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

1 (c).  The rectangle  .

Solution 1 (c).

See text and/or instructor's solution manual.

Answer.  The region in the upper half plane    that lies between the 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  .  

We know that the first quadrant is mapped onto the upper half-plane, so the image    is contained in the upper half plane  .  

From part 1 (a), the image of the line    under the mapping    is the parabola  .

From part 1 (b), the image of the line    under the mapping    is the parabola  .

Therefore, the image of    is the region in the upper half plane    that lies between the parabolas   and  .

We are done.   

Aside.  We can let Mathematica double check our work.

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

This solution is complements of the authors.

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