Exercise 9.  Consider the mapping  .  

Show that the line segment  ,  and the vertical line  , where   are mapped onto orthogonal curves.  

Solution 9.

See text and/or instructor's solution manual.

Solution Method I.   Applying Theorem 10.1,    and  in Section 5.4 we saw that

                    .  

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 Example 10.13 in Section 10.4, we will see that the image of a vertical line    is hyperbola,

and the image of a horizontal line    is an ellipse.

The image of the vertical line    is

(10-23)             .  

Implicit differentiation of   produces the equation   ,  then  .  

The slope at the point    is   

                    

The image of the horizontal segment    is

(10-24)             .  

Implicit differentiation of   produces the equation   ,  then  .  

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   .   

We are really done.

Caveat.  At a point    where     we will have

                    .

Applying Theorem 10.2 we see that the mapping   magnifies angles at the vertex by the factor  .

                      For     at    we have      and   ,  and
                      the mapping   magnifies angles at the vertex    by the factor  .

Remark.  In Section 10.4 we will study some trigonometric mappings, including  .

This solution is complements of the authors.

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