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

:$int\_\{C\}\; (L,\; dx\; +\; M,\; dy)\; =\; iint\_\{D\}\; left(frac\{partial\; M\}\{partial\; x\}\; -\; frac\{partial\; L\}\{partial\; y\}\; ight),\; dA.$

Sometimes a small circle is placed on the integral symbol $left(oint\_\{C\}\; ight)$ 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.

Proof when "D" is a simple region

The following is a proof of the theorem for the simplified area "D", a type I region where "C"₂ and "C"₄ are vertical lines. A similar proof exists for when "D" is a type II region where "C"₁ and "C"₃ are straight lines.

If it can be shown that

:$int\_\{C\}\; L,\; dx\; =\; iint\_\{D\}\; left(-\; frac\{partial\; L\}\{partial\; y\}\; ight),\; dAqquadmathrm\{(1)\}$

and

:$int\_\{C\}\; M,\; dy\; =\; iint\_\{D\}\; left(frac\{partial\; M\}\{partial\; x\}\; ight),\; dAqquadmathrm\{(2)\}$

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

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

:$D\; =\; \{(x,y)|ale\; xle\; b,\; g\_1(x)\; le\; y\; le\; g\_2(x)\}$

where "g"₁ and "g"₂ are continuous functions on ["a", "b"] . Compute the double integral in (1):

:

Now compute the line integral in (1). "C" can be rewritten as the union of four curves: "C"₁, "C"₂, "C"₃, "C"₄.

With "C"₁, use the parametric equations: "x" = "x", "y" = "g"₁("x"), "a" ≤ "x" ≤ "b". Then

:$int\_\{C\_1\}\; L(x,y),\; dx\; =\; int\_a^b\; Big\{L(x,g\_1(x))Big\},\; dx$

With "C"₃, use the parametric equations: "x" = "x", "y" = "g"₂("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 "C"₃ is negated because it goes in the negative direction from "b" to "a", as "C" is oriented positively (counterclockwise). On "C"₂ and "C"₄, "x" remains constant, meaning

:$int\_\{C\_4\}\; L(x,y),\; dx\; =\; int\_\{C\_2\}\; L(x,y),\; dx\; =\; 0$

Therefore,

:

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::$iint\_Dleft(\; ablacdotmathbf\{F\}\; ight)dA=int\_C\; mathbf\{F\}\; cdot\; mathbf\{hat\; n\}\; ,\; ds,$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 $dmathbf\{r\}\; =\; langle\; dx,\; dy\; angle$ 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 $langle\; dy,\; -dx\; angle$. The length of this vector is $sqrt\{dx^2\; +\; dy^2\}\; =\; ds$. So $mathbf\{hat\; n\},ds\; =\; langle\; dy,\; -dx\; angle$.

Now let the components of $mathbf\{F\}\; =\; langle\; P,\; Q\; angle$. 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:$int\_C\; -Q\; dx\; +\; P\; dy\; =\; iint\_\{D\}\; left(frac\{partial\; P\}\{partial\; x\}\; +\; frac\{partial\; Q\}\{partial\; y\}\; ight),\; dA\; =\; iint\_Dleft(\; ablacdotmathbf\{F\}\; ight)dA.$

ee also

* Stokes' theorem
* Divergence theorem
* Planimeter
* Method of image charges - A method used in electrostatics that takes strong advantage of the uniqueness theorem (derived from Green's theorem)
* Green's identities

External links

* [http://mathworld.wolfram.com/GreensTheorem.html Green's Theorem on MathWorld]
* [http://www.mechanisms101.com/greens_theorem_demo.html A flash demo for Green's Theorem]