GRAPHICAL DIFFERENTIATION AND INTEGRATION. 



109 



If we wish to return from the field of transport to that of intensity of the vector 

 A , we have to use the formula 



(c) 



or corresponding to (&') 



(O 



a = r-i 



dn 



a rpda 



dn 



We then use the common differentiating sheet for forming the field of }- or ~r- and 

 afterwards perform the graphical multiplication of this derivative with the scalar T. 



168. Differentiations of Higher Order. Curvature and Divergence of a System 

 of Curves. The processes described of directional or of linear differentiations can 

 be repeated any number of times. By use of the auxiliaries which we have described 



(fiS^rSafe 



Fig. 88. Positive and negative curvature 

 of a curve. 



Fig. 89. Positive and negative divergence of a system of 

 curves. 



we can thus form a derivative of any order. In precisely the same manner the process 

 of integration can be repeated, and will then lead back to the primary function from 

 a derivative of any order. 



A case of special importance is that in which a directional differentiation is 

 succeeded by a linear one. 



In order to consider this case let us suppose that a system of curves 5 is given. 

 By directional differentiation we can derive the angle a which represents the direction 

 of the tangent to these curves and represent the field of this angle by the isogonal 

 curves 

 (a) a = const. 



Upon the field of the scalar a we can perform a linear differentiation, which will 

 then show the variation from place to place of the angle a. Let this linear differentia- 

 tion be performed along the direction of the originally given curves ^ themselves. 

 This derivative 



< < - 1 



represents the change of direction of the tangent per unit length along the curve, i. e., 

 the curvature of the curves s. The differentiation can be performed as described 

 in section 165 by use of the divided sheet of fig. 81, and will give the field of curvature 

 of the given system of curves 5. 



