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 $mathbf\{F\}=psi\; abla\; varphi$: Let on some region "U" in R³ be given functions φ and ψ such that φ is twice continuously differentiable, and ψ is once continuously differentiable, then

:

where $abla^2=\; riangle$ 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, then

: $int\_U\; left(\; psi\; abla^2\; varphi\; -\; varphi\; abla^2\; psi\; ight),\; dV\; =\; oint\_\{partial\; U\}\; left(\; psi\; \{partial\; varphi\; over\; partial\; n\}\; -\; varphi\; \{partial\; psi\; over\; partial\; n\}\; ight),\; dS.$

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":

: $\{partial\; varphi\; over\; partial\; n\}\; =\; abla\; varphi\; cdot\; mathbf\{n\}.$

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:

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

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

: $int\_U\; left\; [\; G(mathbf\{y\},eta)\; abla^2\; psi(mathbf\{y\})\; ight]\; ,\; dV\_mathbf\{y\}\; -\; psi(eta)=\; oint\_\{partial\; U\}\; left\; [\; G(mathbf\{y\},eta)\; \{partial\; psi\; over\; partial\; n\}\; (mathbf\{y\})\; -\; psi(mathbf\{y\})\; \{partial\; G(mathbf\{y\},eta)\; over\; partial\; n\}\; ight]\; ,\; dS\_mathbf\{y\}.$

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

: $psi(eta)=\; oint\_\{partial\; U\}\; left\; [psi(mathbf\{y\})\; \{partial\; G(mathbf\{y\},eta)\; over\; partial\; n\}\; -\; G(mathbf\{y\},eta)\; \{partial\; psi\; over\; partial\; n\}\; (mathbf\{y\})\; ight]\; ,\; dS\_mathbf\{y\}.$

ee also

* Green's functions