E. B. Wilson — Divergence and Curl. 215 



dynamical, or rather a hydrodynamical interpretation is given 

 to V; and this picture, instead of the usual purely geometric 

 figure, becomes the "basis of our reasoning. There is no reason 

 why this imagery should not be invoked if it proves of con- 

 venience in suggesting results or methods of proof : for to the 

 mathematician it is the analysis and not the representation of 

 it which in the last instance establishes the proof upon a firm 

 foundation, whereas to the physicist the picture itself is fre- 

 quently taken as convincing and should therefore be as tangi- 

 ble as possible. 



If now, S be any closed surface in the fluid and da a vector 

 element of the surface, the normal flow of the fluid through 

 da per unit time is the product of the density, the velocity, 

 the area of the element and the cosine of the angle between 

 the velocity and the normal. This flow per unit time is 

 pv.da= V.da, and the total flow through any closed surface /Sis 



ffpv.da =ff V.da. 



s s 



On the other hand, if dr be any element of volume the 

 amount of fluid contained therein is pdr, and the outflow per 

 unit time is —drdp/dt* The total outflow from the whole 

 volume included by <S may be obtained by integrating. 

 Equating the two results, we have 



If in particular this formula be applied to a single element of 

 volume and if the limit be taken as dr = 0, the result 



may be taken as a definition of the divergence of the flux V, 

 provided it be granted, as is physically obvious, that this 

 limit exists and is unique. f Then from (1), 



fff&ivVdT=ff V.da, (3) 



s 



The definition contained in (2) is that already referred to in §1, 

 and equation (3) appears as Gauss's Theorem when once the 

 expression for divT r has been obtained in Cartesian coordinates. 



* Strictly speaking p, v, V, x, y, z are ail functions of the time t as well 

 as of space. But the entire discussion may be considered as taking place 

 at a given moment of time and hence the variation of t may be disregarded 

 except in the sole expression dp/dt. 



f There can be constructed such functions V as will violate this condition : 

 but not in the ordinary run of physical experience where p and V are 

 assumed to be continuous and differentiable. Exceptions might occur, for 

 instance, at the interface of two fluids where the divergence need not be 

 continuous across the interface. 



