532 Prof. V. Karapetoff on Divergence and 



To express the last term through the radius of curvature,, 

 we draw KM' parallel to 00'. Angle K'KM' is a measure 

 for the rate of increase of ds along the normal, or 



O n 



But the same angle c?/5 obtains at the centre of curvature P,, 

 where 



P 



Thus 



aw = * (n> 



^n p 



Substituting this value into eq. (10) we get 



'curiD=t? + ? (12) 



It will thus be seen that the curl at a point in a two- 

 dimensional field depends only upon the flux density, the 

 rate *of change of this density along the normal, and the 

 radius of curvature of the field. The curl is independent of 

 the rate of increase of the flux density in the direction of the 

 field and is also independent of the radius of spread. 



Examining expression (12) for the curl together with 

 eq. (3) for the divergence, it will be seen that these two- 

 expressions together contain all the five characteristics 

 enumerated at the beginning of the article. Moreover, 

 expressions (1), (2), and (3) are formally similar to eqs. (10) 

 and (12), and this fact perhaps helps to see why in vector 

 analysis the same Hamiltonian operator is used for both 

 divergence and curl. 



Special cases : (a) Lamellar flux. — For such a flux there 

 is a potential function <£ such that D = d(j)/ds. In eq. (10), 

 D ds = d<p, and since dcf) between two equipotential surfaces 

 is independent of ??, 



d(D ds) 



or 



=0 



curlD = 0. 



Eq. (12) becomes 



^ + 5=0, d3) 



On p 



