Solution 7.

See text and/or instructor's solution manual.

Solution.

        The transformation    maps the point      onto the point   .

For the function     we have      and   .  

Since   ,   by Theorem 10.1 in Section 10.1 , the mapping is conformal at    and "angles are preserved."

Since    the rotation is  .

        The rays tangent to the curve at    make an angle    and  ,  respectively.

The rays tangent to the image curve at    will also make an angle    and  ,  respectively.

Hence, the rays tangent to the image curve at    will be pointing in the same direction.

        Therefore, that trailing edge of this Joukowski airfoil forms an angle of  0°.

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

Aside.  We can explore some graphs.

The first exploration uses the circles   

and   .

                      The circles    and    in the z-plane.  

                      The line    and the circle    in the Z-plane.  

                      The ray    and the cardioid    in the W-plane.  

                      The arc    and Joukowski airfoil    in the w-plane.  

Aside.   We can explore some more graphs.

The second exploration uses the circles   ,  

and   .

                      The circles    and    in the z-plane.  

                      The line    and the circle    in the Z-plane.  

                      The ray    and the cardioid    in the W-plane.  

                      The arc    and Joukowski airfoil    in the w-plane.  

This solution is complements of the authors.

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