GRAPHICAL DIFFERENTIATION AND INTEGRATION. 121 



By the expression (/) we can construct the field of curl, A. The construction 

 will be perfectly analogous to that of the divergence : 



(i) We perform the graphical differentiation of the intensity-field of the given 

 vector with respect to the positive normal curves to the vector-lines. 



(2) We form the field of curvature of the vector-lines of the given vector (see 

 section 168). 



(3) We perform the graphical multiplication of the latter field with the intensity- 

 field of the given vector. 



(4) We perform the graphical subtraction of the two fields obtained by the 

 operations (3) and (1). 



The expression (/) may be used also for forming the curl of any component of the 

 given vector. If we use cartesian coordinates, the vector-lines of each component- 

 field will be straight lines. The curvature 7 will be equal to zero and the curl of 

 each component-field will be expressed by the first term only. Observing the rule 



?A ?A 



of signs, we get - for the curl of the component A x , and -^ for the field of the 



component A y . Forming the sum we get 

 (g) curl, A = 



9x 9y 



When the field of each component is given, we can construct the field of the 



curl in accordance with this formula. By linear differentiation of the field of A y 



along lines parallel to the axis of X, and of the field of A x along lines parallel to the 



?A ~*A 



axis of Fwe form the fields of the two derivatives - y and - -. Afterwards, by 



Sx Sy J 



graphical subtraction of the latter from the former, we get the field of the curl. 



173. Curl of a Vector in Space. Now let A be any vector in space. We may 

 then define a vector c which has the rectangular components 



, v _9Aj_cAy _M^_M* _SA y ?A X 



{a) c *~ 3y 3z Cy ~lf 3x z ~~?x~ ly~ 



By this definition we see that c is a vector of which each component is the curl of a 

 two-dimensional vector: c x of that which has the components A, and A y \ c y of that 

 which has the components A x and A.; c, of that which has the components A y and 

 A x . We see further that each component of the vector c is normal to that plane 

 which contains the two-dimensional vector from which it is derived. We will agree 

 to represent this vector by curl A, thus 



(a') c = curl A 



Now let us consider any surface a in the three-dimensional field. The vector 

 A will determine a two-dimensional vector in this surface, for which we can write 

 the theorem (e) of the preceding section. But what we have written there as curl, A. 

 conceiving A as the two-dimensional vector contained in the surface, may now be 



