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Math Help - Linear approximation

  1. #1
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    Linear approximation

    Let f:\mathbb R^2\to\mathbb R so that f(x,y)=(s,t)=\left( x+\dfrac{1}{2}\arctan y,y+\dfrac{1}{2}\arctan x \right)

    Find the linear approximation of f^{-1} on a neighborhood of (s_0,t_0)=f(0,1).

    Don't worry about the invertibility, that was another question which I solved, but now I need help with this one, I don't get how to solve it.
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  2. #2
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    By the chain rule, we get that Df(0,1)\circ Df^{-1}(s_0,t_0)=I.
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  3. #3
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    Okay... but, I don't know how to relate it to find the linear approximation.

    If you could help me a bit more.
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  4. #4
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    It's pretty straightforward: Since that relation gives that the derivatives involved are invertible we get Df^{-1}(s_0,t_0)=Df(0,1)^{-1}
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  5. #5
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    Oh, now I get, so I just need to compute the inverse but do I need to multiply it by a vector or something?
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  6. #6
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    The linear approximation to g(x, y) at (x_0, y_0) is g(x_0, y_0)+ D_g(x_0, y_0)(x, y) where the last term is the product of the matrix D_g(x_0, y_0) with the vector (x, y).
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