23.4. A GENERAL GREEN’S THEOREM 441

orientation for it other than this one. In fact there is none. You can see this by looking at thefirst of the two pictures below or by making one and tracing it with a pencil. There is onlyone side to the Mobeus band. An oriented surface must have two sides, one side identifiedby the given unit normal which varies continuously over the surface and the other sideidentified by the negative of this normal. The second picture below was taken by Ouyangwhen he was at meetings in Paris and saw it at a museum.

23.4 A General Green’s TheoremNow suppose U is a region in the uv plane for which Green’s theorem holds and that

V ≡R(U)

where R is C2(U ,R2

)and is one to one, Ru×Rv ̸= 0. Here, to be specific, the u,v axes

are oriented as the x,y axes respectively.

x

y

u

v

Also let F (x,y,z) = (P(x,y) ,Q(x,y) ,0) be a C1 vector field defined near V . Note thatF does not depend on z. Therefore,

∇×F (x,y) = (Qx (x,y)−Px (x,y))k.

You can check this from the definition. Also

R(u,v) =

(x(u,v)y(u,v)

)

and so Ru×Rv, the normal vector to V is∣∣∣∣∣ xu xv

yu yv

∣∣∣∣∣∥∥∥∥∥ xu xv

yu yv

∥∥∥∥∥k