Green's identities

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.

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

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

where $\nabla^{2}$ is the Laplace operator, ${\partial U}$ is the boundary of region U and n is the outward pointing unit normal of surface element dS. This theorem is essentially the higher dimensional equivalent of integration by parts with ψ and the gradient of φ replacing u and v.

Note that Green's first identity above is a special case of the more general identity derived from the divergence theorem by substituting $\mathbf{F}=\psi \mathbf{\Gamma}$:

$\int_U \left( \psi \nabla \cdot \mathbf{\Gamma} + \mathbf{\Gamma} \cdot \nabla \psi\right)\, dV = \oint_{\partial U} \psi \left( \mathbf{\Gamma} \cdot \bold{n} \right)\, dS$

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 φ = G, where G is a fundamental solution of the Laplace equation. This means that:

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

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

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

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

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

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:

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

On manifolds

Green's identities hold on a Riemannian manifold, In this setting, the first two are

$\int_M u\nabla^{2} v\, dV+\int_M\langle\operatorname{grad}\ u, \operatorname{grad}\ v\rangle\, dV = \int_{\partial M} u N v d\tilde{V}$

$\int_M(u\nabla^{2} v - v \nabla^{2} u)\, dV = \int_{\partial M}(u N v - v N u)d\tilde{V}$

where u and v are smooth real-valued functions on M, dV is the volume form compatible with the metric, $d\tilde{V}$ is the induced volume form on the boundary of M, N is oriented unit vector field normal to the boundary, and $\nabla^{2} u := \operatorname{div}(\operatorname{grad}\ u)$ is the Laplacian.

See also

• Green's function

External links

• [1] Green's Identities at Wolfram MathWorld