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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: Let φ and ψ be scalar functions defined on some region U ⊂ R^d, and suppose that φ is twice continuously differentiable, and ψ is once continuously differentiable. Then^[1]

∫U(ψΔφ+∇ψ⋅∇φ)dV=∮∂U⁡ψ(∇φ⋅n)dS=∮∂U⁡ψ∇φ⋅dS{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 dmathbf {S} }

where ∆ is the Laplace operator, ∂U is the boundary of region U, n is the outward pointing unit normal of surface element dS and dS 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 = ψΓ,

∫U(ψ∇⋅Γ+Γ⋅∇ψ)dV=∮∂U⁡ψ(Γ⋅n)dS=∮∂U⁡ψΓ⋅dS .{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 dmathbf {S} ~.}

Green's second identity

If φ and ψ are both twice continuously differentiable on U ⊂ R³, and ε is once continuously differentiable, one may choose F = ψε ∇φ − φε ∇ψ to obtain

∫U[ψ∇⋅(ε∇φ)−φ∇⋅(ε∇ψ)]dV=∮∂U⁡ε(ψ∂φ∂n−φ∂ψ∂n)dS .{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(ψΔφ−φΔψ)dV=∮∂U⁡(ψ∂φ∂n−φ∂ψ∂n)dS.{displaystyle int _{U}left(psi ,Delta varphi -varphi ,Delta 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 normal n to the surface element dS,

∂φ∂n=∇φ⋅n=∇nφ.{displaystyle {partial varphi over partial mathbf {n} }=nabla varphi cdot mathbf {n} =nabla _{mathbf {n} }varphi .}

In particular, this demonstrates that the Laplacian is self-adjoint in the L² inner product for functions vanishing on the boundary.

Green's third identity

Green's third identity derives from the second identity by choosing φ = G, where the Green's function G is taken to be a fundamental solution of the Laplace operator, ∆. This means that:

ΔG(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)=−14π‖x−η‖ .{displaystyle G(mathbf {x} ,{boldsymbol {eta }})={frac {-1}{4pi |mathbf {x} -{boldsymbol {eta }}|}}~.}

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

∫U[G(y,η)Δψ(y)]dVy−ψ(η)=∮∂U⁡[G(y,η)∂ψ∂n(y)−ψ(y)∂G(y,η)∂n]dSy.{displaystyle int _{U}left[G(mathbf {y} ,{boldsymbol {eta }}),Delta psi (mathbf {y} )right],dV_{mathbf {y} }-psi ({boldsymbol {eta }})=oint _{partial U}left[G(mathbf {y} ,{boldsymbol {eta }}){partial psi over partial mathbf {n} }(mathbf {y} )-psi (mathbf {y} ){partial G(mathbf {y} ,{boldsymbol {eta }}) over partial mathbf {n} }right],dS_{mathbf {y} }.}

A simplification arises if ψ is itself a harmonic function, i.e. a solution to the Laplace equation. Then ∇²ψ = 0 and the identity simplifies to

ψ(η)=∮∂U⁡[ψ(y)∂G(y,η)∂n−G(y,η)∂ψ∂n(y)]dSy.{displaystyle psi ({boldsymbol {eta }})=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 }{partial mathbf {n} }}(mathbf {y} )right],dS_{mathbf {y} }.}

The second term in the integral above can be eliminated if G is chosen to be the Green's function for the boundary of the region U where the problem is posed (Dirichlet boundary condition),

ψ(η)=∮∂U⁡ψ(y)∂G(y,η)∂ndSy .{displaystyle psi ({boldsymbol {eta }})=oint _{partial U}psi (mathbf {y} ){frac {partial G(mathbf {y} ,{boldsymbol {eta }})}{partial mathbf {n} }},dS_{mathbf {y} }~.}

This form is used to construct solutions to Dirichlet boundary condition problems. To find solutions for Neumann boundary condition problems, the Green's function with vanishing normal gradient on the boundary is used instead.

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.

On manifolds

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

∫MuΔvdV+∫M⟨∇u,∇v⟩dV=∫∂MuNvdV~∫M(uΔv−vΔu)dV=∫∂M(uNv−vNu)dV~{displaystyle {begin{aligned}int _{M}u,Delta v,dV+int _{M}langle nabla u,nabla vrangle ,dV&=int _{partial M}uNv,d{widetilde {V}}\int _{M}left(u,Delta v-v,Delta uright),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, dV~{displaystyle d{widetilde {V}}} is the induced volume form on the boundary of M, N is oriented unit vector field normal to the boundary, and Δu = div(grad u) is the Laplacian.

Green's vector identity

Green’s second identity establishes a relationship between second and (the divergence of) first order derivatives of two scalar functions. In differential form

pmΔqm−qmΔpm=∇⋅(pm∇qm−qm∇pm),{displaystyle p_{m},Delta q_{m}-q_{m},Delta p_{m}=nabla cdot left(p_{m}nabla q_{m}-q_{m},nabla p_{m}right),}

where p_m and q_m are two arbitrary twice continuously differentiable scalar fields. This identity is of great importance in physics because continuity equations can thus be established for scalar fields such as mass or energy.^[2]

In vector diffraction theory, two versions of Green’s second identity are introduced.

One variant invokes the divergence of a cross product ^[3][4][5] and states a relationship in terms of the curl-curl of the field

P⋅(∇×∇×Q)−Q⋅(∇×∇×P)=∇⋅(Q×(∇×P)−P×(∇×Q)).{displaystyle mathbf {P} cdot left(nabla times nabla times mathbf {Q} right)-mathbf {Q} cdot left(nabla times nabla times mathbf {P} right)=nabla cdot left(mathbf {Q} times left(nabla times mathbf {P} right)-mathbf {P} times left(nabla times mathbf {Q} right)right).}

This equation can be written in terms of the Laplacians,

P⋅ΔQ−Q⋅ΔP+Q⋅[∇(∇⋅P)]−P⋅[∇(∇⋅Q)]=∇⋅(P×(∇×Q)−Q×(∇×P)).{displaystyle mathbf {P} cdot Delta mathbf {Q} -mathbf {Q} cdot Delta mathbf {P} +mathbf {Q} cdot left[nabla left(nabla cdot mathbf {P} right)right]-mathbf {P} cdot left[nabla left(nabla cdot mathbf {Q} right)right]=nabla cdot left(mathbf {P} times left(nabla times mathbf {Q} right)-mathbf {Q} times left(nabla times mathbf {P} right)right).}

However, the terms

Q⋅[∇(∇⋅P)]−P⋅[∇(∇⋅Q)],{displaystyle mathbf {Q} cdot left[nabla left(nabla cdot mathbf {P} right)right]-mathbf {P} cdot left[nabla left(nabla cdot mathbf {Q} right)right],}

could not be readily written in terms of a divergence.

The other approach introduces bi-vectors, this formulation requires a dyadic Green function.^[6][7] The derivation presented here avoids these problems.^[8]

Consider that the scalar fields in Green's second identity are the Cartesian components of vector fields, i.e.

P=∑mpme^m,Q=∑mqme^m.{displaystyle mathbf {P} =sum _{m}p_{m}{hat {mathbf {e} }}_{m},qquad mathbf {Q} =sum _{m}q_{m}{hat {mathbf {e} }}_{m}.}

Summing up the equation for each component, we obtain

∑m[pmΔqm−qmΔpm]=∑m∇⋅(pm∇qm−qm∇pm).{displaystyle sum _{m}left[p_{m}Delta q_{m}-q_{m}Delta p_{m}right]=sum _{m}nabla cdot left(p_{m}nabla q_{m}-q_{m}nabla p_{m}right).}

The LHS according to the definition of the dot product may be written in vector form as

∑m[pmΔqm−qmΔpm]=P⋅ΔQ−Q⋅ΔP.{displaystyle sum _{m}left[p_{m},Delta q_{m}-q_{m},Delta p_{m}right]=mathbf {P} cdot Delta mathbf {Q} -mathbf {Q} cdot Delta mathbf {P} .}

The RHS is a bit more awkward to express in terms of vector operators. Due to the distributivity of the divergence operator over addition, the sum of the divergence is equal to the divergence of the sum, i.e.

∑m∇⋅(pm∇qm−qm∇pm)=∇⋅(∑mpm∇qm−∑mqm∇pm).{displaystyle sum _{m}nabla cdot left(p_{m}nabla q_{m}-q_{m}nabla p_{m}right)=nabla cdot left(sum _{m}p_{m}nabla q_{m}-sum _{m}q_{m}nabla p_{m}right).}

Recall the vector identity for the gradient of a dot product,

∇(P⋅Q)=(P⋅∇)Q+(Q⋅∇)P+P×(∇×Q)+Q×(∇×P),{displaystyle nabla left(mathbf {P} cdot mathbf {Q} right)=left(mathbf {P} cdot nabla right)mathbf {Q} +left(mathbf {Q} cdot nabla right)mathbf {P} +mathbf {P} times left(nabla times mathbf {Q} right)+mathbf {Q} times left(nabla times mathbf {P} right),}

which, written out in vector components is given by

∇(P⋅Q)=∇∑mpmqm=∑mpm∇qm+∑mqm∇pm.{displaystyle nabla left(mathbf {P} cdot mathbf {Q} right)=nabla sum _{m}p_{m}q_{m}=sum _{m}p_{m}nabla q_{m}+sum _{m}q_{m}nabla p_{m}.}

This result is similar to what we wish to evince in vector terms 'except' for the minus sign. Since the differential operators in each term act either over one vector (say pm{displaystyle p_{m}}’s) or the other (qm{displaystyle q_{m}}’s), the contribution to each term must be

∑mpm∇qm=(P⋅∇)Q+P×(∇×Q),{displaystyle sum _{m}p_{m}nabla q_{m}=left(mathbf {P} cdot nabla right)mathbf {Q} +mathbf {P} times left(nabla times mathbf {Q} right),}

∑mqm∇pm=(Q⋅∇)P+Q×(∇×P).{displaystyle sum _{m}q_{m}nabla p_{m}=left(mathbf {Q} cdot nabla right)mathbf {P} +mathbf {Q} times left(nabla times mathbf {P} right).}

These results can be rigorously proven to be correct through evaluation of the vector components. Therefore, the RHS can be written in vector form as

∑mpm∇qm−∑mqm∇pm=(P⋅∇)Q+P×(∇×Q)−(Q⋅∇)P−Q×(∇×P).{displaystyle sum _{m}p_{m}nabla q_{m}-sum _{m}q_{m}nabla p_{m}=left(mathbf {P} cdot nabla right)mathbf {Q} +mathbf {P} times left(nabla times mathbf {Q} right)-left(mathbf {Q} cdot nabla right)mathbf {P} -mathbf {Q} times left(nabla times mathbf {P} right).}

Putting together these two results, a result analogous to Green’s theorem for scalar fields is obtained,

Theorem for vector fields.

P⋅ΔQ−Q⋅ΔP=∇⋅[(P⋅∇)Q+P×(∇×Q)−(Q⋅∇)P−Q×(∇×P)].{displaystyle color {OliveGreen}mathbf {P} cdot Delta mathbf {Q} -mathbf {Q} cdot Delta mathbf {P} =nabla cdot left[left(mathbf {P} cdot nabla right)mathbf {Q} +mathbf {P} times left(nabla times mathbf {Q} right)-left(mathbf {Q} cdot nabla right)mathbf {P} -mathbf {Q} times left(nabla times mathbf {P} right)right].}

The curl of a cross product can be written as

∇×(P×Q)=(Q⋅∇)P−(P⋅∇)Q+P(∇⋅Q)−Q(∇⋅P);{displaystyle nabla times left(mathbf {P} times mathbf {Q} right)=left(mathbf {Q} cdot nabla right)mathbf {P} -left(mathbf {P} cdot nabla right)mathbf {Q} +mathbf {P} left(nabla cdot mathbf {Q} right)-mathbf {Q} left(nabla cdot mathbf {P} right);}

Green’s vector identity can then be rewritten as

P⋅ΔQ−Q⋅ΔP=∇⋅[P(∇⋅Q)−Q(∇⋅P)−∇×(P×Q)+P×(∇×Q)−Q×(∇×P)].{displaystyle mathbf {P} cdot Delta mathbf {Q} -mathbf {Q} cdot Delta mathbf {P} =nabla cdot left[mathbf {P} left(nabla cdot mathbf {Q} right)-mathbf {Q} left(nabla cdot mathbf {P} right)-nabla times left(mathbf {P} times mathbf {Q} right)+mathbf {P} times left(nabla times mathbf {Q} right)-mathbf {Q} times left(nabla times mathbf {P} right)right].}

Since the divergence of a curl is zero, the third term vanishes to yield

Green's vector identity.

P⋅ΔQ−Q⋅ΔP=∇⋅[P(∇⋅Q)−Q(∇⋅P)+P×(∇×Q)−Q×(∇×P)].{displaystyle color {OliveGreen}mathbf {P} cdot Delta mathbf {Q} -mathbf {Q} cdot Delta mathbf {P} =nabla cdot left[mathbf {P} left(nabla cdot mathbf {Q} right)-mathbf {Q} left(nabla cdot mathbf {P} right)+mathbf {P} times left(nabla times mathbf {Q} right)-mathbf {Q} times left(nabla times mathbf {P} right)right].}

With a similar procedure, the Laplacian of the dot product can be expressed in terms of the Laplacians of the factors

Δ(P⋅Q)=P⋅ΔQ−Q⋅ΔP+2∇⋅[(Q⋅∇)P+Q×∇×P].{displaystyle Delta left(mathbf {P} cdot mathbf {Q} right)=mathbf {P} cdot Delta mathbf {Q} -mathbf {Q} cdot Delta mathbf {P} +2nabla cdot left[left(mathbf {Q} cdot nabla right)mathbf {P} +mathbf {Q} times nabla times mathbf {P} right].}

As a corollary, the awkward terms can now be written in terms of a divergence by comparison with the vector Green equation,

P⋅[∇(∇⋅Q)]−Q⋅[∇(∇⋅P)]=∇⋅[P(∇⋅Q)−Q(∇⋅P)].{displaystyle mathbf {P} cdot left[nabla left(nabla cdot mathbf {Q} right)right]-mathbf {Q} cdot left[nabla left(nabla cdot mathbf {P} right)right]=nabla cdot left[mathbf {P} left(nabla cdot mathbf {Q} right)-mathbf {Q} left(nabla cdot mathbf {P} right)right].}

This result can be verified by expanding the divergence of a scalar times a vector on the RHS.

See also

• Green's function


• Kirchhoff integral theorem

References

External links

• 
  Hazewinkel, Michiel, ed. (2001) [1994], "Green formulas", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  


• 
  [2] Green's Identities at Wolfram MathWorld