38 VECTOR ANALYSIS. 



76. From equation (4) we obtain 



where, as elsewhere in these equations, the surface-integral relates to 

 the boundary of the volume-integrals. 



From this, by substitution of Vt for , we derive as a particular case 



fffVt.Vu dv =JfuVt.da"-JffuV.Vt dv ^ffWu.dv-fffrt.Vu dv, 



which is Green's Theorem. The substitution of sVt for o> gives the 

 more general form of this theorem which is due to Thomson, viz., 



JffsVt.Vu dv =ffusVt.d<r -JffuV. [sVt]dv 

 =JftsVu.da- -fffW. [sVu] dv. 

 77. From equation (6) we obtain 



jffV.[rX(*)]dv =JfTX<a.d<r=J)yoo.VxT dv -fffr.Vxu) dv. 

 A particular case is 



dv =<x Vu.do: 



Integration of Differential Equations. 



78. If throughout any continuous space (or in all space) 



Vu=0, 

 then throughout the same space 



u = constant. 



79. If throughout any continuous space (or in all space) 



and in any finite part of that space, or in any finite surface in or 

 bounding it, 



then throughout the whole space 



Vu = 0, and u = constant. 



This will appear from the following considerations : 

 If Vu = in any finite part of the space, u is constant in that part. 

 If u is not constant throughout, let us imagine a sphere situated 

 principally in the part in which u is constant, but projecting slightly 

 into a part in which u has a greater value, or else into a part in 

 which u has a less. The surface-integral of Vu for the part of the 

 spherical surface in the region where u is constant will have the 

 value zero: for the other part of the surface, the integral will be 

 either greater than zero, or less than zero. Therefore the whole 

 surface-integral for the spherical surface will not have the value zero, 

 which is required by the general condition, V. Vu = 0. 



Again, if Vu = only in a surface in or bounding the space in 



