86 



III. GENERAL PRINCIPLES. WORK AND ENERGY. 



and 



dydz dsdy = dScos(nx), 

 dzdx dxdz = dScos (ny), 

 dxdy 8ydx = dScos(nz), 



ar ax 



B which is in the form of a surface 

 integral over the strip of infinite- 

 simal width bounded by the two 

 curves of integration. 



If we again make an infinite- 

 simal transformation, and so continue 

 until the path has swept over any 

 finite portion of a surface S, and 

 sum all the variations of /, we get 



for the final result that the difference in 1 for the two extreme 



paths 1 and 2 is the surface integral 



Fig. 21. 



, /ax dz\ f , , /ar ax\ > J 70 



-f I -5 -TS ) cos (ny) -f I -g -5-7 cos (nz) \ d S 



\dz dx) \ox dy] J \ 



taken over the portion of the surface bounded by the paths 1 and 2 

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

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

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

 We accordingly get the following, known as 



STOKES'S THEOREM. 1 ) The line integral, around any closed contour ; 

 of the tangential component of a vector JR, whose components are 

 X, Y, Z, is equal to the surface integral over any portion of surface 

 bounded by the contour, of the normal component of a vector w, 

 whose components , 77, are related to X, F, Z by the relations 



t_<^_<>Z 



*~ dy dz' 



ax az 



J1 = -^ -pr ) 



02 OX 



ar ax 



1) 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' Analyse, Tom. I, p. 73. 



