Il8 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



This theorem allows us to bring the solenoidal condition section 112 (a) 

 into a new form; for when the surface-integral in equation (e) is zero for every closed 

 surface in the field, the volume-integral must also be identically zero, and this involves 

 (/) div A = o 



This is the differential form of the solenoidal condition. 



The expression - which appears in the equation of definition (d) has a 

 da 9s 



I edit 



similar significance in space as in two dimensions. When the area da of the 



(til cS 



cross-section of the tube is constant, the considered trunk of the tube may be compared 

 to a cylinder. When da varies, the trunk of the tube may be compared to a cone, 



and the derivative - will represent its solid angle. Then - - will represent the 

 ,?.? da Ss 



ratio of this solid angle to the cross-section of the tube and thus be a measure of 



what we may call the divergence of the curves 5 in space. 



In order to express this divergence by the corresponding divergences in two 



dimensions we will consider vector-tubes which are produced in the usual way by the 



intersection of two sets of surfaces of flow (fig. 97). Each tube will then have the 



well-known parallelogrammatic cross-section. If dn is one side in the parallelogram, 



and dz the corresponding height, we have da = dndz, and get 



1 S da _ 1 Sdn , 1 Sdz 



da Ss dn Ss dz Ss 



Introducing this in equation (d), we get this more developed form of the divergence 



r \ j- a 3A , i Sdn , . 1 Sdz 



{g) dlvA= *- M -^^ +A -^ 



The divergence is here given by a trinomial expression, the first two terms of which 



are seen to express the two-dimensional divergence equation (d) of the preceding 



section of the vector A in the surface which contains the curves 5 and n. 



If we resolve the given vector-field into three component-fields, each with vector- 

 lines coinciding with one set of coordinate-curves of a system of curvilinear orthog- 

 onal coordinates, we can write the divergence of each component-field in either 

 of the forms (d) or (g) . In the special case of a cartesian system the vector-lines of 

 each component-field are straight and parallel. Each vector- tube will have a 

 constant cross-section da, or constant base dn and height dz, and only the first term 

 in the second member of formulae (d) or (g) will be different from zero. Therefore, 

 if we call the vectors of the three component-fields A x , A y , A z , and the lengths of 

 arc measured along the vector-lines x, y, and z, we get for the divergence in each 



component-field 



SA SA SA 



div A, = div Aj, = div A z = - 



9x Sy Sz 



When we form the sum, we get the divergence of the resultant-field 



(h) div A = * + y + 2 



Sx Sy Sz 



This is the most generally used expression of the divergence of a vector in space. 



