Exercise 2.  Find the image of the rectangle  ,  under the transformation  .

Solution 2.

Answer.   The image of the rectangle    under    is  .  

Solution.   Recall that      and   .   

Thus      implies that      in turn implies that   ,   thus   .   

Recall that   .   

Thus      implies that   .  

        Therefore,  the image of the rectangle      under      is the portion of the annulus

in first quadrant that lies between the circles  ,   i.e.   .  

We are done.   

          The mapping   .

          Observe the points and their images    

                                                    

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell