# Questions about polar change of coordinates

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.
-
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