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 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 P and Q have continuous partial derivatives on an open region containing D, then

<math>\int_{C} P dx + Q dy = \int\!\!\!\int_{D} \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA<math>

Sometimes the notation

<math>\oint_{C} P dx + Q dy<math>

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

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Proof of Green's Theorem when D is a parametric equations, x = x, y = g₁(x), a ≤ x ≤ b. Therefore:

<math>\int_{C_1} P(x,y) dx = \int_a^b [P(x,g_1(x))] dx<math>

With -C₃, use the parametric equations, x = x, y = g₂(x), a ≤ x ≤ b. Therefore:

<math>\int_{C_3} P(x,y) dx = -\int_{-C_3} P(x,y) dx = - \int_a^b [P(x,g_2(x))] dx<math>

With C₂ and C₄, x is a constant, meaning:

<math> \int_{C_4} P(x,y) dx = \int_{C_2} P(x,y) dx = 0<math>

Therefore,

<math>\int_{C} P dx = \int_{C_1} P(x,y) dx + \int_{C_2} P(x,y) dx + \int_{C_3} P(x,y) + \int_{C_4} P(x,y) dx <math>

<math> = - \int_a^b [P(x,g_2(x))] dx + \int_a^b [P(x,g_1(x))] dx<math>

Combining this with equation 4, we get:

<math>\int_{C} P(x,y) dx = \int\!\!\!\int_{D} \left(- \frac{\partial P}{\partial y}\right) dA<math>

A similar proof can be employed on Eq.2.