Green's theorem
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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 a special case of the more general 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 L and M have continuous partial derivatives on an open region containing D, then
- <math>\int_{C} L\, dx + M\, dy = \int\!\!\!\int_{D} \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right)\, dA<math>
Sometimes the notation
- <math>\oint_{C} L\, dx + M\, dy<math>
is used to indicate the line integral is calculated using the positive orientation of the closed curve C.
Proof of Green's theorem when D is a simple region
If we show Equations 1 and 2
- <math>EQ.1 = \int_{C} L dx = \int\!\!\!\int_{D} \left(- \frac{\partial L}{\partial y}\right) dA<math>
and
- <math>EQ.2 = \int_{C} M\, dy = \int\!\!\!\int_{D} \left(\frac{\partial M}{\partial x}\right)\, dA<math>
are true, we would prove Green's theorem.
If we express D as a region such that:
- <math>D = \{(x,y)|a\le x\le b, g_1(x) \le y \le g_2(x)\}<math>
where g1 and g2 are continuous functions, we can compute the double integral of equation 1:
- <math> EQ.4 = \int\!\!\!\int_{D} \left(\frac{\partial L}{\partial y}\right)\, dA = \int_a^b\!\!\int_{g_1(x)}^{g_2(x)} \left(\frac{\partial L}{\partial y} (x,y)\, dy\, dx \right) = \int_a^b [L(x,g_2(x)) - L(x,g_1(x))]\, dx<math>
Green's-theorem-simple-region.png
Now we break up C as the union of four curves: C1, C2, C3, C4.
With C1, use the parametric equations, x = x, y = g1(x), a ≤ x ≤ b. Therefore:
- <math>\int_{C_1} L(x,y)\, dx = \int_a^b [L(x,g_1(x))]\, dx<math>
With −C3, use the parametric equations, x = x, y = g2(x), a ≤ x ≤ b. Then:
- <math>\int_{C_3} L(x,y)\, dx = -\int_{-C_3} L(x,y)\, dx = - \int_a^b [L(x,g_2(x))]\, dx<math>
With C2 and C4, x is a constant, meaning:
- <math> \int_{C_4} L(x,y)\, dx = \int_{C_2} L(x,y)\, dx = 0<math>
Therefore,
- <math>\int_{C} L\, dx = \int_{C_1} L(x,y)\, dx + \int_{C_2} L(x,y)\, dx + \int_{C_3} L(x,y) + \int_{C_4} L(x,y)\, dx <math>
- <math> = - \int_a^b [L(x,g_2(x))]\, dx + \int_a^b [L(x,g_1(x))]\, dx<math>
Combining this with equation 4, we get:
- <math>\int_{C} L(x,y)\, dx = \int\!\!\!\int_{D} \left(- \frac{\partial L}{\partial y}\right)\, dA<math>
A similar proof can be employed on Eq.2.Green's theorem es:Teorema de Green fr:Théorème de Green it:Teorema di Green ja:グリーンの定理