116, 117] DIVERGENCE THEOREM. 347 



31) 4^(F-F ) 



where the bar over the derivative denotes the mean for all directions 

 at 0, but this mean has just been proved to be equal to -^^JVai 0. 



Consequently dividing by R 2 and taking the limit the terms of higher 

 orders in E disappear and we have 



32) 



R=Q 



The difference in the numerical coefficient in the two equations 27) 

 and 32) is accounted for by the fact that in 32) we have a mean 

 over a surface whereas in 27) we had a mean throughout a volume. 

 Any of the interpretations of the second differential parameter 

 shows that it is also a differential invariant. Thus Green's theorem 

 involves three different sorts of differential invariants. 



117. Divergence. Solenoidal Vectors. If the components 

 of the vector parameter are 



33) 



Pcos(Ps) = = !?, 



we have 



OA\ *TT dX , 3Y . dZ 



34) AV=^ h -a h -o-> 



dx " dy dz 



and the theorem 23) becomes 



35) - ffp cos (Pn) dS = -JJ[X cos (nx) + Fcos (ny) 



If P is everywhere outward from the surface S, the integral is 

 positive, and 



/dx . ar, ^^\^ n 



mean ( - \- -5 \~ -$ ) > 0. 

 \dx ' dy tiz J 



Q -yr Q -T^- O ^ 



Accordingly -= + ^ -- \- -^ is called the divergence of the vector 



<7ic ' oy cz 



point -function whose components are X ; Y, Z, and will be denoted 

 by div. R. 



The theorem as given in equation 35) may be stated as follows, 

 and will be referred to as the DIVERGENCE THEOREM: The mean value 



