V. CURVILINEAR COORDINATES 



5.00 Given three surfaces. 



i. 



=M*,y,z), 



v =Mx,y,z), 



X = 0i (W, V, W), 



y = <t>2(u,v,w), 

 z = 03 (u, v,w): 



\du 



, 



dv 



^V , /^*?Y 



dw) -{toff/' 





dv 



dw 



i i 3 3 



5y aw a?; 6ze; dv dw' 



a^i 30i a0 2 a0 2 a0 3 a0 3 ^ 



a^ du dw du dw du 



a0i a0i a0 2 a0 2 a0 3 a0 3 



" h ' 



5.01 The linear element of arc, ds, is given by: 



70 ,79 ^72 



ds 2 = <fa 2 + rfy 2 + dz 2 = 7^- + 7-5 + TT + 2 ^i 



5.02 The surface elements, areas of parallelograms on the three surfaces, are: 



dv dw / 



Aw du 

 dudv 



99 



