Exercise 1 (b).  Show that    does not satisfy Laplace's equation    in three-dimensional Cartesian space,

and that      satisfies Laplace's equation      in two-dimensional Cartesian space.

Solution 1 (b).

See text and/or instructor's solution manual.

Answer.   For   ,   we get   ,  

and for      we have   .  

Solution.   For the first part, calculate the partial derivatives of     

                    ,     and      

                    ,     and      

                    .  

Then calculate the second partial derivatives  

                    ,     and     
                    
                    ,     and    
                    
                    .   

Then

                      

Hence    does not satisfy Laplace's equation   .

Therefore,     is not a harmonic function.  

          For the second part, calculate the partial derivatives of   .  

                         and     .  

Then calculate the second partial derivatives  

                         and     
                    
                    .  

Then  

                       

Hence    satisfies Laplace's equation   .

Therefore,    is a harmonic function.  

We are done.   

Aside.  We can let Mathematica double check our work.

First, consider   :

Second, consider   :

This solution is complements of the authors.

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