Green's Theorem

From HvWiki

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. It is a special two-dimensional case of the more general Stokes' theorem, and is named after British scientist George Green.

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.

1 Green's first theorem for scalar functions
2 Green's second theorem for scalar functions
3 Green's first theorem for vector functions
4 Green's second theorem for vectors

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Green's first theorem for scalar functions

Starting off by applying the identities to two continuous scalar functions where second derivatives can be defined:

$\phi\nabla^2\psi = \nabla\cdot(\phi\nabla\psi) - \nabla\phi \cdot\nabla\psi \,$

and

$\psi\nabla^2\phi = \nabla\cdot(\psi\nabla\phi) - \nabla\psi \cdot\nabla\phi, \,$

we can construct the difference between these two expressions:

$\phi\nabla^2\psi - \psi\nabla^2\phi = \nabla\cdot(\phi\nabla\psi - \psi\nabla\phi)$

By integrating both sides of this expression and applying the the divergence theorem, we get

$\int_V (\phi\nabla^2\psi - \psi\nabla^2\phi)\, dV = \oint_{\partial V} (\phi\nabla\psi - \psi\nabla\phi)\cdot d\textbf{S}.$

This is Green's first identity for scalar functions.

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Green's second theorem for scalar functions

Related to this is Green's second identity:

$\int_V \phi\nabla^2\psi \, dV = \oint_{\partial V}(\phi\nabla\psi)\cdot d\textbf{S} - \int_V\nabla\phi \cdot\nabla\psi\, dV$

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Green's first theorem for vector functions

As in the scalar case, given two vector functions where second derivatives can be defined, consider the following expression:

$\textbf{A}\cdot\nabla\times\nabla\times\textbf{B} - \textbf{B}\cdot\nabla\times\nabla\times\textbf{A} = \nabla\cdot (\textbf{B}\times\nabla\times\textbf{A} - \textbf{A}\times\nabla \times\textbf{B})$

Again, we integrate and apply the the divergence theorem to the right-hand-side to get:

$\int_V(\textbf{A}\cdot\nabla\times\nabla\times\textbf{B} - \textbf{B}\cdot\nabla\times\nabla\times\textbf{A})\, dV = \oint_{\partial V} (\textbf{B}\times\nabla\times\textbf{A} - \textbf{A}\times\nabla \times\textbf{B})\cdot d\textbf{S}$

This is Green's first vector identity.

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Green's second theorem for vectors

The second identity is derived by similar means and is given by:

$\int_V\textbf{A}\cdot\nabla\times\nabla\times\textbf{B}\, dV = \int_V\nabla\times\textbf{A}\cdot\nabla\times\textbf{B}\, dV - \oint_{\partial V}(\textbf{A}\times\nabla \times\textbf{B})\cdot d\textbf{S}$