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Let $f(r,\alpha)=(r\cos\alpha,r\sin\alpha)$ for $(r,\alpha)$ in $\mathbb{R}^2$ with $r $ non-zero. Which of the following statements are correct?

  1. The linear transformation $Df(r,\alpha)$ is not zero for any $(r,\alpha)$ in $\mathbb{R}^2$ with $r$ non-zero.
  2. $f$ is one-one on $\{(r,\alpha) \in \mathbb{R}^2 \text{ : $r$ is non-zero} \}$.
  3. For any $(r,\alpha)$ in $\mathbb{R}^2$ with $r$ non-zero, $f$ is one-one on a neighbourhood of $(r,\alpha)$.
  4. $Df(r,\alpha)=r^2I$ for any $(r,\alpha)$ in $\mathbb{R}^2$ with $r$ non-zero.
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yes, no, yes, no –  uncookedfalcon Dec 19 '12 at 7:01
    
how to differentiate this? –  antara Dec 19 '12 at 7:03
    
2. Fix an $r$. What happens if you keep increasing $\alpha$? 3. What happens when you adjust $\alpha$ just a small bit in either direction? (How far around do you have to go for $f$ to not be one-to-one?) –  Alexander Gruber Dec 19 '12 at 7:06
    
cant get it sir ,can you plz give an elaborate explnation –  antara Dec 19 '12 at 7:18
    
You need to do some of your own work. –  copper.hat Dec 19 '12 at 7:20

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