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| \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 17:01, 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
and have continuous partial derivatives in that region. Then Green's Theorem states that
![{\displaystyle \oint \limits _{\partial \Omega }(fdx+gdy)=\iint \limits _{\Omega }\left({\frac {\partial g}{\partial x}}-{\frac {\partial f}{\partial y}}\right)dxdy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/309344fba7e068b5ee37bddfc134b57edad8fec0)
The theorem is equivalent to the curl theorem in the plane and can be written in a more compact form as
![{\displaystyle \oint \limits _{\partial \Omega }\mathbf {F} \cdot d\mathbf {S} =\iint \limits _{\Omega }(\nabla \times \mathbf {F} )d\mathbf {A} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2d896e3e3038f31b922095b9a0fcff449e67afc)
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
![{\displaystyle A={\frac {1}{2}}\oint \limits _{\partial \Omega }xdy-ydx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c437c42d5fac37812ed9de25b522e6c5e741022)
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
![{\displaystyle A={\frac {1}{2}}\oint \limits _{\partial \Omega }(xy'-x'y)dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/271aa110285547cb7554570de71ad96d25ae826d)
Statement in three dimensions
Different ways of formulating Green's theorem in three dimensions may be found. One of the more useful formulations is
![{\displaystyle \iiint \limits _{V}{\Big (}\phi {\boldsymbol {\nabla }}^{2}\psi -\psi {\boldsymbol {\nabla }}^{2}\phi {\Big )}\,dV=\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} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33b6db12e9e03b24fd235e79b78bf003942ac625)
Proof
The divergence theorem reads
![{\displaystyle \iiint \limits _{V}\nabla \cdot \mathbf {F} \,dV=\iint \limits _{\partial V}\mathbf {F} \cdot d\mathbf {S} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/442312d869106d4a551b3ad1714346eb256f93ab)
where
is defined by
and
is the outward-pointing unit normal vector field.
Insert
![{\displaystyle \mathbf {F} =\phi {\boldsymbol {\nabla }}\psi -\psi {\boldsymbol {\nabla }}\phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/05a346e6dc686846b3ed6aa6556c02b149136687)
and use
![{\displaystyle {\begin{aligned}{\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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2584c683ba51a3adbb7b832d3b2668d7add44e82)
so that we obtain the result to be proved,
![{\displaystyle \iiint \limits _{V}\phi {\boldsymbol {\nabla }}^{2}\psi -\psi {\boldsymbol {\nabla }}^{2}\phi \,dV=\iint \limits _{\partial V}\phi {\boldsymbol {\nabla }}\psi \cdot d\mathbf {S} -\iint \limits _{\partial V}\psi {\boldsymbol {\nabla }}\phi \cdot d\mathbf {S} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f4c73257cdf58e533f5e324e4f3bd8043cdfa17)