Solution 7.

See text and/or instructor's solution manual.

Answer.  By the chain rule,

                    ,    ,   and  

                    ,    and  

                    .  

Hence,  .

Therefore    is harmonic on D.  

Solution.  Indeed, the proper way to look at the composition is by introducing the intermediate variablesX and Y, and writing

  where    and .   

Here the partial derivatives of and are  ,   ,   ,   and   .  

The partial derivatives of     are    and   which intend to mean differentiate the formula for    with respect to X and Y, respectively.

There is no convenient way to explain that these formulas are to be used with the replacement , and the notation has this weakness.

          ,  
          
similarly

           .

Now these ideas must be used with the functions and which involve the compositions

  where    and .   

    where    and .   

Hence, we obtain

          ,  
          
similarly

           .

This solution is complements of the authors.

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