GRAPHICAL DIFFERENTIATION AND INTEGRATION. 



117 



Now let the total volume be divided into elementary volumes, consisting of 

 infinitely short trunks of infinitely narrow vector-tubes. There will be a transport 

 only through the surface-elements da and da' which form sections of the tube (fig. 96). 

 These sections being normal, we get the transport through them equal respectively 

 to Ada and A' da', and the outflow equal to their difference 

 (c) A' da' -Ada 



Here we can develop A ' and da' as functions of the length of arc 5 along the axis of 

 the tube 



A' = A+ 9 -^ds 

 2s 



da' 



aa-\--zas 



When we introduce this and leave the term of the second order out of consideration, 

 we get the expression of the elementary outflow (c) in the form 



(0 



M dsda+A i*Zds 

 2s 2s 



Introducing the volume of the element dv = dads, this expression may be written 

 / c m (9A , A 1 2da^ 



(3 A . . 1 ?da\. 



Thus for elementary volumes the outflow is proportional to the volume of the 

 element. The factor of proportionality represents the outflow per unit volume, and 



Fig. 96. Fig. 97. 



is called the three-dimensional divergence or simply the divergence of the vector A 



2da 

 ~2s 



(d) 



A- A SA . A I 



div A = \-A--y- 



2s da 



As we can now write each term in the second member of equation (b) in the form 

 div A dr, this second member takes the form of a sum extended to all the elements 

 of volume dr, i. e., the form of a volume-integral. We thus get the important formula 



(e) 



or in words : 



J A n da = J div A dx 



The integral of the normal component of a vector taken over a closed surface 

 is equal to the volume-integral of the divergence of the vector taken in the 

 volume limited by the closed surface (Gauss's theorem). 





fig 



/!.-. 



