Green's identities

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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 identityEdit

This identity is derived from the divergence theorem applied to the vector field Template:Math while using an extension of the product rule that Template:Math: Let Template:Mvar and Template:Mvar be scalar functions defined on some region Template:Math, and suppose that Template:Mvar is twice continuously differentiable, and Template:Mvar is once continuously differentiable. Using the product rule above, but letting Template:Math, integrate Template:Math over Template:Mvar. Then<ref name="strauss">Template:Cite book</ref> <math display="block">\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} </math> where Template:Math is the Laplace operator, Template:Math is the boundary of region Template:Mvar, Template:Math is the outward pointing unit normal to the surface element Template:Math and Template:Math 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 Template:Mvar and the gradient of Template:Mvar replacing Template:Mvar and Template:Mvar.

Note that Green's first identity above is a special case of the more general identity derived from the divergence theorem by substituting Template:Math, <math display="block">\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} ~. </math>

Green's second identityEdit

If Template:Mvar and Template:Mvar are both twice continuously differentiable on Template:Math, and Template:Mvar is once continuously differentiable, one may choose Template:Math to obtain <math display="block"> \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. </math>

For the special case of Template:Math all across Template:Math, then, <math display="block"> \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. </math>

In the equation above, Template:Math is the directional derivative of Template:Mvar in the direction of the outward pointing surface normal Template:Math of the surface element Template:Math, <math display="block"> {\partial \varphi \over \partial \mathbf{n}} = \nabla \varphi \cdot \mathbf{n} = \nabla_\mathbf{n}\varphi.</math>

Explicitly incorporating this definition in the Green's second identity with Template:Math results in <math display="block"> \int_U \left( \psi \, \nabla^2 \varphi - \varphi \, \nabla^2 \psi\right)\, dV = \oint_{\partial U} \left( \psi \nabla \varphi - \varphi \nabla \psi\right)\cdot d\mathbf{S}. </math>

In particular, this demonstrates that the Laplacian is a self-adjoint operator in the Template:Math inner product for functions vanishing on the boundary so that the right hand side of the above identity is zero.

Green's third identityEdit

Green's third identity derives from the second identity by choosing Template:Math, where the Green's function Template:Mvar is taken to be a fundamental solution of the Laplace operator, ∆. This means that: <math display="block"> \Delta G(\mathbf{x},\boldsymbol{\eta}) = \delta(\mathbf{x} - \boldsymbol{\eta}) ~.</math>

For example, in Template:Math, a solution has the form <math display="block">G(\mathbf{x},\boldsymbol{\eta})= \frac{-1}{4 \pi \|\mathbf{x} - \boldsymbol{\eta} \|} ~.</math>

Green's third identity states that if Template:Mvar is a function that is twice continuously differentiable on Template:Mvar, then <math display="block"> \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}.</math>

A simplification arises if Template:Mvar is itself a harmonic function, i.e. a solution to the Laplace equation. Then Template:Math and the identity simplifies to <math display="block">\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}.</math>

The second term in the integral above can be eliminated if Template:Mvar is chosen to be the Green's function that vanishes on the boundary of Template:Mvar (Dirichlet boundary condition), <math display="block">\psi(\boldsymbol{\eta}) = \oint_{\partial U} \psi(\mathbf{y}) \frac{\partial G(\mathbf{y},\boldsymbol{\eta})}{\partial \mathbf{n}} \, dS_\mathbf{y} ~.</math>

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 <ref>Template:Cite book</ref> 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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> First note <math display="block"> \int_U \Delta G(\mathbf{y},\boldsymbol{\eta}) dV_y = 1 = \oint_{\partial U} \frac{\partial G(\mathbf{y},\boldsymbol{\eta})}{\partial \mathbf{n}} dS_y </math> and so <math>\frac{\partial G(\mathbf{y},\boldsymbol{\eta})}{\partial \mathbf{n}}</math> cannot vanish on surface <math>S</math>. A convenient choice is <math>\frac{\partial G(\mathbf{y},\boldsymbol{\eta})}{\partial \mathbf{n}} = 1/\mathcal{A}</math>, where <math>\mathcal{A}</math> is the area of the surface <math>S</math>. The integral can be simplified to <math display="block">\psi(\boldsymbol{\eta})= \langle\psi\rangle_S- \oint_{\partial U} G(\mathbf{y},\boldsymbol{\eta}) \frac{\partial \psi}{\partial \mathbf{n}} (\mathbf{y}) \, dS_\mathbf{y}.</math> where <math>\langle\psi\rangle_S = \frac{1}{\mathcal{A}} \oint_{\partial U} \psi(\mathbf{y}) dS_y </math> is the average value of <math>\psi</math> on surface <math>S</math>.

Furthermore, if <math>\psi</math> is a solution to the Laplace's equation, divergence theorem implies it must satisfy <math>\oint_{\partial U} \frac{\partial \psi}{\partial \mathbf{n}} (\mathbf{y})dS_y = \int_U \Delta \psi(\mathbf{y})dV_y = 0</math>. 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 Template:Mvar is a solution to the Helmholtz equation or wave equation and Template:Mvar 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 manifoldsEdit

Green's identities hold on a Riemannian manifold. In this setting, the first two are <math display="block">\begin{align} \int_M u \,\Delta v\, dV + \int_M \langle\nabla u, \nabla v\rangle\, dV &= \int_{\partial M} u N v \, d\widetilde{V} \\ \int_M \left (u \, \Delta v - v \, \Delta u \right )\, dV &= \int_{\partial M}(u N v - v N u) \, d \widetilde{V} \end{align}</math> where Template:Mvar and Template:Mvar are smooth real-valued functions on Template:Mvar, Template:Mvar is the volume form compatible with the metric, <math>d\widetilde{V}</math> is the induced volume form on the boundary of Template:Mvar, Template:Mvar is the outward oriented unit vector field normal to the boundary, and Template:Math is the Laplacian.

Green's vector identitiesEdit

First vector identityEdit

Using the vector Laplacian identity and the divergence identity,<ref name="Yuffa2021">Template:Cite journal</ref> expand <math>\mathbf{P}\cdot \Delta \mathbf{Q} </math>

<math> \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})] </math>

The last term can be simplified by expanding components <math> \begin{align} \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{align} </math>

The identity can be rewritten as <math> \mathbf{P}\cdot \Delta \mathbf{Q} = \nabla \cdot (\mathbf{P}\times \nabla \times \mathbf{Q}) - (\nabla \times \mathbf{P})\cdot (\nabla \times \mathbf{Q}) + \nabla\cdot[\mathbf{P}(\nabla \cdot \mathbf{Q})] - (\nabla \cdot \mathbf{P})(\nabla \cdot \mathbf{Q}) </math>

In integral form, this is <math> \oint_{\partial U} \mathbf{n}\cdot[\mathbf{P}\times \nabla \times \mathbf{Q} + \mathbf{P}(\nabla \cdot \mathbf{Q}) ] dS = \int_U [ \mathbf{P}\cdot \Delta \mathbf{Q} + (\nabla \times \mathbf{P})\cdot (\nabla \times \mathbf{Q}) + (\nabla \cdot \mathbf{P})(\nabla \cdot \mathbf{Q}) ] dV</math>

Second vector identityEdit

Green's second identity establishes a relationship between second and (the divergence of) first order derivatives of two scalar functions. In differential form <math display="block">p_m \, \Delta q_m-q_m \, \Delta p_m = \nabla\cdot\left(p_m\nabla q_m-q_m \, \nabla p_m\right),</math> where Template:Math and Template: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>Template:Cite journal</ref>

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

One variant invokes the divergence of a cross product <ref>Template:Cite journal</ref><ref>Template:Cite journal</ref><ref>Template:Cite journal</ref> and states a relationship in terms of the curl-curl of the field <math display="block">\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).</math>

This equation can be written in terms of the Laplacians,

<math display="block">\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).</math>

However, the terms <math display="block">\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>Template:Cite journal</ref><ref>Template:Cite journal</ref> The derivation presented here avoids these problems.<ref>Template:Cite journal</ref>

Consider that the scalar fields in Green's second identity are the Cartesian components of vector fields, i.e., <math display="block">\mathbf{P}=\sum_m p_{m}\hat{\mathbf{e}}_m, \qquad \mathbf{Q}=\sum_m q_m \hat{\mathbf{e}}_m.</math>

Summing up the equation for each component, we obtain <math display="block">\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).</math>

The LHS according to the definition of the dot product may be written in vector form as <math display="block">\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}.</math>

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 display="block">\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).</math>

Recall the vector identity for the gradient of a dot product, <math display="block">\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),</math> which, written out in vector components is given by

<math display="block">\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.</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 <math display="block">\sum_m p_m \nabla q_m = \left(\mathbf{P} \cdot \nabla\right) \mathbf{Q} + \mathbf{P} \times \left(\nabla \times \mathbf{Q}\right),</math> <math display="block">\sum_m q_m \nabla p_m = \left(\mathbf{Q} \cdot \nabla\right) \mathbf{P} + \mathbf{Q} \times \left(\nabla \times \mathbf{P}\right).</math>

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

<math display="block"> \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).</math>

Putting together these two results, a result analogous to Green's theorem for scalar fields is obtained,
Theorem for vector fields: <math display="block">\color{OliveGreen}\mathbf{P} \cdot \Delta \mathbf{Q} - \mathbf{Q} \cdot \Delta \mathbf{P} = \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].</math>

The curl of a cross product can be written as <math display="block">\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 <math display="block">\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].</math>

Since the divergence of a curl is zero, the third term vanishes to yield Green's second vector identity: <math display="block">\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].</math>

With a similar procedure, the Laplacian of the dot product can be expressed in terms of the Laplacians of the factors <math display="block"> \Delta \left( \mathbf{P} \cdot \mathbf{Q} \right) = \mathbf{P} \cdot \Delta \mathbf{Q}-\mathbf{Q}\cdot\Delta \mathbf{P} + 2\nabla \cdot \left[ \left( \mathbf{Q} \cdot \nabla \right) \mathbf{P} + \mathbf{Q} \times \nabla \times \mathbf{P} \right].</math>

As a corollary, the awkward terms can now be written in terms of a divergence by comparison with the vector Green equation, <math display="block">\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>

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

Third vector identityEdit

The third vector identity can be derived using the free space scalar Green's function.<ref name="Yuffa2021" /> Take the scalar Green's function definition <math>\Delta G(\mathbf{x},\boldsymbol{\eta}) =\delta(\mathbf{x}-\boldsymbol{\eta})</math>, multiply by <math>\mathbf{P}</math> and subtract <math>G\nabla_i \nabla^i \mathbf{P}</math>.

<math display="block">\nabla_i (\mathbf{P} \nabla^i G - G\nabla^i \mathbf{P}) =\mathbf{P} \delta(\mathbf{x}-\boldsymbol{\eta}) - G \Delta \mathbf{P}</math>

Integrate over volume <math>U</math> and use divergence theorem. <math display="block"> \oint_{\partial U }\bigg(\mathbf{P}(\mathbf{y}) \frac{\partial G(\mathbf{y},\boldsymbol{\eta})}{\partial \mathbf{n}} - G(\mathbf{y},\boldsymbol{\eta})\frac{\partial \mathbf{P}(\mathbf{y})}{\partial \mathbf{n}}\bigg) dS_y +\int_U G(\mathbf{y},\boldsymbol{\eta}) \Delta \mathbf{P}(\mathbf{y}) dV_y = \left\{\begin{matrix} \mathbf{P}(\boldsymbol{\eta}) & \text{for } ~\boldsymbol{\eta}\in U \\ 0 & \text{for } ~\boldsymbol{\eta}\not\in U \\ \end{matrix}\right. </math>

ReferencesEdit

Template:Reflist

External linksEdit

Template:Springer
[1] Green's Identities at Wolfram MathWorld