66 DEFINITE INTEGRALS. [INT. III. 



directions as we leave the point. This interpretation is due to 

 Boussinesq*. 



We may derive the parameter AF by applying Hamilton's 

 operator V twice to V, 



das J dy dzj \ dx J dy dz 



35. Divergence. Solenoidal Vectors. If the components 

 of the vector parameter are 



Pcos(P*) = Z = ^, 

 we have 



AF= 



dx dy dz ' 



and the above theorem becomes 



- JJP cos (Pn) dS = -l l(X cos (nx) + Fcos (ny) + Z cos (nz)) dS 



-/// 



'dX . d_Y d& 

 dy dz. 



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

 integral is positive, and 



fdx dY dz\ _ 



mean = \- -5 h - > 0. 



\ox oy oz/ 



Accordingly r + + is called the divergence of the vec- 



7 dx dy dz 



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

 denoted by div. R. Comparing with 31 we find that the 

 divergence of a vector is minus the scalar part of the V of the 

 vector, 



div. R = - SVE. 



The theorem as just given may be stated as follows, and will 

 be referred to as the DIVERGENCE THEOREM : The mean value of 

 the normal component of any vector point-function outward from 



* Boussinesq, Application des Potentiels a Vetude de Vequilibre et du mouve- 

 ment des solides elastiques, p. 45. 



