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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

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# Looking for a rectifying chart

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