Green's identities explained

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 while using an extension of the product rule that : Let and be scalar functions defined on some region, and suppose that is twice continuously differentiable, and is once continuously differentiable. Using the product rule above, but letting, integrate over . Then^[1] $$\backslash int\_U\; \backslash left(\backslash psi\; \backslash ,\; \backslash Delta\; \backslash varphi\; +\; \backslash nabla\; \backslash psi\; \backslash cdot\; \backslash nabla\; \backslash varphi\; \backslash right)\backslash ,\; dV\; =\; \backslash oint\_\; \backslash psi\; \backslash left(\backslash nabla\; \backslash varphi\; \backslash cdot\; \backslash mathbf\; \backslash right)\backslash ,\; dS=\backslash oint\_\backslash psi\backslash ,\backslash nabla\backslash varphi\backslash cdot\; d\backslash mathbf$$where is the Laplace operator, is the boundary of region, is the outward pointing unit normal to the surface element and 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 and the gradient of replacing and .

Note that Green's first identity above is a special case of the more general identity derived from the divergence theorem by substituting,$$\backslash int\_U\; \backslash left(\backslash psi\; \backslash ,\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; +\; \backslash mathbf\; \backslash cdot\; \backslash nabla\; \backslash psi\backslash right)\backslash ,\; dV\; =\; \backslash oint\_\; \backslash psi\; \backslash left(\backslash mathbf\; \backslash cdot\; \backslash mathbf\; \backslash right)\backslash ,\; dS=\backslash oint\_\backslash psi\backslash mathbf\backslash cdot\; d\backslash mathbf\; ~.$$

Green's second identity

If and are both twice continuously differentiable on, and is once continuously differentiable, one may choose to obtain$$\backslash int\_U\; \backslash left[\backslash psi\; \backslash ,\; \backslash nabla\; \backslash cdot\; \backslash left(\backslash varepsilon\; \backslash ,\; \backslash nabla\; \backslash varphi\; \backslash right)\; -\; \backslash varphi\; \backslash ,\; \backslash nabla\; \backslash cdot\; \backslash left(\backslash varepsilon\; \backslash ,\; \backslash nabla\; \backslash psi\; \backslash right)\; \backslash right]\backslash ,\; dV\; =\; \backslash oint\_\; \backslash varepsilon\; \backslash left(\backslash psi\; -\; \backslash varphi\; \backslash right)\backslash ,\; dS.$$

For the special case of all across, then,$$\backslash int\_U\; \backslash left(\backslash psi\; \backslash ,\; \backslash nabla^2\; \backslash varphi\; -\; \backslash varphi\; \backslash ,\; \backslash nabla^2\; \backslash psi\backslash right)\backslash ,\; dV\; =\; \backslash oint\_\; \backslash left(\backslash psi\; -\; \backslash varphi\; \backslash right)\backslash ,\; dS.$$

In the equation above, is the directional derivative of in the direction of the outward pointing surface normal of the surface element,$$=\; \backslash nabla\; \backslash varphi\; \backslash cdot\; \backslash mathbf\; =\; \backslash nabla\_\backslash mathbf\backslash varphi.$$

Explicitly incorporating this definition in the Green's second identity with results in$$\backslash int\_U\; \backslash left(\backslash psi\; \backslash ,\; \backslash nabla^2\; \backslash varphi\; -\; \backslash varphi\; \backslash ,\; \backslash nabla^2\; \backslash psi\backslash right)\backslash ,\; dV\; =\; \backslash oint\_\; \backslash left(\backslash psi\; \backslash nabla\; \backslash varphi\; -\; \backslash varphi\; \backslash nabla\; \backslash psi\backslash right)\backslash cdot\; d\backslash mathbf.$$

In particular, this demonstrates that the Laplacian is a self-adjoint operator in the 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, where the Green's function is taken to be a fundamental solution of the Laplace operator, ∆. This means that:$$\backslash Delta\; G(\backslash mathbf,\backslash boldsymbol)\; =\; \backslash delta(\backslash mathbf\; -\; \backslash boldsymbol)\; ~.$$

For example, in, a solution has the form$$G(\backslash mathbf,\backslash boldsymbol)=\; \backslash frac\; ~.$$

Green's third identity states that if is a function that is twice continuously differentiable on, then$$\backslash int\_U\; \backslash left[G(\backslash mathbf\{y\},\backslash boldsymbol\{\backslash eta\})\; \backslash ,\; \backslash Delta\; \backslash psi(\backslash mathbf\{y\})\; \backslash right]\; \backslash ,\; dV\_\backslash mathbf\; -\; \backslash psi(\backslash boldsymbol)=\; \backslash oint\_\; \backslash left[G(\backslash mathbf\{y\},\backslash boldsymbol\{\backslash eta\})\; \{\backslash partial\; \backslash psi\; \backslash over\; \backslash partial\; \backslash mathbf\{n\}\}\; (\backslash mathbf\{y\})\; -\; \backslash psi(\backslash mathbf\{y\})\; \{\backslash partial\; G(\backslash mathbf\{y\},\backslash boldsymbol\{\backslash eta\})\; \backslash over\; \backslash partial\; \backslash mathbf\{n\}\}\; \backslash right]\backslash ,\; dS\_\backslash mathbf.$$

A simplification arises if is itself a harmonic function, i.e. a solution to the Laplace equation. Then and the identity simplifies to$$\backslash psi(\backslash boldsymbol)=\; \backslash oint\_\; \backslash left[\backslash psi(\backslash mathbf\{y\})\; \backslash frac\{\backslash partial\; G(\backslash mathbf\{y\},\backslash boldsymbol\{\backslash eta\})\}\{\backslash partial\; \backslash mathbf\{n\}\}\; -\; G(\backslash mathbf\{y\},\backslash boldsymbol\{\backslash eta\})\; \backslash frac\{\backslash partial\; \backslash psi\}\{\backslash partial\; \backslash mathbf\{n\}\}\; (\backslash mathbf\{y\})\; \backslash right]\backslash ,\; dS\_\backslash mathbf.$$

The second term in the integral above can be eliminated if is chosen to be the Green's function that vanishes on the boundary of (Dirichlet boundary condition),$$\backslash psi(\backslash boldsymbol)\; =\; \backslash oint\_\; \backslash psi(\backslash mathbf)\; \backslash frac\; \backslash ,\; dS\_\backslash mathbf\; ~.$$

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 ^[2] for a detailed argument, with an alternative.

It can be further verified that the above identity also applies when is a solution to the Helmholtz equation or wave equation and 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 arewhere and are smooth real-valued functions on, is the volume form compatible with the metric,

d\widetilde{V} is the induced volume form on the boundary of, is the outward oriented unit vector field normal to the boundary, and 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$$p\_m\; \backslash ,\; \backslash Delta\; q\_m-q\_m\; \backslash ,\; \backslash Delta\; p\_m\; =\; \backslash nabla\backslash cdot\backslash left(p\_m\backslash nabla\; q\_m-q\_m\; \backslash ,\; \backslash nabla\; p\_m\backslash right),$$where and 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.^[3]

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

One variant invokes the divergence of a cross product ^{[4] [5] [6]} and states a relationship in terms of the curl-curl of the field$$\backslash mathbf\backslash cdot\backslash left(\backslash nabla\backslash times\backslash nabla\backslash times\backslash mathbf\backslash right)-\backslash mathbf\backslash cdot\backslash left(\backslash nabla\backslash times\backslash nabla\backslash times\; \backslash mathbf\backslash right)\; =\; \backslash nabla\backslash cdot\backslash left(\backslash mathbf\backslash times\backslash left(\backslash nabla\backslash times\backslash mathbf\backslash right)-\backslash mathbf\backslash times\backslash left(\backslash nabla\backslash times\backslash mathbf\backslash right)\backslash right).$$

This equation can be written in terms of the Laplacians,

$$\backslash mathbf\backslash cdot\backslash Delta\; \backslash mathbf-\backslash mathbf\backslash cdot\backslash Delta\; \backslash mathbf\; +\; \backslash mathbf\; \backslash cdot\; \backslash left[\backslash nabla\backslash left(\backslash nabla\backslash cdot\backslash mathbf\{P\}\backslash right)\backslash right]-\backslash mathbf\; \backslash cdot\; \backslash left[\backslash nabla\; \backslash left(\backslash nabla\; \backslash cdot\; \backslash mathbf\{Q\}\backslash right)\backslash right]\; =\; \backslash nabla\; \backslash cdot\; \backslash left(\backslash mathbf\backslash times\; \backslash left(\backslash nabla\backslash times\backslash mathbf\backslash right)\; -\; \backslash mathbf\backslash times\backslash left(\backslash nabla\backslash times\backslash mathbf\backslash right)\backslash right).$$

However, the terms$$\backslash mathbf\backslash cdot\backslash left[\backslash nabla\backslash left(\backslash nabla\backslash cdot\backslash mathbf\{P\}\backslash right)\backslash right]-\backslash mathbf\; \backslash cdot\; \backslash left[\backslash nabla\backslash left(\backslash nabla\backslash cdot\backslash mathbf\{Q\}\backslash right)\backslash right],$$could not be readily written in terms of a divergence.

The other approach introduces bi-vectors, this formulation requires a dyadic Green function.^{[7] [8]} The derivation presented here avoids these problems.^[9]

Consider that the scalar fields in Green's second identity are the Cartesian components of vector fields, i.e.,$$\backslash mathbf=\backslash sum\_m\; p\_\backslash hat\_m,\; \backslash qquad\; \backslash mathbf=\backslash sum\_m\; q\_m\; \backslash hat\_m.$$

Summing up the equation for each component, we obtain$$\backslash sum\_m\; \backslash left[p\_m\backslash Delta\; q\_m\; -\; q\_m\backslash Delta\; p\_m\backslash right]=\backslash sum\_m\; \backslash nabla\; \backslash cdot\; \backslash left(p\_m\; \backslash nabla\; q\_m-q\_m\backslash nabla\; p\_m\; \backslash right).$$

The LHS according to the definition of the dot product may be written in vector form as$$\backslash sum\_m\; \backslash left[p\_m\; \backslash ,\; \backslash Delta\; q\_m-q\_m\; \backslash ,\; \backslash Delta\; p\_m\backslash right]\; =\; \backslash mathbf\; \backslash cdot\; \backslash Delta\backslash mathbf-\backslash mathbf\backslash cdot\backslash Delta\backslash mathbf.$$

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.,$$\backslash sum\_m\; \backslash nabla\backslash cdot\backslash left(p\_m\; \backslash nabla\; q\_m-q\_m\backslash nabla\; p\_m\backslash right)=\; \backslash nabla\; \backslash cdot\; \backslash left(\backslash sum\_m\; p\_m\; \backslash nabla\; q\_m\; -\; \backslash sum\_m\; q\_m\; \backslash nabla\; p\_m\; \backslash right).$$

Recall the vector identity for the gradient of a dot product,$$\backslash nabla\; \backslash left(\backslash mathbf\; \backslash cdot\; \backslash mathbf\; \backslash right)\; =\; \backslash left(\backslash mathbf\; \backslash cdot\; \backslash nabla\; \backslash right)\; \backslash mathbf\; +\; \backslash left(\backslash mathbf\; \backslash cdot\; \backslash nabla\; \backslash right)\; \backslash mathbf\; +\; \backslash mathbf\backslash times\; \backslash left(\backslash nabla\backslash times\backslash mathbf\backslash right)+\backslash mathbf\backslash times\; \backslash left(\backslash nabla\backslash times\backslash mathbf\backslash right),$$which, written out in vector components is given by

$$\backslash nabla\backslash left(\backslash mathbf\backslash cdot\backslash mathbf\backslash right)=\backslash nabla\backslash sum\_m\; p\_m\; q\_m\; =\; \backslash sum\_m\; p\_m\; \backslash nabla\; q\_m\; +\; \backslash sum\_m\; q\_m\; \backslash nabla\; p\_m.$$

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

p _{m} ’s) or the other (q _{m} ’s), the contribution to each term must be$$\backslash sum\_m\; p\_m\; \backslash nabla\; q\_m\; =\; \backslash left(\backslash mathbf\; \backslash cdot\; \backslash nabla\backslash right)\; \backslash mathbf\; +\; \backslash mathbf\; \backslash times\; \backslash left(\backslash nabla\; \backslash times\; \backslash mathbf\backslash right),$$$$\backslash sum\_m\; q\_m\; \backslash nabla\; p\_m\; =\; \backslash left(\backslash mathbf\; \backslash cdot\; \backslash nabla\backslash right)\; \backslash mathbf\; +\; \backslash mathbf\; \backslash times\; \backslash left(\backslash nabla\; \backslash times\; \backslash mathbf\backslash right).$$

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

$$\backslash sum\_m\; p\_m\; \backslash nabla\; q\_m\; -\; \backslash sum\_m\; q\_m\; \backslash nabla\; p\_m\; =\; \backslash left(\backslash mathbf\; \backslash cdot\; \backslash nabla\backslash right)\; \backslash mathbf\; +\; \backslash mathbf\backslash times\; \backslash left(\backslash nabla\backslash times\backslash mathbf\backslash right)-\backslash left(\backslash mathbf\; \backslash cdot\; \backslash nabla\backslash right)\; \backslash mathbf\; -\; \backslash mathbf\backslash times\; \backslash left(\backslash nabla\backslash times\backslash mathbf\backslash right).$$

Putting together these two results, a result analogous to Green's theorem for scalar fields is obtained,
Theorem for vector fields: $$\backslash color\backslash mathbf\; \backslash cdot\; \backslash Delta\; \backslash mathbf\; -\; \backslash mathbf\; \backslash cdot\; \backslash Delta\; \backslash mathbf\; =\; \backslash left[\backslash left(\backslash mathbf\{P\}\; \backslash cdot\; \backslash nabla\backslash right)\; \backslash mathbf\{Q\}\; +\; \backslash mathbf\{P\}\backslash times\; \backslash left(\backslash nabla\backslash times\backslash mathbf\{Q\}\backslash right)-\backslash left(\backslash mathbf\{Q\}\; \backslash cdot\; \backslash nabla\backslash right)\; \backslash mathbf\{P\}\; -\; \backslash mathbf\{Q\}\backslash times\; \backslash left(\backslash nabla\backslash times\backslash mathbf\{P\}\backslash right)\backslash right].$$

The curl of a cross product can be written as$$\backslash nabla\backslash times\backslash left(\backslash mathbf\backslash times\backslash mathbf\backslash right)=\backslash left(\backslash mathbf\backslash cdot\backslash nabla\backslash right)\backslash mathbf-\backslash left(\backslash mathbf\backslash cdot\backslash nabla\backslash right)\backslash mathbf+\backslash mathbf\backslash left(\backslash nabla\backslash cdot\backslash mathbf\backslash right)-\backslash mathbf\backslash left(\backslash nabla\backslash cdot\backslash mathbf\backslash right);$$

Green's vector identity can then be rewritten as$$\backslash mathbf\backslash cdot\backslash Delta\; \backslash mathbf-\backslash mathbf\backslash cdot\backslash Delta\; \backslash mathbf=\; \backslash nabla\; \backslash cdot\; \backslash left[\backslash mathbf\{P\}\; \backslash left(\backslash nabla\backslash cdot\backslash mathbf\{Q\}\backslash right)-\backslash mathbf\{Q\}\; \backslash left(\backslash nabla\; \backslash cdot\; \backslash mathbf\{P\}\backslash right)-\backslash nabla\; \backslash times\; \backslash left(\backslash mathbf\{P\}\; \backslash times\; \backslash mathbf\{Q\}\; \backslash right)\; +\backslash mathbf\{P\}\backslash times\backslash left(\backslash nabla\backslash times\backslash mathbf\{Q\}\backslash right)\; -\; \backslash mathbf\{Q\}\backslash times\; \backslash left(\backslash nabla\backslash times\backslash mathbf\{P\}\backslash right)\backslash right].$$

Since the divergence of a curl is zero, the third term vanishes to yield Green's vector identity:$$\backslash color\backslash mathbf\backslash cdot\backslash Delta\backslash mathbf-\backslash mathbf\; \backslash cdot\; \backslash Delta\; \backslash mathbf\; =\backslash nabla\backslash cdot\backslash left[\backslash mathbf\{P\}\backslash left(\backslash nabla\backslash cdot\backslash mathbf\{Q\}\backslash right)-\backslash mathbf\{Q\}\; \backslash left(\backslash nabla\; \backslash cdot\; \backslash mathbf\{P\}\; \backslash right)\; +\; \backslash mathbf\{P\}\backslash times\; \backslash left(\backslash nabla\backslash times\backslash mathbf\{Q\}\backslash right)\; -\; \backslash mathbf\{Q\}\backslash times\backslash left(\backslash nabla\backslash times\backslash mathbf\{P\}\backslash right)\backslash right].$$

With a similar procedure, the Laplacian of the dot product can be expressed in terms of the Laplacians of the factors$$\backslash Delta\; \backslash left(\backslash mathbf\; \backslash cdot\; \backslash mathbf\; \backslash right)\; =\; \backslash mathbf\; \backslash cdot\; \backslash Delta\; \backslash mathbf-\backslash mathbf\backslash cdot\backslash Delta\; \backslash mathbf\; +\; 2\backslash nabla\; \backslash cdot\; \backslash left[\backslash left(\backslash mathbf\{Q\}\; \backslash cdot\; \backslash nabla\; \backslash right)\; \backslash mathbf\{P\}\; +\; \backslash mathbf\{Q\}\; \backslash times\; \backslash nabla\; \backslash times\; \backslash mathbf\{P\}\; \backslash right].$$

As a corollary, the awkward terms can now be written in terms of a divergence by comparison with the vector Green equation,$$\backslash mathbf\backslash cdot\; \backslash left[\backslash nabla\; \backslash left(\backslash nabla\; \backslash cdot\; \backslash mathbf\{Q\}\; \backslash right)\; \backslash right]\; -\; \backslash mathbf\; \backslash cdot\; \backslash left[\backslash nabla\; \backslash left(\backslash nabla\; \backslash cdot\; \backslash mathbf\{P\}\; \backslash right)\; \backslash right]\; =\; \backslash nabla\; \backslash cdot\backslash left[\backslash mathbf\{P\}\backslash left(\backslash nabla\backslash cdot\backslash mathbf\{Q\}\backslash right)-\backslash mathbf\{Q\}\; \backslash left(\backslash nabla\; \backslash cdot\; \backslash mathbf\{P\}\; \backslash right)\; \backslash right].$$

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
• Lagrange's identity (boundary value problem)

External links

• • http://mathworld.wolfram.com/GreensIdentities.html Green's Identities at Wolfram MathWorld

Notes and References

1. Book: Strauss, Walter. Partial Differential Equations: An Introduction. Wiley.
2. Book: Jackson . John David . Classical Electrodynamics . 1998-08-14 . John Wiley & Sons . 39.
3. Guasti . M Fernández . Complementary fields conservation equation derived from the scalar wave equation . Journal of Physics A: Mathematical and General . IOP Publishing . 37 . 13 . 2004-03-17 . 0305-4470 . 10.1088/0305-4470/37/13/013 . 4107–4121. 2004JPhA...37.4107F .
4. Love. Augustus E. H.. Augustus Edward Hough Love. I. 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 . The Royal Society . 197 . 287–299 . 1901 . 0264-3952 . 10.1098/rsta.1901.0013 . 1–45. free .
5. Stratton . J. A. . Chu . L. J. . Diffraction Theory of Electromagnetic Waves . Physical Review . American Physical Society (APS) . 56 . 1 . 1939-07-01 . 0031-899X . 10.1103/physrev.56.99 . 99–107. 1939PhRv...56...99S .
6. Bruce . Neil C . Double scatter vector-wave Kirchhoff scattering from perfectly conducting surfaces with infinite slopes . Journal of Optics . IOP Publishing . 12 . 8 . 2010-07-22 . 2040-8978 . 10.1088/2040-8978/12/8/085701 . 085701. 2010JOpt...12h5701B . 120636008 .
7. Franz . W . On the Theory of Diffraction . Proceedings of the Physical Society. Section A . IOP Publishing . 63 . 9 . 1950-09-01 . 0370-1298 . 10.1088/0370-1298/63/9/301 . 925–939. 1950PPSA...63..925F .
8. Kirchhoff theory: Scalar, vector, or dyadic? . IEEE Transactions on Antennas and Propagation . Institute of Electrical and Electronics Engineers (IEEE) . 20 . 1 . 1972 . 0096-1973 . 10.1109/tap.1972.1140146 . 114–115 . Chen-To Tai . 1972ITAP...20..114T .
9. Fernández-Guasti . M. . Green's Second Identity for Vector Fields . ISRN Mathematical Physics . Hindawi Limited . 2012 . 2012 . 2090-4681 . 10.5402/2012/973968 . 1–7. free .