Solution 3 (c).

Solution.  Let  [Graphics:../Images/JoukowskiTransModHome_gr_167.gif]  be the angle of inclination of the line   [Graphics:../Images/JoukowskiTransModHome_gr_168.gif].   The slope of the line is  [Graphics:../Images/JoukowskiTransModHome_gr_169.gif]  and  

                    [Graphics:../Images/JoukowskiTransModHome_gr_170.gif].
                    
Therefore

                    [Graphics:../Images/JoukowskiTransModHome_gr_171.gif].  

 

We are done.   

 

Aside.  We can let Mathematica double check our work.

[Graphics:../Images/JoukowskiTransModHome_gr_172.gif]

[Graphics:../Images/JoukowskiTransModHome_gr_173.gif]


[Graphics:../Images/JoukowskiTransModHome_gr_174.gif]

[Graphics:../Images/JoukowskiTransModHome_gr_175.gif]


[Graphics:../Images/JoukowskiTransModHome_gr_176.gif]

[Graphics:../Images/JoukowskiTransModHome_gr_177.gif]

So that    [Graphics:../Images/JoukowskiTransModHome_gr_178.gif].

 

 

[Graphics:../Images/JoukowskiTransModHome_gr_179.gif]
[Graphics:../Images/JoukowskiTransModHome_gr_180.gif]


[Graphics:../Images/JoukowskiTransModHome_gr_181.gif]
[Graphics:../Images/JoukowskiTransModHome_gr_182.gif]


[Graphics:../Images/JoukowskiTransModHome_gr_183.gif]
[Graphics:../Images/JoukowskiTransModHome_gr_184.gif]


[Graphics:../Images/JoukowskiTransModHome_gr_185.gif]
[Graphics:../Images/JoukowskiTransModHome_gr_186.gif]


We are really done.   

 

Aside.  We can explore some graphs.

                    [Graphics:../Images/JoukowskiTransModHome_gr_187.gif]          [Graphics:../Images/JoukowskiTransModHome_gr_188.gif]

                    The image of the circle  [Graphics:../Images/JoukowskiTransModHome_gr_189.gif]   under the mapping   

                    [Graphics:../Images/JoukowskiTransModHome_gr_190.gif]    is the line    [Graphics:../Images/JoukowskiTransModHome_gr_191.gif]    that passes through the origin.   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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