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. It is the two-dimensional special case of the more general Stokes' theorem, and is named after British mathematician George Green.

Let C be a positively oriented, piecewise smooth, simple closed curve in the plane R2, and let D be the region bounded by C. If L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there, then

\int_{C} (L\, dx + M\, dy) = \iint_{D} \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right)\, dA.

Sometimes a small circle is placed on the integral symbol \left(\oint_{C}\right) to indicate that the curve C is closed. For positive orientation, an arrow pointing in the counterclockwise direction may be drawn in this circle.

In physics, Green's theorem is mostly used to solve two-dimensional flow integrals, stating that the sum of fluid outflows at any point inside a volume is equal to the total outflow summed about an enclosing area.

Contents

Proof when D is a simple region

If D is a simple region with its boundary consisting of the curves C1, C2, C3, C4, Green's theorem can be demonstrated.

The following is a proof of the theorem for the simplified area D, a type I region where C2 and C4 are vertical lines. A similar proof exists for when D is a type II region where C1 and C3 are straight lines.

If it can be shown that

\int_{C} L\, dx = \iint_{D} \left(- \frac{\partial L}{\partial y}\right)\, dA\qquad\mathrm{(1)}

and

\int_{C} M\, dy = \iint_{D} \left(\frac{\partial M}{\partial x}\right)\, dA\qquad\mathrm{(2)}

are true, then Green's theorem is proven in the first case.

Define the type I region D as pictured on the right by:

D = \{(x,y)|a\le x\le b, g_1(x) \le y \le g_2(x)\}

where g1 and g2 are continuous functions on a, b. Compute the double integral in (1):

\iint_{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 \Big\{L(x,g_2(x)) - L(x,g_1(x)) \Big\} \, dx\qquad\mathrm{(3)}

Now compute the line integral in (1). 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. Then

\int_{C_1} L(x,y)\, dx = \int_a^b \Big\{L(x,g_1(x))\Big\}\, dx

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

\int_{C_3} L(x,y)\, dx = -\int_{-C_3} L(x,y)\, dx = - \int_a^b [L(x,g_2(x))]\, dx

The integral over C3 is negated because it goes in the negative direction from b to a, as C is oriented positively (counterclockwise). On C2 and C4, x remains constant, meaning

\int_{C_4} L(x,y)\, dx = \int_{C_2} L(x,y)\, dx = 0

Therefore,

\int_{C} L\, dx = \int_{C_1} L(x,y)\, dx + \int_{C_2} L(x,y)\, dx + \int_{C_3} L(x,y)\, dx + \int_{C_4} L(x,y)\, dx
= -\int_a^b [L(x,g_2(x))]\, dx + \int_a^b [L(x,g_1(x))]\, dx\qquad\mathrm{(4)}

Combining (3) with (4), we get (1). Similar computations give (2).

Relationship to the divergence theorem

Green's theorem is equivalent to the following two-dimensional analogue of the divergence theorem:

\iint_D\left(\nabla\cdot\mathbf{F}\right)dA=\int_C \mathbf{F} \cdot \mathbf{\hat n} \, ds,

where \mathbf{\hat n} is the outward-pointing unit normal vector on the boundary.

To see this, consider the unit normal in the right side of the equation. Since d\mathbf{r} = \langle dx, dy\rangle is a vector pointing tangential along a curve, and the curve C is the positively-oriented (i.e. counterclockwise) curve along the boundary, an outward normal would be a vector which points 90° to the right, which would be \langle dy, -dx\rangle. The length of this vector is \sqrt{dx^2 + dy^2} = ds. So \mathbf{\hat n}\,ds = \langle dy, -dx\rangle.

Now let the components of \mathbf{F} = \langle P, Q\rangle. Then the right hand side becomes

\int_C \mathbf{F} \cdot \mathbf{\hat n} \, ds = \int_C P dy - Q dx

which by Green's theorem becomes

\int_C -Q dx + P dy = \iint_{D} \left(\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}\right)\, dA = \iint_D\left(\nabla\cdot\mathbf{F}\right)dA.

See also

External links

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  • This page was last modified on 18 November 2008, at 21:02.

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