a circle

    [itex]y^2 + x^2 = r^2[/itex]

    Also are there any closed loop functions that aren't multi variable, i.e in the form y=f(x) and not z=f(x,y)

    Is there a way to tell that a function is a closed loop without plotting it?


    PS, I'm not asking about closed loop integrals here, just functions where the line joins back to itself.
  2. jcsd
  3. Feb 7, 2013 #2
    Not in form of y=f(x), since will be multiple values.

    More general is to present it by pair of functions x=x(t), y=y(t)

    z=x^2+y^2 may be represented as x=sin(t), y=cos(t)

    For any pair of functions x(t) and y(t), with pair of values t0, t1,
    such that x(t0)=x(t1) and y(t0)=y(t1),

    there is a loop of length t1 - t0
  4. Feb 8, 2013 #3
    Ah okay thank you, this has cleared a few things up, but what do you mean by:
    This confuses me, do you mean y=f(t) and x=f(t)? And what do you mean by pair values?
  5. Feb 9, 2013 #4
    For example, will take that z=x^2+y^2 <=> x=sin(t), y=cos(t)

    Pair of functions: x=sin(t), y=cos(t)
    Pair of values: t0=0, t1=2*pi

    sin(t0) = sin(0) = 0 = sin(2*pi) = sin(t1)

    cos(t0) = cos(0) = 1 = cos(2*pi) = cos(t1)
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