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Looking for a rectifying chart

  1. Jun 29, 2011 #1
    given task
    Find a rectifying chart at (0,0) of the vector field
    X: R^2 -> TR^2 , (x,y) -> X(x,y)
    with X(x,y):= (cos^2(x+y)+sin^2(x-y))d/dx + (cos^2(x+y)-sin^2(x-y))d/dy .

    attempt of solution
    We know that there exists a rectifying chart since X(0,0)=(1,1).
    A transversal direction would be (1,-1).

    Therefore we get the differential equations:
    dx/dt= cos^2(x+y)+sin^2(x-y) (1)
    and
    dy/dt= cos^2(x+y)-sin^2(x-y) (2)

    Next we have to find theire solutions. Change of variables and integration lead me to the general solution of (1):

    t+const = sec(2y)*tan^-1(sin(x-y)*sec(x+y))

    => tan(t*sin(2y))+const = sin(x-y)/sin(x+y)
    where I got stuck... //Recall we have to solve this equation for x and (2) for y.

    Maybe I should change my coordinate system at the very beginning, such that x+y=a and b=x-y? What do you think? Any ideas?

    Kind regards,
    mihyaeru
     
  2. jcsd
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