Solution 2 (a).

See text and/or instructor's solution manual.

Use the relations   [Graphics:../Images/JoukowskiTransModHome_gr_16.gif],   and  

                    [Graphics:../Images/JoukowskiTransModHome_gr_17.gif]    and    [Graphics:../Images/JoukowskiTransModHome_gr_18.gif],     

and polar coordinates    [Graphics:../Images/JoukowskiTransModHome_gr_19.gif],    [Graphics:../Images/JoukowskiTransModHome_gr_20.gif],    and    [Graphics:../Images/JoukowskiTransModHome_gr_21.gif]   

and then get the equations    

                    [Graphics:../Images/JoukowskiTransModHome_gr_22.gif]    and    [Graphics:../Images/JoukowskiTransModHome_gr_23.gif].  

Solve these equations for   [Graphics:../Images/JoukowskiTransModHome_gr_24.gif],   [Graphics:../Images/JoukowskiTransModHome_gr_25.gif]   and get     

                    [Graphics:../Images/JoukowskiTransModHome_gr_26.gif]    and    [Graphics:../Images/JoukowskiTransModHome_gr_27.gif],    respectively.   

Use the trigonometric identity   [Graphics:../Images/JoukowskiTransModHome_gr_28.gif]   to eliminate [Graphics:../Images/JoukowskiTransModHome_gr_29.gif],  

                    [Graphics:../Images/JoukowskiTransModHome_gr_30.gif],  

and  obtain the equation of an ellipse   

                    [Graphics:../Images/JoukowskiTransModHome_gr_31.gif].  

 

We are done.   

 

Aside.  We can let Mathematica double check our work.

[Graphics:../Images/JoukowskiTransModHome_gr_32.gif]

[Graphics:../Images/JoukowskiTransModHome_gr_33.gif]

[Graphics:../Images/JoukowskiTransModHome_gr_34.gif]


[Graphics:../Images/JoukowskiTransModHome_gr_35.gif]

[Graphics:../Images/JoukowskiTransModHome_gr_36.gif]


[Graphics:../Images/JoukowskiTransModHome_gr_37.gif]

[Graphics:../Images/JoukowskiTransModHome_gr_38.gif]


[Graphics:../Images/JoukowskiTransModHome_gr_39.gif]

[Graphics:../Images/JoukowskiTransModHome_gr_40.gif]


[Graphics:../Images/JoukowskiTransModHome_gr_41.gif]

[Graphics:../Images/JoukowskiTransModHome_gr_42.gif]


We are really done.   

 

Aside.  We can graph the images of some circles under the mapping   [Graphics:../Images/JoukowskiTransModHome_gr_43.gif].  

                    [Graphics:../Images/JoukowskiTransModHome_gr_44.gif]          [Graphics:../Images/JoukowskiTransModHome_gr_45.gif]

                    The image of the circles   [Graphics:../Images/JoukowskiTransModHome_gr_46.gif],   (for  [Graphics:../Images/JoukowskiTransModHome_gr_47.gif]),   are mapped onto the ellipses   

                    the ellipses   [Graphics:../Images/JoukowskiTransModHome_gr_48.gif]   by the Joukowski transformation   [Graphics:../Images/JoukowskiTransModHome_gr_49.gif].   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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