Exercise 8.  Consider the mapping  .  

Show that the lines    are mapped onto orthogonal curves.

Solution 8.

See text and/or instructor's solution manual.

Solution Method I.   Applying Theorem 10.1,    and  ,  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 5.1, we saw that the image of the line    is the circle   ,  

and hence the image points will satisfy the equation      which can be written as   .  

Implicit differentiation of   produces the equation   ,  then  .

The slope at the point    is   

                    .  

The image of the line    is the ray   .  Implicit differentiation of   produces  

                     

The slope at the point    is  

                    .  

Since  ,  the curves are orthogonal.

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