- Green's theorem
In

physics andmathematics ,**Green's theorem**gives the relationship between aline integral around a simple closed curve "C" and adouble integral over the plane region "D" bounded by "C". It is the two-dimensional special case of the more generalStokes' theorem , and is named after British scientistGeorge Green .Let "C" be a positively oriented,

piecewise smooth ,simple closed curve in the plane**R**, 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^{2}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}arecontinuous function s 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 theparametric equation s: "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*]

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