56 DEFINITE INTEGRALS. [iNT. III. 



surface S, and sum all the variations of /, we get for the final 

 result that the difference in / for the two extreme paths 1 and 2 

 is the surface integral 



fiX dZ\ ft 7 dX\ :1 -: 



+ a -- 5-) cos (ny) + {-= -- 5- 006(910) Y dS 



\dz dxj \da) dyj '] 



taken over the portion of the surface bounded by the paths 1 and 

 2 from A to B. Now I I may be considered the integral from B 

 to A along the path 1, so that / 2 /i is the integral around the 

 closed path which forms the contour of the portion of surface 8. 

 We accordingly get the following, known as 



STOKES'S THEOREM*. The line integral, around any closed 

 contour, of the tangential component of a vector R, whose com- 

 ponents are X, Y, Z, is equal to the surface integral over any 

 portion of surface bounded by the contour, of the normal com- 

 ponent of a vector &>, whose components f, 77, f are related to 

 X, Y, Z by the relations 



8^_8F 



fm ty a* ' 



dx dz 



== _ 



3# dy ' 



The normal must be drawn toward that side of the surface that 

 shall make the rotation of a right-handed screw advancing along 

 the normal agree with the direction of traversing the closed 

 contour of integration. 



(ltd* = (Xdoi + Ydy + Zdz = jjco cos (am) dS 



= 1 1 (| cos (nx) 4- 97 cos (ny) + f cos (nz)) dS. 



The vector o> related to the vector point-function R by the differ- 

 ential equations above is called the rotation, spin (Clifford), or 

 curl (Maxwell and Heaviside) of R. Such vectors are of frequent 



* The proof here given is from the author's notes on the lectures of Professor 

 von Helmholtz. A similar treatment is given by Picard, Traite d 1 Analyse, Tom. i, 

 p. 73. 



