Exercise 4.  Find the image of the annulus    under  .

Solution 4.

Answer.   The image of      under      is the rectangular set   .  

Solution.   Use       and the formulas

                        and    .   

Then      implies that      implies that   ,

which in turn implies that    .    

Also,     implies that  .

        Therefore, the image of the annulus      under the mapping   

is the rectangular set   .  

We are done.   

          The mapping    .

          Observe the points and their images    

                                                  

We are really done.   

Aside.   If it is important then we can include the following details.

          Here the values        and        are calculated as limits

                    

                    

This solution is complements of the authors.

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