Exercise 12.  If    is a mapping, where    is not analytic, then what behavior would one expect regarding the angles between curves?

Solution 12.

See text and/or instructor's solution manual.

Solution.   In general we would not expect that angles are preserved.

Illustrated Solution 12 Part I.   If     be analytic at     then    is not analytic at and is not a conformal mapping at .

However, as shown in Exercise 11    preserves the magnitude, but reverses the sense, of angles at  .  

For illustration purposes we can revisit Exercises 6 and 7 where we looked at     and  .

The mappings    and    will preserves the magnitude, but reverses the sense, of angles.

Aside.  We can let Mathematica illustrate our work.

Revisit Exercise 6.  Consider the mappings    and  .

                                   At       preserves angles.

                                   At       preserves the magnitude, but reverses the sense of angles.  

Revisit Exercise 7.  Consider the mappings    and  .

                                   At       preserves angles.

                                   At       preserves the magnitude, but reverses the sense of angles.  

Illustrated Solution 12 Part II.   In the exercises in Section 3.2 we studied several mappings    where    is not analytic and not conformal.

For illustration purposes we can revisit Exercise 1 (g), Exercise 11 (a) and Exercise 11 (c) in Section 3.2.  

Aside.  We can let Mathematica illustrate our work.

Revisit Exercise 1 (g).   .  

                    The image of this orthogonal grid under    is not an orthogonal grid.

Revisit Exercise 11 (a).  .

                    The image of this orthogonal grid under    is not an orthogonal grid.

Revisit Exercise 11 (b).  .

                    The image of this orthogonal grid under    is not an orthogonal grid.

This solution is complements of the authors.

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