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Let $U$ be the open set $(0,∞)×(0,2π)$ in the $(r,θ)$-plane $\mathbb{R}^2$.

We define $F : U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^2$ by $F (r, θ ) = (r \cos θ , r \sin θ )$.

How to decide whether F is orientation-preserving or orientation-reversing as a diffeomorphism onto its image?

I know that an automorphism $f : V \rightarrow V$ is said to be orientation-preserving (or orientation-reversing) if it maps a basis to another basis of the same (or the opposite) orientation.

But I dont know how to apply this definition to this question.

Please explain the reason why $F$ is orientation-preserving or orientation-reversing in an explicit and instructive way. Thank you.

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    $\begingroup$ Please explain explicitly and informatively what you know (definitions would be a start), and what you've tried. You know the "routine" by now. $\endgroup$
    – amWhy
    Commented May 7, 2013 at 17:15
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    $\begingroup$ @amWhy for president! $\endgroup$
    – Git Gud
    Commented May 7, 2013 at 17:17
  • $\begingroup$ Okay. Sorry I forgot to write my konwledge. @amWhy $\endgroup$
    – 1190
    Commented May 7, 2013 at 17:17

1 Answer 1

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HINT:

If the Jacobian matrix has negative determinant then it is orientation reversing. If it has positive determinant then it is orientation preserving.

The Jacobian matrix, in this case, is the two-by-two matrix whose columns are $F_{r}$ and $F_{\theta}$. Can you find the partial derivatives, put them in a matrix and find its determinant?

You may find that the Jacobian is singular at some point.

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  • $\begingroup$ I know this rules. But I cannot apply these to my question. I cannot show. Please can you teach me how to show/apply these rules to my question? $\endgroup$
    – 1190
    Commented May 7, 2013 at 17:27
  • $\begingroup$ What? I am confused too much:( $\endgroup$
    – 1190
    Commented May 7, 2013 at 17:28
  • $\begingroup$ You have $F = (r\cos\theta,r\sin\theta)$. What are the two partial derivatives $F_{r}$ and $F_{\theta}$? (They will be vectors.) $\endgroup$ Commented May 7, 2013 at 17:29
  • $\begingroup$ $F_{θ}=-rsinθdθ+rcosθdθ$ and $F_{r}=cosθdr+sinθdr$ $\endgroup$
    – 1190
    Commented May 7, 2013 at 17:32
  • $\begingroup$ I found these. Okay! Then, what Will I do? $\endgroup$
    – 1190
    Commented May 7, 2013 at 17:32

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