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## Homework Statement

Given a certain poincare map, show that it is area preserving for all values of r

## Homework Equations

[tex] \binom{x_{n+1}}{y_{n+1}}=\begin{pmatrix}e^{r} & 0 \\ 0 & e^{-r} \end{pmatrix}\begin{pmatrix}cos(\phi+I_{n}) & -sin(\phi+I_{n}) \\ sin(\phi+I_{n}) & cos(\phi+I_{n}) \end{pmatrix}\begin{pmatrix}x_{n}\\ y_{n} \end{pmatrix}[/tex]

where

[tex]I_{n}=x_{n}^2+y_{n}^2[/tex]

## The Attempt at a Solution

I tried to prove that the determinant [tex] \binom{x_{n+1}}{y_{n+1}}=\begin{pmatrix}e^{r} & 0 \\ 0 & e^{-r} \end{pmatrix}\begin{pmatrix}cos(\phi+I_{n}) & -sin(\phi+I_{n}) \\ sin(\phi+I_{n}) & cos(\phi+I_{n}) \end{pmatrix}[/tex] is equal to 1, I believe that is wrong since the I in the matrix is X^2+Y^2, so I was wondering should i use the Poisson bracket method?