Green's Theorem: Difference between revisions

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Green's Theorem is a vector identity that is equivalent to the [[curl theorem]] in two dimensions. It relates the line integral around a simple closed curve <math>\partial\Omega</math> with the doubble integral over the plane region <math>\Omega\,</math>.
{{subpages}}
'''Green's Theorem''' is a vector identity that is equivalent to the [[curl theorem]] in two dimensions. It relates the line integral around a simple closed curve <math>\partial\Omega</math> with the double integral over the plane region <math>\Omega\,</math>.


The theorem is named after the british mathematician [[George Green]]. It can be applied to variuos fields in physics, among others flow integrals.
The theorem is named after the British mathematician [[George Green]]. It can be applied to various fields in physics, among others flow integrals.


== Mathematical Statement ==
== Mathematical Statement in two dimensions==
Let <math>\Omega\,</math> be a region in <math>\R^2</math> with a positively oriented, piecewise smooth, simple closed boundary <math>\partial\Omega</math>. <math>f(x,y)</math> and <math>g(x,y)</math> are functions defined on a open region containing <math>\Omega\,</math> and have continuous [[partial derivative|partial derivatives]] in that region. Then Green's Theorem states that
Let <math>\Omega\,</math> be a region in <math>\R^2</math> with a positively oriented, piecewise smooth, simple closed boundary <math>\partial\Omega</math>. <math>f(x,y)</math> and <math>g(x,y)</math> are functions defined on a open region containing <math>\Omega\,</math> and have continuous [[partial derivative|partial derivatives]] in that region. Then Green's Theorem states that
: <math>
: <math>
Line 14: Line 15:
</math>
</math>


== Applications ==
=== Application:  Area Calculation ===
=== Area Calculation ===
Green's theorem is very useful when it comes to calculating the area of a region. If we take <math>f(x,y)=y</math> and <math>g(x,y)=x</math>, the area of the region <math>\Omega\,</math>, with boundary <math>\partial\Omega</math> can be calculated by
Green's theorem is very useful when it comes to calculating the area of a region. If we take <math>f(x,y)=y</math> and <math>g(x,y)=x</math>, the area of the region <math>\Omega\,</math>, with boundary <math>\partial\Omega</math> can be calculated by
: <math>
: <math>
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This formula gives a relationship between the area of a region and the line integral around its boundary.
This formula gives a relationship between the area of a region and the line integral around its boundary.


If the curve is parametrisized as <math>\left(x(t),y(t)\right)</math>, the area formula becomes
If the curve is parametrized as <math>\left(x(t),y(t)\right)</math>, the area formula becomes
: <math>
: <math>
A=\frac{1}{2}\oint\limits_{\partial \Omega}(xy'-x'y)dt
A=\frac{1}{2}\oint\limits_{\partial \Omega}(xy'-x'y)dt
</math>
</math>
==Statement in three dimensions==
Different ways of formulating Green's theorem in three dimensions may be found. One of the more useful formulations is
: <math>
\iiint\limits_V \Big( \phi \boldsymbol{\nabla}^2\psi - \psi \boldsymbol{\nabla}^2\phi\Big)\, d V =
\iint\limits_{\partial V} \big(\phi \boldsymbol{\nabla}\psi\big) \cdot d\mathbf{S} - \iint\limits_{\partial V} \big(\psi \boldsymbol{\nabla}\phi\big)  \cdot d\mathbf{S}.
</math>
===Proof===
The [[divergence theorem]] reads
: <math >\iiint\limits_V \nabla \cdot \mathbf{F} \, d V =
\iint\limits_{\partial V}\mathbf{F} \cdot d\mathbf{S}
</math>
where <math>d\mathbf{S}</math> is defined by <math>d\mathbf{S}=\mathbf{n} \, dS</math> and <math>\mathbf{n}</math> is the outward-pointing unit normal vector field.
Insert
:<math>
\mathbf{F} = \phi \boldsymbol{\nabla}\psi - \psi \boldsymbol{\nabla}\phi
</math>
and use
:<math>
\begin{align}
\boldsymbol{\nabla}\cdot \mathbf{F} &= \big(\boldsymbol{\nabla}\phi\big)\cdot \big(\boldsymbol{\nabla}\psi\big)
-\big(\boldsymbol{\nabla}\psi\big)\cdot \big( \boldsymbol{\nabla}\phi\big)
+ \phi \boldsymbol{\nabla}^2\psi - \psi \boldsymbol{\nabla}^2\phi \\
&= \phi \boldsymbol{\nabla}^2\psi - \psi \boldsymbol{\nabla}^2\phi
\end{align}
</math>
so that we obtain the result to be proved,
: <math>
\iiint\limits_V  \phi \boldsymbol{\nabla}^2\psi - \psi \boldsymbol{\nabla}^2\phi\, d V =
\iint\limits_{\partial V}\phi \boldsymbol{\nabla}\psi \cdot d\mathbf{S} - \iint\limits_{\partial V}\psi \boldsymbol{\nabla}\phi  \cdot d\mathbf{S} .
</math>[[Category:Suggestion Bot Tag]]

Latest revision as of 17:01, 23 August 2024

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Green's Theorem is a vector identity that is equivalent to the curl theorem in two dimensions. It relates the line integral around a simple closed curve with the double integral over the plane region .

The theorem is named after the British mathematician George Green. It can be applied to various fields in physics, among others flow integrals.

Mathematical Statement in two dimensions

Let be a region in with a positively oriented, piecewise smooth, simple closed boundary . and are functions defined on a open region containing and have continuous partial derivatives in that region. Then Green's Theorem states that

The theorem is equivalent to the curl theorem in the plane and can be written in a more compact form as

Application: Area Calculation

Green's theorem is very useful when it comes to calculating the area of a region. If we take and , the area of the region , with boundary can be calculated by

This formula gives a relationship between the area of a region and the line integral around its boundary.

If the curve is parametrized as , the area formula becomes

Statement in three dimensions

Different ways of formulating Green's theorem in three dimensions may be found. One of the more useful formulations is

Proof

The divergence theorem reads

where is defined by and is the outward-pointing unit normal vector field.

Insert

and use

so that we obtain the result to be proved,