Green formulas

Formulas of the integral calculus of functions of several variables, connecting the values of an -fold integral over a domain in an -dimensional Euclidean space with an -fold integral along the piecewise-smooth boundary of this domain. The Green formulas are obtained by integration by parts of integrals of the divergence of a vector field that is continuous in and that is continuously differentiable in .

In the simplest Green formula,

(1)

the curvilinear integral along the contour is expressed as a double integral over the domain . Here the domain is oriented in a natural manner, while an induced orientation, known as counterclockwise, is taken along the boundary . Formula (1) has a simple hydrodynamic meaning: The flow across the boundary of a liquid flowing on a plane at rate is equal to the integral over of the intensity (divergence) of the sources and sinks distributed over . In this sense the Green formula (1) resembles the Ostrogradski formula (see also Stokes formula).

Formula (1) is sometimes attributed to C.F. Gauss and B. Riemann, but none of its usual appellations corresponds to historical truth; it was in fact encountered as early as the 18th century in the analytical studies of L. Euler and others.

G. Green [1] must be credited with the following formulas of potential theory:

(2)

the preparatory Green formula, and

(3)

Here is a domain in , , is the volume element of , is the surface element of , is the unit outer (co-)normal to ,

is the operator of differentiation in the direction of the (co-)vector , and

is the Laplace operator.

Formulas (2) and (3) are also valid if is a domain in , , is the volume element in , is the -dimensional volume element of , and

is the Laplace operator in independent variables.

The generalizations of the Green formulas (2) and (3) for linear partial differential operators with sufficiently smooth coefficients have the following form:

1) If

are (real) adjoint second-order differential operators, , then

where is the unit (co-)vector of the outer normal to :

is the operator of differentiation in the direction of the so-called co-normal

of the operator .

2) If

then

where is the co-normal of , and

3) If

are (real) adjoint differential operators of order , is an integer multi-index of length , , , and , then

(4)

Here the boundary integral can be written as the bilinear sum

where , are certain linear differential operators of orders , , .

Green's formulas play an important role in analysis and, particularly, in the theory of boundary value problems for differential operators (both ordinary and partial differential operators) of the second or higher orders. For functions , which are sufficiently smooth in , Green's formulas (2) and (4) serve as the source of several relations which are useful in the study of solutions of boundary value problems, in the clarification of the type of boundary value problems, in obtaining explicit solutions, etc. Thus, the function in (2), which is harmonic in , satisfies the Gauss theorem

for . For sufficiently smooth functions , in , and the function

which, for , has the same singularity as the fundamental solution of the Laplace operator, the following Green formulas are valid:

(5)

(6)

Here,

where is the area of the -dimensional unit sphere in . Here it is assumed, for , that the boundary has a continuous tangent plane in some neighbourhood of .

Formulas (5) and (6) serve to obtain integral representations of the solutions of basic boundary value problems in potential theory (cf. Harmonic function; Green function; Poisson formula). Thus, they are used to obtain the Green formula, or Green integral,

(7)

for a function which is harmonic in . This integral plays an important role in the theory of harmonic functions (cf. Harmonic function). Formulas resembling (5) and (6), which give integral representations of the solution of the Cauchy problem or of the mixed problem, are also valid for a normal hyperbolic operator of order two. See Kirchhoff formula; Riemann method; Riemann function. For Green formulas in the theory of boundary value problems see also [4]–[9].

References

[1]	G. Green, "An essay on the application of mathematical analysis to the theories of electricity and magnetism" , Nottingham (1828) (Reprint: Mathematical papers, Chelsea, reprint, 1970, pp. 1–82)
[2]	J. Maxwell, "Selected works on the theory of electromagnetic fields" , Moscow (1954) (In Russian; translated from English)
[3]	V.I. Smirnov, "A course of higher mathematics" , 2 , Addison-Wesley (1964) (Translated from Russian)
[4]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)
[5]	V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian)
[6]	S.L. Sobolev, "Partial differential equations of mathematical physics" , Pergamon (1964) (Translated from Russian)
[7]	C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)
[8]	N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Interscience (1963)
[9]	J.L. Lions, E. Magenes, "Non-homogenous boundary value problems and applications" , 1–2 , Springer (1972) (Translated from French)