132, 133] GREEN'S THEOREM IN CURVILINEAR COORDINATES. 



379 



On the surface, since 



0. In T, since 



and 



are zero, 4u = 0. Accordingly J(u) = 0. But, as we have shown, 

 this can only be if u = const. But on S, u = 0, hence, throughout 

 r 9 u = and v = v 1 . 



133. Green's Theorem in Orthogonal Curvilinear Co- 

 ordinates. We shall now consider Green's theorem in terms of 

 any orthogonal coordinates, beginning with the special case forming 

 the divergence theorem, 117, 35). 



28) -if [X cos (nx) + Tcos 



+ Zco8(n#y\ 



~/ff 13 - 



dY 



dz\, 



^-\dr. 



Instead of the components X, Y, Z, let us consider the projec- 

 tions P 1; P 2 , P 3 of a vector P along the directions of the tangents 

 to the coordinate lines q_uq%,q% at any point. Then projecting along 

 the normal n to S, we have the integrand in the surface integral 



29) 



If we divide the volume T 

 up into elementary curved 

 prisms bounded by level sur- 

 faces of q 2 and g 3 , as in the 

 case of rectangular coordinates 

 (Fig. 136), we have, at each 

 case of cutting into or out 

 of S respectively, 



P t cos (nn ) -f- P 2 cos (nn 2 ) + P 3 cos 



dS 



{ , 



dScos(nni) 



where dS^ is the area of the 

 part out by the prism from 

 the level surface q v 

 Now by 114, 

 ~ dq z dq s 



accordingly 



30) - 



Fig. 136. 



the change from the double to the triple integral involving the same 

 considerations as in the proof given for rectangular coordinates in 

 115. 



