GRAPHICAL DIFFERENTIATION AND INTEGRATION. 115 



We have thus expressed the outflow through the contour of an elementary area as 

 the product of the area of the elements and a factor which must then represent 

 the outflow per unit area. We shall call this outflow per unit area the two-dimensional 

 divergence of the vector A and introduce the notation 



fj\ j- a c ^ 1 a x ?d n 



{d) dl ^ A= ^ +A -dn^ 



We can now write every term in the sum which forms the second member of 

 equation (b) in the form divider. The sum then takes the form of an integral 

 extended to all the elements of area da; that is, we get the formula 



(e) jA n ds = f div 2 A do- 

 or expressed in words : 



The integral of the normal component of a two-dimensional vector taken around 



a closed curve is equal to the integral of the two-dimensional divergence 



of this vector taken over the area bordered by the closed curve. 



The two-dimensional divergence, or the outflow per unit area, can be found by 



a process of differentiation given by equation (//). The last term has a simple 



geometrical sense. As dn represents the elementary distance between two curves 



cdn 

 s, the derivative will evidently represent the elementary angle da between the 



C S 



tangents of two curves s which have the distance dn from each other (see fig. 95). 



Thus we get , , 



& 1 ddn _ da _ 



(f) dn 9s ~ dn ~ 



where 8 is the divergence of the system of curves 5 as defined in section 168. Thus 

 the two-dimensional divergence of the vector A can be written in the form 



(g) div 2 A= S -+A8 



cS 



where 8 is the divergence of the vector-lines. When in this formula we give the 

 vector A the constant scalar value 1, we get div. A 8, which shows that the diver- 

 gence of a system of curves is equal to the divergence of a unit vector which has 

 these curves as vector-lines. 



By formula (g) we have reduced the construction of the field of divergence of a 

 two-dimensional vector to graphical differentiations which we have performed 

 already. We shall find it by the following series of operations: 



(1) We perform the graphical differentiation of the intensity -field of the given 

 vector with respect to its vector-lines. (See fig. 83, where we can interpret the 

 curves .s as the vector-lines and the given scalar field as the intensity-field of the 

 given vector.) 



(2) We form the field of divergence of the vector-lines, using either of the two 

 developed methods according as the isogons of the vector or the vector-lines them- 

 selves are given. (See section 168.) 



(3) We perform the graphical multiplication of the intensity-field of the vector 

 and the divergence-field of its vector-lines. 



