218 E. B. Wilson — Divergence and Curl. , 



where the line integral is taken around the curve bounding 

 the surface S. If finally the expression for curl V be evalu- 

 ated in Cartesian coordinates by applying (7) to an infinitesi- 

 mal square situated successively in each coordinate plane,* the 

 equation (8) will be seen to be merely a statement of Stokes's 

 Theorem. 



4. An appeal to the theory of linear vector functions, 

 which are all too little studied, will afford a relief from the 

 mere plausibility on which the statements at the end of § 3 

 were founded, and will give an entirely new basis for the 

 evaluation in %, y, z of the curl and divergence. Moreover the 

 analytic nature of the argument will be more in evidence and 

 less recourse to intuition will be necessary. 



Consider a point, P whose vector coordinate is r. The veloc- 

 ity of this point is V (to follow the notation of § 3). The 

 velocities in the neighborhood of P may be expanded about 

 that point by Taylor's Series. This gives 



V+dV+ higher powers. 



By the fundamental relation defining v f this becomes 



V-\-dv.\j V+ higher powers in dr.\ (9) 



After the lapse of a time 8t, r has become r', which may be 

 written 



f' — v+ V8t + higher powers in &.§ 



The vector r+dr becomes r' + Sr'. If higher powers alike 

 in 8t and in dr be neglected, 



r' + dr' = r+dr + V8l + dr. v YU. 



This shows that there has been a general translation through 

 the distance VSt and that the deformation in the neighbor- 

 hood of r is expressed by the relation 



dr' = dr.{I+ v YU). (10) 



* The proof is entirely analogous to that given above for the case of the 

 divergence. It should be noted that the method of procedure in § 2 and 

 § 3 is directly the opposite to that followed in the treatise on Vector Analysis. 

 Instead of starting with the analytic expressions in Cartesian coordinates for 

 the divergence and curl and then obtaining the theorems of Gauss and 

 Stokes, we here start with such definitions of div Y and curl V as make 

 these theorems immediately obvious and then apply the definitions to find- 

 ing the analytic expressions. This last step is merely for the purpose of 

 comparison. It would be possible to remain entirely within the domain of 

 vector analysis. 



f Equation (2), p. 404 of Vector Analysis. 



% The theorem of the total differential is supposed to apply, namely that 

 the remainder of the series may be written as dr.'i, where 'f is a 

 dyadic which vanishes with dr and hence may be discarded as an infinitesi- 

 mal of higher order. 



§ It should be noted that <5t and dr are wholly independent ; each may 

 be made as small as desired without affecting the other. Hence a product 

 6tdv is not of the second order when compared with 6t or dr. 



