Exercise 10.  Consider the mapping  ,  where    denotes the principal branch of the logarithm function.  

Show that the positive -axis and the vertical line    are mapped onto orthogonal curves.  

Solution 10.

See text and/or instructor's solution manual.

Solution Method I.   Applying Theorem 10.1,     and   .   

Then    hence    is conformal at  .  

The lines and intersect orthogonally at the point ,

therefore their image curves will intersect orthogonally at the point .

Solution Method II.   In Section 5.2 found that the image of a vertical line    is

                    ,   and  

and the image of a horizontal line    is

                    ,
                    
and their tangent vectors are  

                    ,   and

                    .

At the point    the tangent vectors to the curves   are

  and  ,  respectively,  and we have

                    .  

Therefore, the lines    are mapped onto orthogonal curves.

We are done.   

Aside.  We can let Mathematica illustrate our work.

                                   The transformation   .   

                                   The transformation   .   

Remark.  In Section 10.3 and Section 10.4 we will study some composite mappings involving  .

This solution is complements of the authors.

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