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

1 (a).  The horizontal line  .

Solution 1 (a).

See text and/or instructor's solution manual.

Answer.    .  

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  .  

If  ,  then  .  

Solve the second equation    for    and substitute it into the first equation    and get  ,  

and then we see that  .  

We are done.   

Aside.  We can let Mathematica double check our work.

                    The image of the line    under the mapping    is the parabola  .

This solution is complements of the authors.

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