Surface integral

An integral over a surface. Let a surface in the three-dimensional Euclidean space , possibly with self-intersections, be specified by the vector representation

(1)

in Cartesian coordinates , where is a continuously-differentiable vector function defined on the closure of a two-dimensional Jordan-measurable domain lying in the plane with Cartesian coordinates and . Let

be the coefficients of the first fundamental form of . If is a function defined on , i.e. a function , then the following defines the surface integral of the first kind (or the integral over the surface area):

(2)

This definition is independent of the representation of the surface. A surface integral of the first kind is the limit of corresponding Riemann sums, which may be described in terms related to the surface. For example, if is a Riemann-integrable function, if is a decomposition of into parts that are images under the mapping (1) of sets forming a decomposition of (see Multiple integral) and if

is the area of , then

where is the fineness (mesh) of and . If is explicitly specified in the form , , (2) becomes

If there are no singular points on a surface with vector representation (1), i.e. if , then can be oriented by selecting a continuous unit normal on it, for example

For an oriented surface one defines the surface integrals of the second kind by

(3)

where surface integrals of the first kind appear on the right-hand side. If is the surface with orientation opposite to that of , then

Similar equations apply to the other surface integrals of the second kind in (3). Surface integrals of the second kind resemble those of the first kind in being the limits of Riemann sums, which can be described in terms of the surface.

If

then

Similar formulas apply to the other surface integrals of the second kind in (3).

In particular, for the case of a surface , ,

The first of these integrals is called an integral over the "upper" side of and the second over the "lower" side. This terminology is used because the vector

forms an acute angle with the -axis in the case of an explicit specification of , i.e. when , and , i.e. it is directed "upwards" , while in the second case it forms an obtuse angle and is thus directed "downwards" .

If a smooth surface is the boundary of a bounded domain and denotes its orientation by means of the outward normal, hence is determined by the inward normal, with respect to the domain, then the surface integrals of the second kind over the oriented surface are called surface integrals with respect to the outside of the surface, while those over are called surface integrals with respect to the inside.

The surface integrals for a piecewise-smooth surface that can be divided into a finite number of parts each of which has a vector representation (1) are defined as the sums of the surface integrals over the corresponding parts. Thus, surface integrals defined with respect to piecewise-smooth surfaces are not dependent on the method of dividing the surfaces into those parts.

The Ostrogradski formula establishes a relationship between a triple integral over a three-dimensional bounded domain and the surface integrals over its boundary, while the Stokes formula gives a relationship between a surface integral and the curvilinear integral over the contour representing the boundary.

The surface integral is equal to the area of . If a mass with density is distributed on , then the surface integral is equal to the entire mass. If is a vector function specified on a surface that is oriented by means of the unit normal , then the surface integral

is called the flux of the vector field through . Clearly, this is independent of the choice of a coordinate system in . One also uses surface integrals to describe a double-layer potential or a simple-layer potential.

If the surface is a differentiable -manifold, if the continuously-differentiable non-negative functions , , form a partition of unity on , i.e. the support of each function is contained in some chart of and for each point , and if is a function defined on , then by definition

(4)

where each integral on the right-hand side is understood in the sense of (2). If is an oriented two-dimensional manifold, then

(5)

The other types of surface integrals of the second kind in (3) are defined similarly. The definitions (4) and (5) are independent of the choice of a partition of unity in .

References

[1]	V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian)
[2]	L.D. Kudryavtsev, "Mathematical analysis" , Moscow (1973) (In Russian)
[3]	S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian)
[4]	B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, "Modern geometry" , Springer (1987) (Translated from Russian)
[5]	A.S. Mishchenko, A.T. Fomenko, "A course of differential geometry and topology" , MIR (1988) (Translated from Russian)

Comments

Ostrogradski's formula is usually called Gauss' formula in the West. The phrase divergence theorem is also used.

In vector notation one can write

The distinction between surface integrals of the first and the second kind is not common in Western literature.

Integrals over differentiable -dimensional manifolds can be conveniently defined using -forms. See, e.g., [a4] and Integration on manifolds.

References

[a1]	G.E. Shilov, "Mathematical analysis" , 1–2 , M.I.T. (1974) (Translated from Russian)
[a2]	T.M. Apostol, "Calculus" , 2 , Waltham (1969)
[a3]	T.M. Apostol, "Mathematical analysis" , Addison-Wesley (1974)
[a4]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a5]	M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish pp. 1–5
[a6]	J.J. Stoker, "Differential geometry" , Wiley (Interscience) (1969)
[a7]	D.J. Struik, "Lectures on classical differential geometry" , Addison-Wesley (1961)
[a8]	R.C. Buck, "Advanced calculus" , McGraw-Hill (1965)
[a9]	W. Fleming, "Functions of several variables" , Springer (1977)
[a10]	J. Marsden, A. Weinstein, "Calculus" , 3 , Springer (1988)