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, …, s_n, and select in the ith part a point M_i. If these parts are sufficiently small, their masses are approximately equal to f(M_i)s_i, and the mass of the entire surface is approximately equal to

The smaller the parts s_i, 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.