214: E. B. Wilson — Divergence and Curl. 



Art. XXII. — On Divergence and Curl • by Edwin Bidwkll 



Wilson. 



1. In the work on Vector Analysis,* which I had the honor 

 to write for the late Professor Gibbs, I developed the theory 

 of the divergence and curl of a vector function of position in 

 space in very much the same way as that followed by Professor 

 Gibbs in his lectures.f This discussion may seem to be ele- 

 mentary, natural, and sufficient, if regarded from the point of 

 view of a mathematician developing a vectorial analysis. To 

 the physicist, however, the conceptions of divergence and 

 curl are of prime importance ; and no student can make any 

 satisfactory headway in hydrodynamics or electricity without 

 first getting a thorough grasp — a grasp as intuitive and physical 

 as possible — of these fundamental conceptions. For this 

 reason it becomes desirable to treat the problem from several 

 points of view. In what follows I set forth some methods 

 which seem from experience in teaching to be useful in sharp- 

 ening up the meaning of divergence and curl with the con- 

 nected integral theorems of Gauss and Stokes. 



In the treatise cited above, it is mentioned on pages 187 and 

 194 that certain integral expressions, namely 



Lim 

 dv 



t oil f.f fAa ] and iT= o h£kjf- dr ] 



may serve respectively as definitions for the divergence and 

 curl of the vector function /. It is the purpose of the present 

 paper to discuss these definitions, first without, and second 

 with the use of linear vector functions. At first sight there 

 appears an apparent anomaly in defining a differentiating oper- 

 ator such as divergence or curl by means of integrations. 

 There is, however, no real reason why integral calculus, whether 

 of vectors or scalars, should not precede a great part of the 

 differential calculus — and for theoretical purposes such prece- 

 dence is often valuable. 



2. Let V(x, y, z) be any vector function of position in space, 

 and let X(x, y,z), Y(x, y, z), Z(x, y, z) be its components, so 

 that 



V= Xi+Tj + Zk. 



Visualize V as the flux of a fluid, that is, as the product of 

 the density p and the velocity in path v. In this manner a 



* Vector Analysis, a Text-book for the use of Students of Mathematics 

 and Physics, founded upon the Lectures of J. Willard Gibbs. New York : 

 Charles Scribner's Sons, 1901. 



f Vector Analysis, §§ 71-72, pp. 150-157 ; §§81-82, pp. 184-193. 



