Exercise 18.  Show the following concerning the exponential map  .  

18 (a).  The image of the line  ,  where    is a spiral.

Solution 18 (a).

See text and/or instructor's solution manual.

Solution.    .  

Substitute     into the equations    and    and obtain

                    

This set is the parametric representation of a spiral.

We are done.   

Aside.  We can let Mathematica double check our work.

                                        The image of a portion of the line   ,
                                        is a portion of the spiral   .

This solution is complements of the authors.

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