Exercise 1.  Find the images of the mapping    in each case, and sketch the mapping.

1 (e).  The infinite strip .

Solution 1 (e).

See text and/or instructor's solution manual.

Answer.  The image is the region bounded by the two parabolas   and .

Solution.   The mapping is    and

the point    in the xy-plane is mapped to the point    in the uv-plane,

and we get Equations (2-9);    and  .  

For any x, we have  .  

If     then  .   If    then  .  

Therefore the image of    is the region bounded by the two parabolas   and .

The complete details for showing that the strip    is mapped between these two parabolas might be:

For any x, we have  .   So

           

Similarly

          

Therefore, if   the image of the vertical line will lie between the two parabolas   and .

We are done.   

Aside.  We can let Mathematica double check our work.

                    The image of    under the mapping    is the region bounded by   and .

This solution is complements of the authors.

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