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} $ .
Answer
Answers can be viewed only if
 The questioner was satisfied and accepted the answer, or
 The answer was disputed, but the judge evaluated it as 100% correct.
1 Attachment

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?

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.
The answer is accepted.
 answered
 205 views
 $10.00
Related Questions
 Multivariable Calculus Problem Set
 Calc 3 Question
 Prove that $\int_{\infty}^{\infty}\frac{\cos ax}{x^4+1}dx=\frac{\pi}{2}e^{\frac{a}{\sqrt{2}}}(\cos \frac{a}{\sqrt{2}}+\sin \frac{a}{\sqrt{2}} )$
 Multivariable Calculus Questions
 Rose curve
 Prove that $p_B :\prod_{\alpha \in A} X_\alpha \to \prod_{\alpha \in B} X_\alpha$ is a continuous map
 Evaluate $\iint_{\partial W} F \cdot dS$
 Find the coordinates of the point $(1,1,1)$ in Spherical coordinates