Green's Theorem: Difference between revisions
imported>Paul Wormer (appended 3D case) |
mNo edit summary |
||
| (2 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
{{subpages}} | {{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>. | '''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 various 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 15: | Line 15: | ||
</math> | </math> | ||
=== 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> | ||
| Line 23: | Line 22: | ||
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 | 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 | Different ways of formulating Green's theorem in three dimensions may be found. One of the more useful formulations is | ||
: <math> | : <math> | ||
| Line 59: | Line 58: | ||
\iiint\limits_V \phi \boldsymbol{\nabla}^2\psi - \psi \boldsymbol{\nabla}^2\phi\, d V = | \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} . | \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> | </math>[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 16:01, 23 August 2024
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega\,} and have continuous partial derivatives in that region. Then Green's Theorem states that
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oint\limits_{\partial\Omega}(fdx+gdy)=\iint\limits_\Omega \left(\frac{\partial g}{\partial x}-\frac{\partial f}{\partial y}\right)dxdy }
The theorem is equivalent to the curl theorem in the plane and can be written in a more compact form as
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oint\limits_{\partial \Omega}\mathbf{F}\cdot d\mathbf{S}=\iint\limits_\Omega (\nabla\times\mathbf{F})d\mathbf{A} }
Application: Area Calculation
Green's theorem is very useful when it comes to calculating the area of a region. If we take Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x,y)=y} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x,y)=x} , the area of the region Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega\,} , with boundary Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial\Omega} can be calculated by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=\frac{1}{2}\oint\limits_{\partial \Omega} xdy-ydx }
This formula gives a relationship between the area of a region and the line integral around its boundary.
If the curve is parametrized as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(x(t),y(t)\right)} , the area formula becomes
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=\frac{1}{2}\oint\limits_{\partial \Omega}(xy'-x'y)dt }
Statement in three dimensions
Different ways of formulating Green's theorem in three dimensions may be found. One of the more useful formulations is
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}. }
Proof
The divergence theorem reads
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iiint\limits_V \nabla \cdot \mathbf{F} \, d V = \iint\limits_{\partial V}\mathbf{F} \cdot d\mathbf{S} }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\mathbf{S}} is defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\mathbf{S}=\mathbf{n} \, dS} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{n}} is the outward-pointing unit normal vector field.
Insert
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F} = \phi \boldsymbol{\nabla}\psi - \psi \boldsymbol{\nabla}\phi }
and use
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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} }
so that we obtain the result to be proved,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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} . }