Green's theorem

From Academic Kids

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>
Missing image
Green's-theorem-simple-region.png
If D is the simple region so that x ∈ [a, b] and g1(x) < y < g2(x) and the boundary of D is divided into the curves C1, C2, C3, C4, we can demonstrate Green's theorem.

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), axb. 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), axb. 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:Thorme de Green it:Teorema di Green ja:グリーンの定理

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