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 statement is the following. 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

[\int_ L\, dx + M\, dy = \iint_ \left(\frac - \frac\right)\, dA.]

Sometimes a small circle is placed on top of the integral symbol:

[\oint_]

This indicates that the curve C is closed. To indicate positive orientation, an arrow pointing in the counter-clockwise direction is sometimes drawn in the circle over the integral symbol.

Proof of Green's theorem when D is a simple region

We will prove the theorem for the simplified area D where C₂ and C₄ are vertical lines, however the theorem remains valid for any area D as defined above.

If it can be shown that

[\int_ L\, dx = \iint_ \left(- \frac\right) dA\qquad\mathrm]

and

[\int_ M\, dy = \iint_ \left(\frac\right)\, dA\qquad\mathrm]

are true, then Green's theorem is proven.

We define a region D that is simple enough for our purposes. If region D is expressed such that:

[D = \]

where g₁ and g₂ are continuous functions, the double integral in (1) can be computed:

[ \iint_ \left(\frac\right)\, dA]	[=\int_a^b\!\!\int_^ \left[frac (x,y), dy, dx right] ]
	[ = \int_a^b \Big\ \, dx\qquad\mathrm]

Now C can be rewritten 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:

[\int_ L(x,y)\, dx = \int_a^b \Big\\, dx]

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

[\int_ L(x,y)\, dx = -\int_ L(x,y)\, dx = - \int_a^b [L(x,g_2(x))]\, dx]

On C₂ and C₄, x remains constant, meaning

[ \int_ L(x,y)\, dx = \int_ L(x,y)\, dx = 0]

Therefore,

[ \int_ L\, dx ]	[ = \int_ L(x,y)\, dx + \int_ L(x,y)\, dx + \int_ L(x,y) + \int_ L(x,y)\, dx ]
	[ = -\int_a^b [L(x,g_2(x))]\, dx + \int_a^b [L(x,g_1(x))]\, dx\qquad\mathrm]

Combining (3) with (4), we get:

[\int_ L(x,y)\, dx = \iint_ \left(- \frac\right)\, dA]

A similar proof can be employed on (2).

Green's theorem

Proof of Green's theorem when D is a simple region

See also