E. B. Wilson — Divergence and Curl. 2L9 



This dyadic <&=!+ VvSt, which determines the changes in 

 lines, is strikingly like that which I obtained about a year ago 

 in treating a problem in continuous groups,* and the treatment 

 of this problem will therefore follow the other to some extent. 

 To obtain the expression in volume it is merely necessary to 

 note that dr / =dT^> 3 . 



$,=(7+vP&),=Z, + J,-Jr vVSt + I: (vF) 2 (^) 2 + (vF) 3 (S*) 3 . 



Disregarding higher powers in St, noting that I 2 and I are 

 equivalent and that 



I: vVSt= v VSt = v.V8t, 



s 

 we have dr — dr (1 + v . V8t). 



Hence the rate of increase of volume is v . V- As the den- 

 sity has been taken as unity, this is the rate of diminution of 

 density, that is it is the divergence. Hence 



,. „ Tr bX bY bZ 



dlV V= V.V= —— + -r— + — 



dx by bz 



and 



= 11 V.da — II {XdydzAr Ydzdx + Zdxdy). 



s s 



Thus the expression for the divergence in Cartesian coordin- 

 ates has been obtained without applying (2) to the infinites- 

 imal cube. 



Consider next the curl. The distance each point has 

 moved is 



r' + dr'—(r + dr) = V8t + dr.v VSt. 



Hence the velocity of each point, which is this value divided 

 by St, is 



V= V + dr. v V,\ 



where a zero has been affixed to V to denote that V a is the 

 velocity at a particular point P. Then 



/ VMr =f V Q Mr+fdr. v VMr, 



o o o 



where S has been prefixed to dr to denote the increment of dr 

 along an infinitesimal curve surrounding the point P. Separ- 



* Sur le groupe qui laisse invariante l'aire gauche. Nouvelles Annales de 

 Mathematiques, vol. v, ser. 3, pp. 163-170, June, 1905. 

 f This is merely equation (9). 



