18.4 A General Green’s Theorem
Now suppose U is a region in the uv plane for which Green’s theorem holds and that
where R is C2
and is one to one,
Ru ×Rv≠0. Here, to be specific, the
u,v axes are oriented as the
x,y axes respectively.
Also let F
=
be a
C1 vector field defined near
V . Note that
F does not
depend on
z. Therefore,
∇ × F(x,y) = (Qx (x,y)− Px(x,y))k.
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You can check this from the definition. Also
( )
x (u,v)
R (u,v) = y(u,v)
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and so Ru × Rv, the normal vector to V is
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|| xu xv ||
|| yu yv||
∥∥--------∥∥-k
∥∥ xu xv ∥∥
∥ yu yv∥
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Suppose
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||xu xv ||
||y y || > 0
u v
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so the unit normal is then just k. Then Stoke’s theorem applied to this special case yields
∫ ∫ || ||
F⋅dR= (Qx(x(u,v),y(u,v))− Px(x(u,v),y(u,v)))k⋅k || xu xv ||dA
∂V U | yu yv |
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Now by the change of variables formula, this equals
∫
= (Qx (x,y)− Px (x,y))dA
V
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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
, where
≠0,
R is one to one, and twice continuously differentiable
with
Ru × Rv in the direction of
k, then Green’s theorem holds for
V also.
This verifies the following theorem.
Theorem 18.4.1 (Green’s Theorem) Let V be an open set in the plane and let ∂V be piecewise smooth and let
F
=
be a C1 vector field defined near V. Then if V is oriented counter clockwise, it
is often
the case that
∫ ∫ ( ∂Q ∂P )
F⋅dR = ∂x-(x,y)− ∂y-(x,y) dA. (18.4)
∂V V
| (18.4) |
In particular, if there exists U such as the simple convex in both directions case considered earlier for which
Green’s theorem holds, and V = R
where R :
U → V is C2 such that ≠0
and
Rx ×Ry is in the direction of k, then 18.4 is valid where the orientation around ∂V is consistent with the
orientation around U.
This is a very general version of Green’s theorem which will include most of what will be of
interest.