Green's identities: Definition from Answers.com

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In mathematics, Green's identities are a set of three identities in vector calculus. They are named after the mathematician George Green, who discovered Green's theorem.

1 Green's first identity
2 Green's second identity
3 Green's third identity
4 See also

Green's first identity

This identity is derived from the divergence theorem applied to the vector field $\mathbf{F}=\psi \nabla \varphi$ : Let φ and ψ be scalar functions defined on some region U in R³, and suppose that φ is twice continuously differentiable, and ψ is once continuously differentiable. Then

$\int_U \left( \psi \nabla^2 \varphi\right)\, dV = \oint_{\partial U} \psi \left( \nabla \varphi \cdot \bold{n} \right)\, dS - \int_U \left( \nabla \varphi \cdot \nabla \psi\right)\, dV,$

where $\nabla^2=\Delta$ is the Laplace operator, ${\partial U}$ is the boundary of region U and n is the outward pointing unit normal of surface element dS.

Green's second identity

If φ and ψ are both twice continuously differentiable on U in R³, and ε is once continuously differentiable:

$\int_U \left[ \psi \nabla \cdot \left( \epsilon \nabla \varphi \right) - \varphi \nabla \cdot \left( \epsilon \nabla \psi \right) \right]\, dV = \oint_{\partial U} \epsilon \left( \psi {\partial \varphi \over \partial n} - \varphi {\partial \psi \over \partial n}\right)\, dS.$

For the special case of $ε = 1$ all across U in R³ then:

$\int_U \left( \psi \nabla^2 \varphi - \varphi \nabla^2 \psi\right)\, dV = \oint_{\partial U} \left( \psi {\partial \varphi \over \partial n} - \varphi {\partial \psi \over \partial n}\right)\, dS.$

In the equation above ∂φ / ∂n is the directional derivative of φ in the direction of the outward pointing normal n to the surface element dS:

${\partial \varphi \over \partial n} = \nabla \varphi \cdot \mathbf{n}.$

Green's third identity

Green's third identity derives from the second identity by choosing $\varphi=G$ , where G is a fundamental solution, or Green's function, of the Laplace equation. This means that:

$\nabla^2 G(\mathbf{x},\eta) = \delta(\mathbf{x} - \eta).$

For example in $\mathbb{R}^3$ , the fundamental solution has the form:

$G(\mathbf{x},\eta)={-1 \over 4 \pi\|\mathbf{x} - \eta \|}.$

Green's third identity states that if ψ is a function that is twice continuously differentiable on U, then

$\int_U \left[ G(\mathbf{y},\eta) \nabla^2 \psi(\mathbf{y})\right]\, dV_\mathbf{y} - \psi(\eta)= \oint_{\partial U} \left[ G(\mathbf{y},\eta) {\partial \psi \over \partial n} (\mathbf{y}) - \psi(\mathbf{y}) {\partial G(\mathbf{y},\eta) \over \partial n} \right]\, dS_\mathbf{y}.$

A further simplification arises if ψ is itself a harmonic function, i.e. a solution to the Laplace equation. Then $\nabla^2\psi = 0$ and the identity simplifies to:

$\psi(\eta)= \oint_{\partial U} \left[\psi(\mathbf{y}) {\partial G(\mathbf{y},\eta) \over \partial n} - G(\mathbf{y},\eta) {\partial \psi \over \partial n} (\mathbf{y}) \right]\, dS_\mathbf{y}.$

	Green formula
	List of multivariable calculus topics
	Green's theorem

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Green's identities

Contents

Green's first identity

Green's second identity

Green's third identity

See also