Exercise 13.  For the temperature    in the upper half-disk  ,  

show that the isothermals   ,   for  ,   are portions of circles that pass through the points    and  ,  as illustrated in Figure 11.33.  

Solution 13.

See text and/or instructor's solution manual.

Solution.   Start with the equation      then get  

                        

Now substitute    and obtain

                       

Therefore, the isothermals    are portions of circles that pass through the points    and  .  

We are done.   

Aside.  We can let Mathematica double check our work.

In the equation     replace   .

Now solve for  Q.

Now replace      and  

We are really done.   

Aside.   We can graph the function   .   

                    A contour graph of the function   ,

                    where     for   .

We are really really done.   

                    A graph of the function   .

                    A graph of the function   .

                     Another way to view   .

                    A graph of the function   .

This solution is complements of the authors.

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