NationMaster - Encyclopedia: 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 two-dimensional case of the more general Stokes' theorem. The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ... In mathematical analysis, there is a serious distinction between a double integral and an iterated integral. ... The title page to George Greens original essay on what is now known as Greens theorem. ... Stokess theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ...

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 In mathematics, a function f(x) of a real number variable x is defined piecewise, if f(x) is given by different expressions on various intervals. ... In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ... In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant. ...

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

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

$oint_{C}$

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. In mathematics, a positively oriented curve is a planar simple closed curve (that is, a curve in the plane whose starting point is also the end point and which has no other self-intersections) such that when travelling on it one always has the curve interior to the left (and...

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. An example of Green's theorem is a burp: The sum of the gas production (outflow) within the volume-boundary of a stomach is equal to the outflow of gas through the area-boundary that is the opening to the esophagus while issuing a burp (outflow).

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

If D is the simple region so that x ∈ [a, b] and g₁(x) < y < g₂(x) and the boundary of D is divided into the curves C₁, C₂, C₃, C₄, we can demonstrate Green's theorem.

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. Image File history File links Download high resolution version (795x747, 38 KB) Illustration of a simple region for proving Greens theorem File links The following pages link to this file: Greens theorem ... Image File history File links Download high resolution version (795x747, 38 KB) Illustration of a simple region for proving Greens theorem File links The following pages link to this file: Greens theorem ...

If it can be shown that

$int_{C} L, dx = iint_{D} left(- frac{partial L}{partial y}right) dAqquadmathrm{(1)}$

and

$int_{C} M, dy = iint_{D} left(frac{partial M}{partial x}right), dAqquadmathrm{(2)}$

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:

$D = {(x,y)|ale xle b, g_1(x) le y le g_2(x)}$

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

$iint_{D} left(frac{partial L}{partial y}right), dA$	$=int_a^b!!int_{g_1(x)}^{g_2(x)} left[frac{partial L}{partial y} (x,y), dy, dx right]$
	$= int_a^b Big{L[x,g_2(x)] - L[x,g_1(x)] Big} , dxqquadmathrm{(3)}$

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: Graph of a butterfly curve, a parametric equation discovered by Temple H. Fay In mathematics, a parametric equation explicitly relates two or more variables in terms of one or more independent parameters. ...

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

With −C₃, use the parametric equations, x = x, y = g₂(x), a ≤ x ≤ b. Then: Graph of a butterfly curve, a parametric equation discovered by Temple H. Fay In mathematics, a parametric equation explicitly relates two or more variables in terms of one or more independent parameters. ...

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

On C₂ and C₄, x remains constant, meaning

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

Therefore,

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

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

$int_{C} L(x,y), dx = iint_{D} left(- frac{partial L}{partial y}right), dA$

A similar proof can be employed on (2).

Categories: Vector calculus | Mathematical theorems Planimeter A planimeter is a technical drawing instrument used to measure the surface area of an arbitrary two dimensional shape. ... The method of image charges (also known as the method of images and method of mirror charges) is a basic problem solving tool in electrostatics. ...

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