Green in plane digression
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A brief Word note by Phil dated 5.22.16, in a support folder for his tensor and wedge product document. It writes Green's theorem in the plane in differential-form notation (the integral of dα over M equals the integral of α over ∂M). It then derives a volume-integral-of-a-gradient theorem from the divergence theorem using a constant vector a, and specializes to R2 and R3. The text is fragmentary, with some symbols lost.
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Green in plane digression PhL 5.22.16
Digression on Green's Theorem in a Plane
I am searching for this equation
∫M dα = ∫∂M α
∫M (∂xg - ∂yf) dxdy =∫∂M (fdx + gdy) Green's Theorem in the plane
// α = 1-form, dα = 2-form, M = planar area, ∂M = its bounding curve
∫M (∂1f2 - ∂2f1) dx1dx2 =∫∂M (f1dx1 + f2dx2) Green's Theorem in the plane
// α = 1-form, dα = 2-form, M = planar area, ∂M = its bounding curve
This looks like a curl application. Hold off.
so certainly α is a 1 form
If we take
f = g a where a is a fixed vector
then
f = (ga) = g a + g a = g a
and then the divergence theorem reads
∫V g a dV = ∫S g a dA
(∫V g dV) a = (∫S g dA) a .
Since this is true for any constant vector a, we must have
∫V g dV = ∫S g dA
Writing dA = dA .
∫V g dV = ∫S g dA // "volume integral of a gradient theorem"
For R2 this reads
∫A g dA = ∫C g ds
where A = Area and C = Curve. Taking ds =
On the other hand, selecting a = gives
∫V ∂xg dV = ∫S g a dA
For R3 this is known as Gauss's Theorem.
For R2 one has
f = ∂xf1 + ∂yf2
∫M (∂xg - ∂yf) dxdy =∫∂M (fdx + gdy)