Prove that $f$ is a diffeomorphism $C^∞$, that maps... (More inside)

Prove that $f:(x, y) ∈ R^2$ - {(0, 0)} → ($\frac{x}{x^2 +y^2} $ ; $\frac{y}{x^2+y^2} $) ∈ $R^2$ - {(0, 0)} is a diffeomorphism $C^∞$, that maps each circle of radius $r > 0$ centered in the origin to a cocentric circle with radius $\frac{1}{r} $ .

Multivariable Calculus Manifolds Functional Analysis
Matchmaticians Alumnus Matchmaticians Alumnus
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Erdos Erdos
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  • Matchmaticians Alumnus Matchmaticians Alumnus
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    Hey, just one question, I don't really understand how that last part proves the last argument (that f maps each circle of radius r > 0 centered in the origin to a concentric circle with radius 1/r). Would you be able to clarify that?

  • Erdos Erdos
    0

    Note that if |(x,y)| =s, then |f(x,y)|=|(u,v)|=1/s. When s varies between (0,r), 1/s covers (1/r, infinity). I hope this helps.

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