In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act. 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 F = ψ ∇ φ {\displaystyle \mathbf {F} =\psi \nabla \varphi } while using an extension of the product rule that ∇ ⋅ ( ψ X ) = ∇ ψ ⋅ X + ψ ∇ ⋅ X {\displaystyle \nabla \cdot \left(\psi \mathbf {X} \right)=\nabla \psi \cdot \mathbf {X} +\psi \nabla \cdot \mathbf {X} } , where ψ {\displaystyle \psi } and φ {\displaystyle \varphi } are scalar functions and X {\displaystyle \mathbf {X} } is a vector field. Let ψ {\displaystyle \psi } and φ {\displaystyle \varphi } be scalar functions defined on some region U ⊂ Rd, and suppose that ψ {\displaystyle \psi } is once continuously differentiable and φ {\displaystyle \varphi } is twice continuously differentiable. Using the product rule above with letting X = ∇φ, integration of ∇ ⋅ F = ∇ ⋅ ( ψ ∇ φ ) {\displaystyle \nabla \cdot \mathbf {F} =\nabla \cdot \left(\psi \nabla \varphi \right)} over U results in
∫ U ( ψ Δ φ + ∇ ψ ⋅ ∇ φ ) d V = ∮ ∂ U ψ ( ∇ φ ⋅ n ) d S = ∮ ∂ U ψ ∇ φ ⋅ d S {\displaystyle \int _{U}\left(\psi \,\Delta \varphi +\nabla \psi \cdot \nabla \varphi \right)\,dV=\oint _{\partial U}\psi \left(\nabla \varphi \cdot \mathbf {n} \right)\,dS=\oint _{\partial U}\psi \,\nabla \varphi \cdot d\mathbf {S} }
where Δ {\displaystyle \Delta } (or ∇ 2 {\displaystyle \nabla ^{2}} ) in Δ f = ∇ 2 f = ∇ ⋅ ∇ f {\displaystyle \Delta f=\nabla ^{2}f=\nabla \cdot \nabla f} is the Laplace operator, ∂U is the boundary of region U, n is the outward pointing unit normal to the surface element dS, and dS = ndS is the oriented surface element. This theorem is a special case of the divergence theorem, and 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 F = ψ Γ {\displaystyle \mathbf {F} =\psi \mathbf {\Gamma } } where ψ {\displaystyle \psi } is a scalar function and Γ {\displaystyle \mathbf {\Gamma } } is a vector field (in the above, Γ {\displaystyle \mathbf {\Gamma } } is specified as ∇ φ {\displaystyle \nabla \varphi } ),
∫ U ( ψ ∇ ⋅ Γ + Γ ⋅ ∇ ψ ) d V = ∮ ∂ U ψ ( Γ ⋅ n ) d S = ∮ ∂ U ψ Γ ⋅ d S . {\displaystyle \int _{U}\left(\psi \,\nabla \cdot \mathbf {\Gamma } +\mathbf {\Gamma } \cdot \nabla \psi \right)\,dV=\oint _{\partial U}\psi \left(\mathbf {\Gamma } \cdot \mathbf {n} \right)\,dS=\oint _{\partial U}\psi \mathbf {\Gamma } \cdot d\mathbf {S} ~.}
Green's second identity If both φ {\displaystyle \varphi } and ψ {\displaystyle \psi } are twice continuously differentiable scalar functions on U ⊂ R3, and ε is a once continuously differentiable scalar function, then one may choose F = ψ ε ∇ φ − φ ε ∇ ψ {\displaystyle \mathbf {F} =\psi \varepsilon \nabla \varphi -\varphi \varepsilon \nabla \psi } in the divergence theorem where
∇ ⋅ F = ∇ ⋅ ( ψ ε ∇ φ − φ ε ∇ ψ ) = ∇ ( ψ ε ) ⋅ ∇ φ + ψ ε ∇ 2 φ − ∇ ( φ ε ) ⋅ ∇ ψ − φ ε ∇ 2 ψ = ( ε ∇ ψ + ψ ∇ ε ) ⋅ ∇ φ + ψ ε ∇ 2 φ − ( ε ∇ φ + φ ∇ ε ) ⋅ ∇ ψ − φ ε ∇ 2 ψ = ψ ∇ ε ⋅ ∇ φ + ψ ε ∇ 2 φ − φ ∇ ε ⋅ ∇ ψ − φ ε ∇ 2 ψ = ψ ∇ ⋅ ( ε ∇ φ ) − φ ∇ ⋅ ( ε ∇ ψ ) {\displaystyle {\begin{array}{lcl}\nabla \cdot \mathbf {F} &=&\nabla \cdot \left(\psi \varepsilon \nabla \varphi -\varphi \varepsilon \nabla \psi \right)\\&=&\nabla \left(\psi \varepsilon \right)\cdot \nabla \varphi +\psi \varepsilon \nabla ^{2}\varphi -\nabla \left(\varphi \varepsilon \right)\cdot \nabla \psi -\varphi \varepsilon \nabla ^{2}\psi \\&=&\left(\varepsilon \nabla \psi +\psi \nabla \varepsilon \right)\cdot \nabla \varphi +\psi \varepsilon \nabla ^{2}\varphi -\left(\varepsilon \nabla \varphi +\varphi \nabla \varepsilon \right)\cdot \nabla \psi -\varphi \varepsilon \nabla ^{2}\psi \\&=&\psi \nabla \varepsilon \cdot \nabla \varphi +\psi \varepsilon \nabla ^{2}\varphi -\varphi \nabla \varepsilon \cdot \nabla \psi -\varphi \varepsilon \nabla ^{2}\psi \\&=&\psi \nabla \cdot \left(\varepsilon \nabla \varphi \right)-\varphi \nabla \cdot \left(\varepsilon \nabla \psi \right)\end{array}}}
is obtained with help from vector calculus identities. As a result,
∫ U [ ψ ∇ ⋅ ( ε ∇ φ ) − φ ∇ ⋅ ( ε ∇ ψ ) ] d V = ∮ ∂ U ε ( ψ ∂ φ ∂ n − φ ∂ ψ ∂ n ) d S . {\displaystyle \int _{U}\left[\psi \,\nabla \cdot \left(\varepsilon \,\nabla \varphi \right)-\varphi \,\nabla \cdot \left(\varepsilon \,\nabla \psi \right)\right]\,dV=\oint _{\partial U}\varepsilon \left(\psi {\partial \varphi \over \partial \mathbf {n} }-\varphi {\partial \psi \over \partial \mathbf {n} }\right)\,dS.}
For the special case of ε = 1 all across U ⊂ R3, then,
∫ U ( ψ ∇ 2 φ − φ ∇ 2 ψ ) d V = ∮ ∂ U ( ψ ∂ φ ∂ n − φ ∂ ψ ∂ n ) d S . {\displaystyle \int _{U}\left(\psi \,\nabla ^{2}\varphi -\varphi \,\nabla ^{2}\psi \right)\,dV=\oint _{\partial U}\left(\psi {\partial \varphi \over \partial \mathbf {n} }-\varphi {\partial \psi \over \partial \mathbf {n} }\right)\,dS.}
In the equation above, ∂φ/∂n is the directional derivative of φ in the direction of the outward pointing surface normal n of the surface element dS,
∂ φ ∂ n = ∇ φ ⋅ n = ∇ n φ . {\displaystyle {\partial \varphi \over \partial \mathbf {n} }=\nabla \varphi \cdot \mathbf {n} =\nabla _{\mathbf {n} }\varphi .}
Explicitly incorporating this definition in the Green's second identity with ε = 1 results in
∫ U ( ψ Δ φ − φ Δ ψ ) d V = ∮ ∂ U ( ψ ∇ φ − φ ∇ ψ ) ⋅ d S . {\displaystyle \int _{U}\left(\psi \,\Delta \varphi -\varphi \,\Delta \psi \right)\,dV=\oint _{\partial U}\left(\psi \nabla \varphi -\varphi \nabla \psi \right)\cdot d\mathbf {S} .}
Where Δ = ∇ 2 {\displaystyle \Delta =\nabla ^{2}} is the Laplacian. In particular, this demonstrates that the Laplacian is a self-adjoint operator in the L2 inner product for functions vanishing on the boundary so that the right hand side of the above identity is zero.
Green's third identity Green's third identity derives from the second identity by choosing φ = G, where Green's function G is taken to be a fundamental solution of the Laplace operator, ∆. This means that:
Δ G ( x , η ) = δ ( x − η ) . {\displaystyle \Delta G(\mathbf {x} ,{\boldsymbol {\eta }})=\delta (\mathbf {x} -{\boldsymbol {\eta }})~.}
For example, in R3, a solution has the form
G ( x , η ) = − 1 4 π ‖ x − η ‖ . {\displaystyle G(\mathbf {x} ,{\boldsymbol {\eta }})={\frac {-1}{4\pi \|\mathbf {x} -{\boldsymbol {\eta }}\|}}~.}
Green's third identity states that if ψ is a function that is twice continuously differentiable on U, then
∫ U [ δ ( y − η ) ψ ( η ) − G ( y , η ) Δ ψ ( y ) ] d V y = ∮ ∂ U [ ψ ( y ) ∂ G ( y , η ) ∂ n − G ( y , η ) ∂ ψ ( y ) ∂ n ] d S y . {\displaystyle \int _{U}\left[\delta (\mathbf {y} -{\boldsymbol {\eta }})\psi ({\boldsymbol {\eta }})-G(\mathbf {y} ,{\boldsymbol {\eta }})\,\Delta \psi (\mathbf {y} )\right]\,dV_{\mathbf {y} }=\oint _{\partial U}\left[\psi (\mathbf {y} ){\partial G(\mathbf {y} ,{\boldsymbol {\eta }}) \over \partial \mathbf {n} }-G(\mathbf {y} ,{\boldsymbol {\eta }}){\partial \psi (\mathbf {y} ) \over \partial \mathbf {n} }\right]\,dS_{\mathbf {y} }.}
Note that the integral over δ ( y − η ) ψ ( η ) {\displaystyle \delta (\mathbf {y} -{\boldsymbol {\eta }})\psi ({\boldsymbol {\eta }})} reproduces ψ {\displaystyle \psi } when η ∈ U {\displaystyle {\boldsymbol {\eta }}\in U} and gives 0 otherwise. A simplification arises if ψ is itself a harmonic function, i.e. a solution to the Laplace equation in U {\displaystyle U} . Then Δ ψ = 0 {\displaystyle \Delta \psi =0} and the identity simplifies to
∮ ∂ U [ ψ ( y ) ∂ G ( y , η ) ∂ n − G ( y , η ) ∂ ψ ( y ) ∂ n ] d S y = { ψ ( η ) η ∈ U 0 η ∉ U . {\displaystyle \oint _{\partial U}\left[\psi (\mathbf {y} ){\frac {\partial G(\mathbf {y} ,{\boldsymbol {\eta }})}{\partial \mathbf {n} }}-G(\mathbf {y} ,{\boldsymbol {\eta }}){\frac {\partial \psi (\mathbf {y} )}{\partial \mathbf {n} }}\right]\,dS_{\mathbf {y} }={\begin{cases}\psi ({\boldsymbol {\eta }})\qquad &{\boldsymbol {\eta }}\in U\\0&{\boldsymbol {\eta }}\notin U\end{cases}}.}
The second term in the integral above can be eliminated if G is chosen to be Green's function that vanishes on the boundary of U (Dirichlet boundary condition),
∮ ∂ U ψ ( y ) ∂ G ( y , η ) ∂ n d S y = { ψ ( η ) η ∈ U 0 η ∉ U . {\displaystyle \oint _{\partial U}\psi (\mathbf {y} ){\frac {\partial G(\mathbf {y} ,{\boldsymbol {\eta }})}{\partial \mathbf {n} }}\,dS_{\mathbf {y} }={\begin{cases}\psi ({\boldsymbol {\eta }})\qquad &{\boldsymbol {\eta }}\in U\\0&{\boldsymbol {\eta }}\notin U\end{cases}}.}
This form is used to construct solutions to Dirichlet boundary condition problems. Solutions for Neumann boundary condition problems may also be simplified, though the Divergence theorem applied to the differential equation defining Green's functions shows that the Green's function cannot integrate to zero on the boundary, and hence cannot vanish on the boundary. See Green's functions for the Laplacian or for a detailed argument, with an alternative. For the Neumann boundary condition, an appropriate choice of Green's function can be made to simplify the integral. First note
∫ U Δ G ( y , η ) d V y = 1 = ∮ ∂ U ∂ G ( y , η ) ∂ n d S y η ∈ U . {\displaystyle \int _{U}\Delta G(\mathbf {y} ,{\boldsymbol {\eta }})dV_{\mathbf {y} }=1=\oint _{\partial U}{\frac {\partial G(\mathbf {y} ,{\boldsymbol {\eta }})}{\partial \mathbf {n} }}dS_{\mathbf {y} }~\qquad \qquad {\boldsymbol {\eta }}\in U.} and so ∂ G ( y , η ) ∂ n {\displaystyle {\frac {\partial G(\mathbf {y} ,{\boldsymbol {\eta }})}{\partial \mathbf {n} }}} cannot vanish on surface S {\displaystyle S} . A convenient choice is ∂ G ( y , η ) ∂ n = 1 A {\displaystyle {\frac {\partial G(\mathbf {y} ,{\boldsymbol {\eta }})}{\partial \mathbf {n} }}={\frac {1}{\mathcal {A}}}} , where A {\displaystyle {\mathcal {A}}} is the area of the surface S {\displaystyle S} . The integral can be simplified to
ψ ( η ) = ⟨ ψ ⟩ S − ∮ ∂ U G ( y , η ) ∂ ψ ( y ) ∂ n d S y . {\displaystyle \psi ({\boldsymbol {\eta }})=\langle \psi \rangle _{S}-\oint _{\partial U}G(\mathbf {y} ,{\boldsymbol {\eta }}){\frac {\partial \psi (\mathbf {y} )}{\partial \mathbf {n} }}\,dS_{\mathbf {y} }.}
where ⟨ ψ ⟩ S = 1 A ∮ ∂ U ψ ( y ) d S y {\displaystyle \langle \psi \rangle _{S}={\frac {1}{\mathcal {A}}}\oint _{\partial U}\psi (\mathbf {y} )dS_{\mathbf {y} }} is the average value of ψ {\displaystyle \psi } on surface S {\displaystyle S} . Furthermore, if ψ {\displaystyle \psi } is a solution to the Laplace's equation, divergence theorem implies it must satisfy ∮ ∂ U ∂ ψ ( y ) ∂ n d S y = ∫ U Δ ψ ( y ) d V y = 0 {\displaystyle \oint _{\partial U}{\frac {\partial \psi (\mathbf {y} )}{\partial \mathbf {n} }}dS_{\mathbf {y} }=\int _{U}\Delta \psi (\mathbf {y} )dV_{\mathbf {y} }=0} . This is a necessary condition for the Neumann boundary problem to have a solution. It can be further verified that the above identity also applies when ψ is a solution to the Helmholtz equation or wave equation and G is the appropriate Green's function. In such a context, this identity is the mathematical expression of the Huygens principle, and leads to Kirchhoff's diffraction formula and other approximations.
On manifolds Green's identities hold on a Riemannian manifold. In this setting, the first two are
∫ M u Δ v d V + ∫ M ⟨ ∇ u , ∇ v ⟩ d V = ∫ ∂ M u N v d V ~ ∫ M ( u Δ v − v Δ u ) d V = ∫ ∂ M ( u N v − v N u ) d V ~ {\displaystyle {\begin{aligned}\int _{M}u\,\Delta v\,dV+\int _{M}\langle \nabla u,\nabla v\rangle \,dV&=\int _{\partial M}uNv\,d{\widetilde {V}}\\\int _{M}\left(u\,\Delta v-v\,\Delta u\right)\,dV&=\int _{\partial M}(uNv-vNu)\,d{\widetilde {V}}\end{aligned}}}
where u and v are smooth real-valued functions on M, dV is the volume form compatible with the metric, d V ~ {\displaystyle d{\widetilde {V}}} is the induced volume form on the boundary of M, N is the outward oriented unit vector field normal to the boundary, and Δu = div(grad u) is the Laplacian.
Green's vector identities
First vector identity Using the vector Laplacian identity and the divergence identity, expand P ⋅ Δ Q {\displaystyle \mathbf {P} \cdot \Delta \mathbf {Q} }
P ⋅ Δ Q = ∇ ⋅ ( P × ∇ × Q ) − ( ∇ × P ) ⋅ ( ∇ × Q ) + P ⋅ [ ∇ ( ∇ ⋅ Q ) ] {\displaystyle \mathbf {P} \cdot \Delta \mathbf {Q} =\nabla \cdot (\mathbf {P} \times \nabla \times \mathbf {Q} )-(\nabla \times \mathbf {P} )\cdot (\nabla \times \mathbf {Q} )+\mathbf {P} \cdot [\nabla (\nabla \cdot \mathbf {Q} )]}
The last term can be simplified by expanding components
P ⋅ [ ∇ ( ∇ ⋅ Q ) ] = P i [ ∇ i ( ∇ j Q j ) ] = ∇ i [ P i ( ∇ j Q j ) ] − ( ∇ i P i ) ( ∇ j Q j ) = ∇ ⋅ [ P ( ∇ ⋅ Q ) ] − ( ∇ ⋅ P ) ( ∇ ⋅ Q ) {\displaystyle {\begin{aligned}\mathbf {P} \cdot [\nabla (\nabla \cdot \mathbf {Q} )]&=P^{i}[\nabla _{i}(\nabla _{j}Q^{j})]\\&=\nabla _{i}[P^{i}(\nabla _{j}Q^{j})]-(\nabla _{i}P^{i})(\nabla _{j}Q^{j})\\&=\nabla \cdot [\mathbf {P} (\nabla \cdot \mathbf {Q} )]-(\nabla \cdot \mathbf {P} )(\nabla \cdot \mathbf {Q} )\end{aligned}}}
The identity can be rewritten as
P ⋅ Δ Q = ∇ ⋅ ( P × ∇ × Q ) − ( ∇ × P ) ⋅ ( ∇ × Q ) + ∇ ⋅ [ P ( ∇ ⋅ Q ) ] − ( ∇ ⋅ P ) ( ∇ ⋅ Q ) {\displaystyle \mathbf {P} \cdot \Delta \mathbf