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 only be viewed under the following conditions:
- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.
1 Attachment
Erdos
4.8K
-
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.
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- answered
- 1545 views
- $10.00
Related Questions
- Double, Triple, and Change in Variables of Integrals Problems
- Line Integral
- Compute $\iint_D \frac{dx dy}{\sqrt{1+x+2y}}$ on $D=[0,1]\times [0,1]$
- Stokes' theorem $\int_S \nabla \times F \cdot dS= \int_C F\cdot dr$ verification
- Exercise 4.33 from Spivak's Calculus on Manifolds.
- Integrate $\int_0^1\int_{\sqrt{x}}^{1}e^{y^3}dydx$
- Stoke's Theorem
- Prove that $\int_{-\infty}^{\infty}\frac{\cos ax}{x^4+1}dx=\frac{\pi}{2}e^{-\frac{a}{\sqrt{2}}}(\cos \frac{a}{\sqrt{2}}+\sin \frac{a}{\sqrt{2}} )$
