530 Prof. V. Karapetoff on Divergence and 



Since we are dealing with infinitesimal dimensions, OB 

 may be taken to be a straight line, and O'B' coincides with 

 the tangent O'K/. Drawing a line O'L' parallel to OB 

 (or to OK) triangle O'K'L' is obtained, in which angle dot 

 represents the rate o£ increase of dn with s, or 



dcc = ^. 

 OS 



But the same angle obtains at Q, where du = dn/H, so that 

 yte have 



o(dn) __ dn 



^W "IT* 



Substituting this expression into eq. (2) we finally obtain 



divD =f + H ^ 



In other words, the divergence at a point in a two- 

 dimensional field depends only upon the flux density, the 

 rate of change of this density along the tangent, and the 

 radius of spread. The divergence is independent of the 

 radius of curvature of the field and of the rate of increase 

 of the flux density along the normal. 



Special cases : (a) Solenoidal flux. — For such a flux D dn 

 is the same at 00' as at BB' ; thus, 



*%*-* to 



or I) dn = const. 



This means that for any two points, 1 and 2, along a line of 

 force, 



D 1 _ dn 2 



D 2 dn x ' 



Eq. (3) shows shows that divD = 0, when 



g+5- <=> 



By analogy with the potential function for a lamellar flux, 

 a, flux function, i/r, may be introduced for the solenoidal flux. 

 This function is defined by the equation 



D=M, (5 a) 



On 



and the existence of such a function is a condition for the 



