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| 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]].
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| ==Green's first identity==
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| This identity is derived from the [[divergence theorem]] applied to the vector field <math>\mathbf{F}=\psi \nabla \varphi </math>: Let φ and ψ be scalar functions defined on some region ''U'' in '''R'''<sup>3</sup>, and suppose that φ is twice [[continuously differentiable]], and ψ is once continuously differentiable. Then<ref name="strauss">{{cite book|last=Strauss|first=Walter|title=Partial Differential Equations: An Introduction|publisher=Wiley}}</ref>
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| : <math>\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 </math>
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| where <math>\nabla^{2}</math> is the [[Laplace operator]], <math>{\partial U}</math> 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.
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| Note that Green's first identity above is a special case of the more general identity derived from the [[divergence theorem]] by substituting <math>\mathbf{F}=\psi \mathbf{\Gamma}</math>:
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| : <math>\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. </math>
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| ==Green's second identity==
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| If φ and ψ are both twice continuously differentiable on ''U'' in '''R'''<sup>3</sup>, and ε is once continuously differentiable, we can choose <math>\mathbf{F}=\psi \epsilon \nabla \varphi - \varphi \epsilon \nabla \psi</math> and obtain:
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| : <math> \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. </math>
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| For the special case of <math> \epsilon = 1 </math> all across ''U'' in '''R'''<sup>3</sup> then:
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| : <math> \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. </math>
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| 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'':
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| : <math> {\partial \varphi \over \partial n} = \nabla \varphi \cdot \mathbf{n}.</math>
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| ==Green's third identity==
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| Green's third identity derives from the second identity by choosing <math>\varphi=G</math>, where G is a [[Green's function]] of the [[Laplace operator]]. This means that:
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| :<math> \nabla^2 G(\mathbf{x},\mathbf{\eta}) = \delta(\mathbf{x} - \mathbf{\eta}).</math>
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| For example in <math>\mathbb{R}^3</math>, a solution has the form:
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| :<math>G(\mathbf{x},\mathbf{\eta})={-1 \over 4 \pi\|\mathbf{x} - \mathbf{\eta} \|}.</math>
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| Green's third identity states that if ψ is a function that is twice continuously differentiable on ''U'', then
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| : <math> \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}.</math>
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| A simplification arises if ψ is itself a [[harmonic function]], i.e. a solution to the [[Laplace equation]]. Then <math>\nabla^2\psi = 0</math> and the identity simplifies to:
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| : <math> \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}.</math>
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| The second term in the integral above can be eliminated if we choose G to be the Green's function that vanishes on the boundary of region ''U'' ([[Dirichlet boundary condition]]):
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| : <math> \psi(\mathbf{\eta})= \oint_{\partial U} \psi(\mathbf{y}) {\partial G(\mathbf{y},\mathbf{\eta}) \over \partial n} \, dS_\mathbf{y}.</math>
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| 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.
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| 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]].
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| ==On manifolds==
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| Green's identities hold on a Riemannian manifold, In this setting, the first two are
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| :<math>\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}</math>
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| :<math>\int_M(u\nabla^{2} v - v \nabla^{2} u)\, dV = \int_{\partial M}(u N v - v N u)d\tilde{V}</math>
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| where u and v are smooth real-valued functions on M, dV is the volume form compatible with the metric, <math>d\tilde{V}</math> is the induced volume form on the boundary of M, N is oriented unit vector field normal to the boundary, and <math>\nabla^{2} u := \operatorname{div}(\operatorname{grad}\ u)</math> is the Laplacian.
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| ==Green's vector identity==
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| Green’s second identity establishes a relationship between second and (the divergence of) first order derivatives of two scalar functions. In differential form
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| <center><math>p_{m}\nabla^{2}q_{m}-q_{m}\nabla^{2}p_{m}=\nabla\cdot\left(p_{m}\nabla q_{m}-q_{m}\nabla p_{m}\right),</math></center>
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| where <math>p_{m}</math> and <math>q_{m}</math> 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.<ref>M. Fernández-Guasti. Complementary fields conservation equation derived from the scalar wave equation. ''J. Phys. A: Math. Gen.'', 37:4107–4121, 2004.</ref> Although the second Green’s identity is always presented in vector analysis, only a scalar version is found on textbooks. Even in the specialized literature, a vector version is not easily found. In vector diffraction theory, two versions of Green’s second identity are introduced. One variant invokes the divergence of a cross product <ref>A. E. H. Love. The Integration of the Equations of Propagation of Electric Waves. ''Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character'', 197:pp. 1–45, 1901.</ref><ref>J. A. Stratton and L. J. Chu. Diffraction Theory of Electromagnetic Waves. ''Phys. Rev.'', 56(1):99–107, Jul 1939.</ref><ref>N. C. Bruce. Double scatter vector-wave Kirchhoff scattering from perfectly conducting surfaces with infinite slopes. ''Journal of Optics'', 12(8):085701, 2010.</ref> and states a relationship in terms of the curl-curl of the field <math>\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\nabla\times\mathbf{P}-\mathbf{P}\times\nabla\times\mathbf{Q}\right)</math>. This equation can be written in terms of the Laplacians:
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| <math>\mathbf{P}\cdot\nabla^{2}\mathbf{Q}-\mathbf{Q}\cdot\nabla^{2}\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\nabla\times\mathbf{Q}-\mathbf{Q}\times\nabla\times\mathbf{P}\right).</math>
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| However, the terms <math>\mathbf{Q}\cdot\left[\nabla\left(\nabla\cdot\mathbf{P}\right)\right]-\mathbf{P}\cdot\left[\nabla\left(\nabla\cdot\mathbf{Q}\right)\right]</math>, could not be readily written in terms of a divergence. The other approach introduces bi-vectors, this formulation requires a dyadic Green function.<ref>W. Franz, On the Theory of Diffraction. ''Proceedings of the Physical Society. Section A'', 63(9):925, 1950.</ref><ref>Chen-To Tai. Kirchhoff theory: Scalar, vector, or dyadic? ''Antennas and Propagation, IEEE Transactions on'', 20(1):114–115, jan 1972.</ref> The derivation presented here avoids these problems.<ref>M. Fernández-Guasti. Green's second identity for vector fields. ISRN Mathematical Physics, 2012:7, 2012. Article ID: 973968. [http://www.isrn.com/journals/mp/2012/973968]</ref>
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| Consider that the scalar fields in Green's second identity are the Cartesian components of vector fields, i.e. <math>\mathbf{P}=\sum\limits _{m}p_{m}\hat{\mathbf{e}}_{m}</math> and <math>\mathbf{Q}=\sum\limits _{m}q_{m}\hat{\mathbf{e}}_{m}</math>. Summing up the equation for each component, we obtain
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| <center><math>\sum\limits _{m}\left[p_{m}\nabla^{2}q_{m}-q_{m}\nabla^{2}p_{m}\right]=\sum\limits _{m}\left[\nabla\cdot\left(p_{m}\nabla q_{m}-q_{m}\nabla p_{m}\right)\right].</math></center>
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| The LHS according to the definition of the dot product may be written in vector form as
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| <center><math>\sum\limits _{m}\left[p_{m}\nabla^{2}q_{m}-q_{m}\nabla^{2}p_{m}\right]=\mathbf{P}\cdot\nabla^{2}\mathbf{Q}-\mathbf{Q}\cdot\nabla^{2}\mathbf{P}.</math></center>
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| 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. <math>\sum\limits _{m}\left[\nabla\cdot\left(p_{m}\nabla q_{m}-q_{m}\nabla p_{m}\right)\right]=\nabla\cdot\left(\sum\limits _{m}p_{m}\nabla q_{m}-\sum\limits _{m}q_{m}\nabla p_{m}\right)</math>. Recall the vector identity for the gradient of a dot product
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| <math>\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\nabla\times\mathbf{Q}+\mathbf{Q}\times\nabla\times\mathbf{P}</math>,
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| which, written out in vector components is given by <math>\nabla\left(\mathbf{P}\cdot\mathbf{Q}\right)=\nabla\sum\limits _{m}p_{m}q_{m}=\sum\limits _{m}p_{m}\nabla q_{m}+\sum\limits _{m}q_{m}\nabla p_{m}.</math> 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 <math>p_{m}</math>’s) or the other (<math>q_{m}</math>’s), the contribution to each term must be
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| <center><math>
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| \sum\limits _{m}p_{m}\nabla q_{m} = \left(\mathbf{P}\cdot\nabla\right)\mathbf{Q}+\mathbf{P}\times\nabla\times\mathbf{Q},</math></center>
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| <center><math>\sum\limits _{m}q_{m}\nabla p_{m} = \left(\mathbf{Q}\cdot\nabla\right)\mathbf{P}+\mathbf{Q}\times\nabla\times\mathbf{P}.</math></center>
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| These results can be rigorously proven to be correct through [http://luz.izt.uam.mx/mediawiki/index.php/Green%27s_vector_identity evaluation of the vector components]. Therefore, the RHS can be written in vector form as
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| <center><math>\sum\limits _{m}p_{m}\nabla q_{m}-\sum\limits _{m}q_{m}\nabla p_{m}=\left(\mathbf{P}\cdot\nabla\right)\mathbf{Q}+\mathbf{P}\times\nabla\times\mathbf{Q}-\left(\mathbf{Q}\cdot\nabla\right)\mathbf{P}-\mathbf{Q}\times\nabla\times\mathbf{P}.</math></center>
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| Putting together these two results, a '''theorem for vector fields''' analogous to Green’s theorem for scalar fields is obtained
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| <center><math>\color{OliveGreen}
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| \mathbf{P}\cdot\nabla^{2}\mathbf{Q}-\mathbf{Q}\cdot\nabla^{2}\mathbf{P}=\nabla\cdot\left[\left(\mathbf{P}\cdot\nabla\right)\mathbf{Q}+\mathbf{P}\times\nabla\times\mathbf{Q}-\left(\mathbf{Q}\cdot\nabla\right)\mathbf{P}-\mathbf{Q}\times\nabla\times\mathbf{P}\right].</math></center>
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| The curl of a cross product can be written as <math>\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)</math>; Green’s vector identity can then be rewritten as
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| <center><math>\mathbf{P}\cdot\nabla^{2}\mathbf{Q}-\mathbf{Q}\cdot\nabla^{2}\mathbf{P}=
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| \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\nabla\times\mathbf{Q}-\mathbf{Q}\times\nabla\times\mathbf{P}\right].</math></center>
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| Since the divergence of a curl is zero, the third term vanishes and ''Green’s vector identity'' is
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| <center><math>\color{OliveGreen}\mathbf{P}\cdot\nabla^{2}\mathbf{Q}-\mathbf{Q}\cdot\nabla^{2}\mathbf{P}=\nabla\cdot\left[\mathbf{P}\left(\nabla\cdot\mathbf{Q}\right)-\mathbf{Q}\left(\nabla\cdot\mathbf{P}\right)+\mathbf{P}\times\nabla\times\mathbf{Q}-\mathbf{Q}\times\nabla\times\mathbf{P}\right].</math></center>
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| With a similar porcedure, the Laplacian of the dot product can be expressed in terms of the Laplacians of the factors
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| <center><math>\nabla^{2}\left(\mathbf{P}\cdot\mathbf{Q}\right)=\mathbf{P}\cdot\nabla^{2}\mathbf{Q}-\mathbf{Q}\cdot\nabla^{2}\mathbf{P}+2\nabla\cdot\left[\left(\mathbf{Q}\cdot\nabla\right)\mathbf{P}+\mathbf{Q}\times\nabla\times\mathbf{P}\right].</math></center>
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| As a corollary, the awkward terms can now be written in terms of a divergence by comparison with the vector Green equation
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| <math>\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].</math>
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| This result can be verified by expanding the divergence of a scalar times a vector on the RHS.
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| ==See also==
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| * [[Green's function]]
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| * [[Kirchhoff integral theorem]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| * {{springer|title=Green formulas|id=p/g045080}}
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| *[http://mathworld.wolfram.com/GreensIdentities.html] Green's Identities at Wolfram MathWorld
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| {{DEFAULTSORT:Green's Identities}}
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| [[Category:Vector calculus]]
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| [[Category:Mathematical identities]]
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