Now suppose U is a region in the uv plane for which Green’s theorem holds and that

V ≡ R (U )

where R is C^{2}

(U-,ℝ2)

and is one to one, R_{u}×R_{v}≠0. Here, to be specific, the u,v axes are oriented as the
x,y axes respectively.

PICT

Also let F

(x,y,z)

=

(P (x,y),Q (x,y),0)

be a C^{1} vector field defined near V . Note that F 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 R_{u}× R_{v}, the normal vector to V is

| |
|| xu xv ||
|| yu yv||
∥∥--------∥∥-k
∥∥ xu xv ∥∥
∥ yu yv∥

Suppose

| |
||xu xv ||
||y y || > 0
u v

so the unit normal is then just k. Then Stoke’s theorem applied to this special case yields

| |
∫ ∫ || xu xv ||
∂V F⋅dR= U (Qx (x(u,v),y(u,v))− Px(x(u,v),y(u,v)))k ⋅k || yu yv ||dm2(u,v)

Now by the change of variables formula, this equals

∫
= V (Qx (x,y)− Px(x,y))dm2

This is just Green’s theorem for V . Thus if U is a region for which Green’s theorem holds and if V is
another region, V = R

(U )

, where

|R × R |
u v

≠0, R is one to one, and twice continuously differentiable
with R_{u}× R_{v} in the direction of k, then Green’s theorem holds for V also.

This verifies the following theorem.

Theorem 12.3.1(Green’s Theorem) Let V be an openset in the plane and let ∂V be piecewise smooth and letF

(x,y)

=

(P (x,y) ,Q (x,y))

be a C^{1}vector field defined near V. Then if ∂V is oriented counter clockwise, it isoften^{1}the case that

∫ ∫ ( )
F ⋅dR = ∂Q-(x,y)− ∂P- (x,y) dm . (12.5)
∂V V ∂x ∂y 2

(12.5)

In particular, if there exists U such as the simple convex in both directions case considered earlier for whichGreen’s theorem holds, and V = R

(U )

where R : U → V is C^{2}

(-- 2)
U,ℝ

such that

|Rx × Ry |

≠0 andR_{x}×R_{y}is in the direction ofk, then 12.5is valid where the orientation around ∂V is consistent with theorientation around ∂U.

This is a very general version of Green’s theorem which will include most of what will be of interest.
However, it suffers from several shortcomings. First the region V is defined as R

(U )

where R ∈ C^{2}

(U,ℝ2)

and ∂U is piecewise C^{1}. What if you just have a bounded variation function R : S^{1}→ ℝ^{2}
where S^{1} denotes the unit circle? Can you give Green’s theorem for the inside of this curve? Of
course the first step is to verify that there is an inside and an outside such that R

(S1)

is the
boundary of both. This is the Jordan curve theorem presented earlier. Next you need to deal with
the general case of line integrals with respect to bounded variation curves which is not done
above.

The more general result presented in the next section will allow for all of these considerations and will
also allow for much more general vector fields F. It is not necessary to assume F ∈ C^{2}

(U,ℝ2 )

for example.
In fact, it suffices to have F be C^{1} on the open set U and the partial derivatives of F be in L^{1}

(U )

, and you
only need F continuous on U. This is significantly more general, and the presentation in the following
section is sufficient to include these things. In fact, it is possible to do all of this for even more general
situations.