Il6 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



(4) We perform the graphical addition of the two fields obtained by the 

 operations (1) and (3). 



The construction described will be of great importance for the kinematic 

 diagnosis of air- and sea-motions. 



Other expressions of the divergence will also be useful. If the vector-lines 



happen to run at an invariable distance dn from each other, we shall have the 



divergence of the vector-lines equal to zero, and the divergence of the vector will 



SA 

 be given by one term only, . Now, when we express the field of the vector A by 



3s 



the fields of its two cartesian components A x and A y , the component-fields have 



straight and parallel lines of flow. The divergence of the two component-fields 



S A 2 A 



will be respectively - and - and their sum will give the divergence of the 



Sx Sy 



resultant field. 



(*) div 2 A - Ms + 3 A> 



Sx Sy 



When the fields of the rectangular components A x and A y are given, this expression 



gives a simple construction of the fields of divergence. By graphical differentiation 



S A ^a 



we form separately the fields of - - and of - y , and then by graphical addition that 



Sx Sy 



of div 2 A. 



Inasmuch as coordinate-methods should be used on our charts, it must be 

 remembered that the meridians are not equidistant coordinate-curves. The diver- 

 gence of the meridians must be taken into account when the divergence of a 

 velocity-field should be formed separately from charts of the south-north com- 

 ponent and of the west-east component of the wind. 



171. Divergence of a Vector in Space. The two-dimensional divergence which 

 we can represent on our charts will have its importance as part of the three-dimen- 

 sional divergence of that vector in space of which the two-dimensional vector is a 

 component. We shall therefore also consider the divergence of a vector in space. 



Transport in the vector-field in space is represented by the surface-integral 

 of the normal component of the vector 



(a) j Ada 



In the case when the surface a is closed the transport will represent the outflow 

 of the volume bounded by the closed surface (see section 1 1 1 ) . 



If we divide a given volume into any number of parts and form the sum of 

 the outflows out of each part, the transport through the dividing surfaces will 

 cancel, and we find that the outflow in three dimensions has the same additive 

 property as it has in two dimensions. This property can be expressed by the formula 



(b) JAda = ?,JA n d<j 



where the integral appearing as the first member is extended to the limiting surface 

 of the total volume, and the integrals appearing in the second member are extended 

 to the limiting surfaces of the different parts into which the total volume is divided. 



