604 CHAPTER 30. STOKES AND GREEN’S THEOREMS
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)−Py (x,y))k.
You can check this from the definition. Also
R(u,v) =(
x(u,v)y(u,v)
)and so, from the definition of Ru ×Rv, the desired unit normal vector to V is
xuyv − xvyu
|xuyv − xvyu|k
Suppose xuyv − xvyu > 0. Then the unit normal is k. Then Stoke’s theorem applied to thisspecial case yields∫
∂VF ·dR=
∫U(Qx (x(u,v) ,y(u,v))−Py (x(u,v) ,y(u,v)))k ·k
∣∣∣∣ xu xvyu yv
∣∣∣∣dA
Now by the change of variables formula, this equals∫
V (Qx (x,y)−Py (x,y))dA. This is justGreen’s theorem for V . Thus if U is a region for which Green’s theorem holds and if V isanother region, V =R(U) , where |Ru ×Rv| ̸= 0, R is one to one, and twice continuouslydifferentiable with Ru ×Rv in the direction of k, then Green’s theorem holds for V also.
This verifies the following theorem.
Theorem 30.4.1 (Green’s Theorem) Let V be an open set in the plane for which thedivergence theorem holds and let ∂V be piecewise smooth and F (x,y) = (P(x,y) ,Q(x,y))be a C1 vector field defined near V. Then if V is oriented counter clockwise,∫
∂VF ·dR=
∫V
(∂Q∂x
(x,y)− ∂P∂y
(x,y))
dA. (30.2)
In particular, if there exists U for which the divergence theorem holds and V = R(U)where R : U → V is C2
(U ,R2
)such that
∣∣Rx ×Ry∣∣ ̸= 0 and Rx ×Ry is in the direction
of k, then 30.2 is valid where the orientation around ∂V is consistent with the orientationaround U.
This is a very general version of Green’s theorem which will include most if not allof what will be of interest. However, there are more general versions of this importanttheorem. 1
1For a general version see the advanced calculus book by Apostol. Also see my book on calculus of real andcomplex variables. The general versions involve the concept of a rectifiable Jordan curve. You need to be ableto take the area integral and to take the line integral around the boundary. This general version of this theoremappeared in 1951. Green lived in the early 1800’s.