156 General Analytical Theorems [CH. vn 



Thus Green's Theorem assumes the form 



lljdivVdxdydz = -2 jJNdS (97). 



A vector F which is such that div F = at every point within a certain 

 region is said to be " solenoidal " within that region. If F is solenoidal 

 within any region, Green's Theorem shews that 



where the integral is taken over any closed surface inside the region within 

 which F is solenoidal. Two instances of a solenoidal vector have so far 

 occurred in this book the electric intensity in free space, and the polarisa- 

 tion in an uncharged dielectric. 



178. Integration through space external to closed surfaces. Let the 

 outer surface be a sphere at infinity, say a sphere of radius r, where r is 

 to be made infinite in the limit. The value of 



1 1 



(lu + mv + nw) dS 



taken over this sphere will vanish if u, v, and w vanish more rapidly at 

 infinity than . Thus, if this condition is satisfied, we have that 



where the volume integration is taken through all space external to certain 

 closed surfaces, and the surface-integration is taken over these surfaces, 

 I, m, n being the direction-cosines of the outward normal. 



179. Integration through the interior of a closed surface. Let the inner 

 surfaces in fig. 53 all disappear, then we have 



where the volume integration is throughout the space inside a closed surface, 

 and the surface integration is over this area, I, m, n being the direction- 

 cosines of the inward normal to the surface. 



180. Integration through a region in which u, v, w are discontinuous. 

 The only case of discontinuity of u, v, w which possesses any physical impor- 

 tance is that in which u, v, w change discontinuously in value in crossing 

 certain surfaces, these being finite in number. To treat this case, we enclose 

 each surface of discontinuity inside a surface drawn so as to fit it closely on 



