30, 31] DEFINITE INTEGRALS. 53 



normal to the surface S drawn always toward the same side of the 

 surface, it is easily seen that the area of the projection of the 

 element dS on the X F-plane is dS^y or dS z = dS cos (nz). We may 

 accordingly write the surface integral, 



f (p) cos () dS = II f (p) dS, = jjf(p) dxdy, 



with the understanding that in the last form the integral is to be 

 taken over the projection of the surface S on the X F-plane, in 

 such a manner that the projection is to be divided into regions for 

 each of which the normal to 8 in the corresponding portions of 8 

 points either always towards the X F-plane or always away from it, 

 and that those parts of the integral for which the normal points 

 in opposite directions are given opposite signs. It will be seen 

 that this corresponds exactly to the interpretation of the line- 

 integral in terms of x, when x changes its direction of variation. 

 The first form of the integral above, with 8 as the variable of 

 integration, is preferable, its meaning being unambiguous. 



31. Dependence of Line Integral on Path. Stokes's 

 Theorem. Curl. The line integrals with which we shall have 

 most to do are integrals of a vector point-function. If R is a 

 vector function of the point, whose projections are X, F, Z, 

 functions of x, y, z, the component of R along the tangent to the 

 curve AB at any point is, since the direction cosines of the 



dx dy dz 



tangent are -y- , -f- \ -j- , 

 ds* ds' ds' 



r / T> T \ IT- dx Tr dy dz 



R cos (R, ds) = X -Y- + Y-2- + Z -y . 

 ds ds ds 



The line integral of this resolved component 



dz 



- r + -f- + 

 A 



may be written 



( B -D /D j \ j f s f-v-dx ^ r dy , 7 dz 



Rcos(R, ds).ds=\ (X- r + Y-f- + Z J - 



} A i A \ ds ds ds 



/; 



with the understanding of the previous section. 



The functions X, F, Z, being given for every point x, y, z, the 

 integral / will in general depend on the form of the curve AB. 

 If we make an infinitesimal transformation of the curve, the 



