PARTIAL DIFFERENTIATION 
151 
small circle about this point, when described in the counter-clockwise 
sense, goes over into a small oval about (?/ 0 , v 0 ), likewise described in 
the counter-clockwise sense. But when the sense is reversed, the 
Jacobian is negative. 
34. Let 
u =f(x, y, *), v = y, z), w = ip(x, y, z), 
d(u, v, w) ^ 0 
d(x, y, z) 
the usual conditions of continuity being assumed in the neighbor 
hood of (x 0 , y 0 , z 0 ). Show that the Jacobian is positive at (» 0 , y 0 , z 0 ) 
if the positive directions of the curvilinear coordinates (w, v, w) are 
oriented there as the positive directions of the (x, y, ^-coordinates; 
otherwise, the Jacobian is negative. 
35. If F(u, v, x, y) and v, x, y) are two functions which satisfy 
the conditions of § 10, show that 
8_(F l ^_ 
d(u,v) = d(x,y) 
d(x,y) d(F,$) 
8 (u, v) 
Is the corresponding theorem true in the general case, n — n?