Surface Integral

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surface integral

[′sər·fəs ′int·ə·grəl]
The integral of a function of several variables with respect to surface area over a surface in the domain of the function.

Surface Integral


the integral of a function defined on some surface. For example, the problem of calculating the mass distributed over a surface S with a variable surface density f(M) leads to a surface integral. To calculate this mass, we divide the surface into parts with areas s1, s2, …, sn, and select in the ith part a point Mi. If these parts are sufficiently small, their masses are approximately equal to f(Mi)si, and the mass of the entire surface is approximately equal to

The smaller the parts si, the more exact is this value. The exact value of the mass of the surface is therefore

if the dimensions and the areas of the parts approach zero. Certain problems in physics result in similar limits. These limits are called surface integrals of the first kind for the function f(M) with respect to a surface S and are symbolized

∫∫(s) f(M) ds = ∫∫(s) f(x, y, z)ds

The evaluation of such integrals reduces to the calculation of double integrals.

Limits of similar sums are also encountered in other problems in physics, for example, in determining the flow of a fluid across a surface S. In these problems, however, the areas of the projections of the parts on the three coordinate planes are used instead of the areas of the parts. Here S is assumed to be oriented— that is, one of the directions of the normals is indicated as positive; the area of the projection is assigned a plus or minus sign, depending on whether the angle between the positive direction of the normal and the axis perpendicular to the plane of the projections is acute or obtuse. The limits of sums of this type are surface integrals over oriented surfaces, and are called surface integrals of the second kind and are symbolized

∫∫(s) P dy dz + Q dz dx + R dx dy

In contrast to surface integrals of the first kind, the sign of surface integrals of the second kind depends on the orientation of the surface S.

M. V. Ostrogradskii established an important theorem that gives the relation between the surface integral of the second kind over a closed oriented surface S and the triple integral over the volume Vbounded by S. It follows from this theorem that if the functions P, Q, and R have continuous partial derivatives and the equation

is satisfied in V, then the surface integrals of the second kind over all oriented surfaces contained in V and having the same boundary are equal. In this case, functions P1, Q1, and R1 can be found such that

Stokes’ theorem expresses the line integral over a closed curve in terms of the surface integral of the second kind over an oriented surface bounded by this curve.


Nikol’skii, S. M. Kurs matematicheskogo analiza, vol. 2. Moscow, 1973.
Il’in, V. A., and E. G. Pozniak. Osnovy matematicheskogo analiza, part 2. Moscow, 1973.
Kudriavtsev, L. D. Matematicheskii analiz, 2nd ed. vol. 2. Moscow, 1973.
References in periodicals archive ?
The time average wall shear stress (TAWSS) and its surface integral at bifurcation sites were quantified.
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Michielssen, "Time domain adaptive integral method for surface integral equations," IEEE Trans.
Actually, the method of moments (MoM) solutions of surface integral equations (SIEs) are widely used for numerical analysis of EM radiation and scattering problems [1-9].
Surface integral equations were formulated and solved by the method of moments (MoM) for the two-dimensional (2D) [6], and three-dimensional (3D) [7,8] scattering problems.
Determination of mechanical work Wet of elastic deformation is made by correlating it with the surface integral [S.
Utah) discusses error analysis of the moment method, or boundary element method, for solving surface integral equations of electromagnetic and acoustic radiation and scattering.
j]) = 0, is transformed into the RHS surface integral extended along a surface denoted by S' and called the homologous surface of S with reference to x.
It means that the surface integral in (6b) is positive and the first boundary is losing linear momentum.
k] can be evaluated by inner product of L (f (r)) and A(r) plus a surface integral containing boundary conditions for some cases of differential operators:
g] can be rewritten as a surface integral by use of Stoke's Theorem yielding [[phi].
1,2] Analysis methods of the inner electromagnetic properties of applied DR devices such as the perfect magnetic conducting wall method, [3] the moment method based on the surface integral techniques [4,5] and general mode matching approaches [6,7] have been studied.

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