GRAPHICAL DIFFERENTIATION AND INTEGRATION. 



113 



Fig. 92 represents the ascendant of the same field of which fig. 83 represents 

 a partial derivative. 



From the field of the ascendant (a) we can derive that of any other derivative 



ca 



F *-Is 



w 



as we have 



(J) F S = F cosd 



where is the angle between the directions n and 5. This algebraic method of 

 finding the partial derivative F s will be convenient if the direction of the ascendant 

 F is represented by isogonal curves <p = const., and the direction of 5 by isogonal 

 curves a = const. We shall then pass from the field of F to that of F s by the follow- 

 ing operations (compare sections 149, 156). 



(1) By graphical subtraction we form 

 the field of the angle 6 = <p a. In the 

 drawing of these auxiliary curves special 

 attention should be attached to the drawing 

 of the curves 6 = 16, and 6 = 48, which will 

 be curves F s = o in the resultant field. 



(2) By use of these auxiliary curves and 

 the curves F = const., we derive the field of 

 the scalar value of F s according to equation 

 (c) by use of the first of tables K. 



By this process we can thus derive the 

 partial derivative of fig. 83 from the ascen- 

 dant-field of fig. 92. 



If we know the field of the ascendant or 

 the gradient and the value of the scalar a at 

 a single point, we can reconstruct the field of 

 the scalar a. The simplest method will be 

 this: We first perform a linear integration 

 along the particular curve n which passes 



through the point where we have the known value of a. By this integration we 

 find a series of points through which equiscalar curves representing the required 

 integer values of a shall pass. Through these points we may then by directional 

 integration draw the curves a = const., normal to the vector-curves of the ascendant 

 or the gradient. 



170. Divergence of a Two-Dimensional Vector. We have considered already 

 the "transport" in the two-dimensional field (section 119), i. e., the integral of 

 the normal component A n of a vector taken along a curve. 



(a) jA H ds 



In the special case of a closed curve the transport directed outward was called the 

 " outflow " out from the area limited by the curve. 



Fig. 92. Scalar field a = i2, n, 10, . . . (fine con- 

 tinuous lines), vector-lines of the ascendant 

 (thick lines with arrow-heads), and intensity- 

 curves of the ascendant (stippled curves). 



