158 General Analytical Theorems [CH. vn 



Special Form of Greens Theorem. 



181. An important case of the theorem occurs when u, v, w have the 

 special values 



, 

 v = <^ , 



dy 



w = 3>^ , 



dz 



where <I> and M* are any functions of x, y and z. The value of (lu + mv + nw) 

 is now 



=- + m 



or <f> 



a^\ 



n 3- 

 as/ 



where - denotes differentiation along the normal, of which the direction- 



on 



cosines are I, m, n. 

 We also have 



du dv dw a f . dv\ a ( , a^) a , 



__ I ___ I __ = _ Jcb _ I J -- J<J) _ L -i -- Jcb _ 



a# a a* d ^ 



_ 



" a# aa; + 9y 8y dz dz H V8^ + 8y 2 "aF/ ' 

 Thus the theorem becomes 



9 ^^ 



This theorem is true for all values of <l> and "SP, so that we may inter- 

 change 4> and "SP", and the equation remains true. Subtracting the equation 

 so obtained from equation (100), we get 



- ^ ^ ...... (101). 



APPLICATIONS OF GREEN'S THEOREM. 



182. In equation (101), put <J> = 1 and = 7, where F denotes the 

 electrostatic potential. We obtain 



dS .................. (102). 



