How can I convert an annular sector to a rectangular region?

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I have an annular region $v=\{(r,\theta): r_1<r<r_2, a<\theta<b\} $. How can I transform this region to a rectangular region $u=\{(x,y): -1<x<1, m<y<n\} $? Thanks..

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$$ x = 2\frac{r - r_1}{r_2 - r_1} - 1 ~~~\mbox{and}~~~ y = \theta $$