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Magnetic field lines

  1. Sep 6, 2010 #1
    Suppose that a function [itex]B:\mathbb{R}^n\to\mathbb{R}^n[/itex] and [itex]c:\mathbb{R}\to\mathbb{R}^n[/itex] are defined such that [itex]c[/itex] is differentiable, and

    [tex]
    \dot{c}(t) = B(c(t))
    [/tex]

    for all [itex]t[/itex]. The question is that what must be assumed of [itex]B[/itex], so that it would become possible to prove that

    [tex]
    c(T)=c(0)
    [/tex]

    with some [itex]T\neq 0[/itex]?
     
  2. jcsd
  3. Sep 7, 2010 #2
    Do you mean prove that c(dot)(T) = c(dot)(0)?
     
  4. Sep 7, 2010 #3
    No. I mean that the curve comes back to where it started from. (Not that it would point in the same direction at least twice.)
     
  5. Sep 7, 2010 #4
    [tex]\oint _{\partial S}B \bullet ndS = 0[/tex] or B must be divergence free.
     
  6. Sep 7, 2010 #5
    That answer is incorrect.

    [itex]n=2[/itex], [itex]B(x)=(x_1,-x_2)[/itex], [itex]c(t)=(e^t,e^{-t})[/itex] give a counter example.
     
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