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.

1 Green's first identity
2 Green's second identity
3 Green's third identity
4 On manifolds
5 See also
6 External links
7 Reference

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^[1]

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

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 $\mathbf{F}=\psi \mathbf{\Gamma}$ :

$\int_U \left( \psi \nabla \cdot \mathbf{\Gamma} %2B \mathbf{\Gamma} \cdot \nabla \psi\right)\, dV = \oint_{\partial U} \psi \left( \mathbf{\Gamma} \cdot \bold{n} \right)\, dS$

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 $\epsilon = 1$ 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 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

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

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

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.

External links

[1] Green's Identities at Wolfram MathWorld

Reference

^ Strauss, Walter. Partial Differential Equations: An Introduction. Wiley.