Extra Example 1.  Show that the partial derivatives of  [Graphics:Images/IntegralRepresentationMod_gr_146.gif] are harmonic functions.  

Explore Extra Solution 1.

[Graphics:../Images/IntegralRepresentationMod_gr_147.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_148.gif]

 

 

According to Corollary 6.3, all of its partial derivatives  [Graphics:../Images/IntegralRepresentationMod_gr_149.gif], [Graphics:../Images/IntegralRepresentationMod_gr_150.gif], [Graphics:../Images/IntegralRepresentationMod_gr_151.gif], [Graphics:../Images/IntegralRepresentationMod_gr_152.gif], [Graphics:../Images/IntegralRepresentationMod_gr_153.gif]  exists and are harmonic functions.  

We can use Mathematica to verify Laplace's equation for some of these functions.

[Graphics:../Images/IntegralRepresentationMod_gr_154.gif]





[Graphics:../Images/IntegralRepresentationMod_gr_155.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_156.gif]

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_157.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_158.gif]

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_159.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_160.gif]

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_161.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_162.gif]

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_163.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_164.gif]

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_165.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_166.gif]

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_167.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_168.gif]

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_169.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_170.gif]

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_171.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_172.gif]

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_173.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_174.gif]

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_175.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_176.gif]

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_177.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_178.gif]

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_179.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_180.gif]

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_181.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_182.gif]

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_183.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_184.gif]

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_185.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_186.gif]

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_187.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_188.gif]

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_189.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_190.gif]

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_191.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_192.gif]

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_193.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_194.gif]

 

 

[Graphics:../Images/IntegralRepresentationMod_gr_195.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_196.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2006 John H. Mathews, Russell W. Howell