Green's theorem
In physics and mathematics, Green's Theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. Green's Theorem was named after British scientist George Green, and is based on Stokes' theorem. The theorem states:
- Let C be a positively oriented, piecewise smooth, simple closed curve in the plane and let D be the region bounded by C. If P and Q have continuous partial deriviatives on an open region containing D, then
Sometimes the notation
is used to indicate the line integral is calculuated using the positive orientation of the closed curve C.
Proof of Green's Theorem, General Edition
Proof of Green's Theorem when D is a simple region
If we show Equations 1 and 2
and
are true, we would prove Green's Theorem.
If we express D as a region such that:
where g1 and g2 are continuous functions, we can compute the double integral of equation 1:
Now we break up C as the union of four curves: C1, C2, C3, C4.
- (Pic could be added here to see how C could be broken up and help explain following proof)
With C1, use the parametric equations, x = x, y = g1(x), a ≤ x ≤ b. Therefore:
With -C3, use the parametric equations, x = x, y = g2(x), a ≤ x ≤ b. Therefore:
With C2 and C4, x is a constant, meaning:
Therefore,
Combining this with equation 4, we get:
A similar proof can be employed on Eq.2.
Referenced By
1828 in science | Cyclic process | George Green | Isoperimetry | List of calculus topics | List of mathematical proofs | List of mathematical topics (G-I) | List of mathematical topics (G-Z) | List of proofs | Mathematical timeline | Mikhail Vasilievich Ostrogradsky | Polygon | Polygons | Stoke's theorem | Stokes' theorem | Stokes theorem | Timeline of mathematics
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