536 Prof. V. Karapetoff on Divergence and 



2 b. Curl in three dimensions. 



The vector of a curl, generally speaking, is not normal to 

 the original vector D. It is therefore convenient to deduce 

 expressions for the projections of the curl upon the three 

 axes of coordinates t, n, and b, referred to above. The 

 expressions so obtained may in some cases give a better 

 insight into the physical nature of the curl than the usual 

 expressions of its projections upon a fixed system of 

 coordinates. 



(a) The component of the curl along the binormal. — This 

 component is perpendicular to the projection of the line of 

 force shown in fig. 2 : its value may be deduced in exactly 

 the same way as is done for the two-dimensional field above. 

 We thus get an expression similar to eq. (12): 



(c„rlD) s :=g + 5 (23) 



The subscript b indicates a component of the curl along the 

 binormal, and p is the radius of curvature of the line of 

 force at 0. This component of the curl represents the 

 whole curl in a two-dimensional field. 



(b) The component of the curl along the normal. — This 

 component is perpendicular to the projection of the line of 

 force shown in fig. 3. By analogy with eq. (23) we have 



(curlD)„=^- + ? .... (24) 



where t is the radius of tortuosity * of the line of force 

 at 0. 



(c) The component of the curl in the direction of vector D 

 itself. — This component of the curl is perpendicular to the 

 projection of the line of force shown in fig. 4. Since it is 

 not possible to speak of the curvature of the line of force in 

 the plane of fig. 4, no expression similar to eqs. (23) and 

 (24) can be written for this component of the curl. It is 

 convenient to apply to this plane the usual expression for 

 the component of a curl in Cartesian coordinates. Using the 

 directions of b and n as the axes of coordinates, we get 



(cm-lD),= ^---^, .... (25) 



* The term tortuosity is used in this article in place of the more usual 

 term torsion, in order to avoid a possible ambiguity in application, for 

 example, to the theory of elasticity, where the term torsion has a 

 meaning of its own. 



