Surface integral

The surface integral or surface integral is a generalization of the integral term on flat or curved surfaces. The domain of integration is therefore not a one-dimensional interval, but a two-dimensional quantity in three-dimensional space. For a more general presentation in with see: Integration on manifolds.

It is generally made ​​between a scalar and a vectorial surface integral, depending on the shape of the integrand and the so-called surface element. they read

• 2.1 The scalar surface integral
• 2.2 The vectorial surface integral

Terms and Definitions

In integrating over the area of ​​surface contact parameterizations in place of the variable of integration and the surface elements at the place of the infinitesimal ( infinitesimally small ) interval width.

Parameterization

As a two-dimensional quantity can represent a surface as a function of two variables ( parameterized ). Is a set whose boundary contains no double points, continuously differentiable, not infinitely long and also a picture of the is, they say, is parameterization of the surface as is. At this point it should be noted that a large part of the difficulty in dealing with surface integrals associated with the parameterization. It is a priori not clear that different parameterizations produce the same value for the integral. A change of coordinates for surface integrals is not trivial and is therefore motivation for the use of differential forms.

In general, leaves a space in the two parameters in the following form and represent:

On the surface the form of curves and the coordinate lines. These coat the surface with a coordinate grid, which pass through each point two coordinate lines. Thus, each point has coordinates on the clear surface.

Example 1: Parameterization

The surface of a sphere with radius can be parameterized as follows: is the rectangle and

This parameterization satisfies the sphere equation (see also spherical coordinates ). here is the polar angle ( or mostly ), and the azimuth angle ( or generally referred to ).

Example 2: Explicit representation

Is a function of, and the area indicated in the mold, and the two parameters; the parameterization of the surface thus looks as follows:

Surface element

If the one-dimensional case, which is the width of an infinitely small interval, so it is natural to replace it in the two-dimensional case by the surface of an infinitely small surface element. Due to the parametrization described in the previous section can be at any point on the surface of two tangents place (see also: Curvilinear coordinates): Once the tangent that arises when one leaves constant and varies minimally, and once with interchanged variables. So that means two tangents to the two coordinate lines in the considered point. These tangents can be expressed by two infinitesimal tangent vectors ( is the parameterized form of the surface ):

In the following compact notation for the partial derivatives is used:

Are these tangents at any point on the surface parallel, then one speaks of a regular parametrization. The cross product of the tangent vectors is then a vector whose length is equal to zero.

The two tangent vectors lie in the tangent plane of the surface at the point considered. The area of ​​the plane spanned by two tangent vectors parallelogram now corresponds exactly to the amount of their cross product.

Is now a regular parameterization of the surface, so we define:

• Scalar surface element

• Vectorial surface element

According to the characteristics of the cross product of the vector normal to the surface element surface, its amount corresponds exactly to the size of the infinitesimal surface element.

In the above presented form the vectorial surface element is not well defined, since its direction depends on whether one or calculated. Both options are antiparallel to each other. Looking at closed surfaces, one usually agreed that the outwardly facing surface vectorial element is to be used.

Example 1: Parameterization

The surface of the sphere with radius R, as shown above, can be parameterized by the polar angle and the azimuth angle. The surface element is obtained from the following calculation:

In the normal vector are two possible solutions (), depending on the order of and in the cross product. Typically, selected here, the positive solution in which the convex surface facing away from the ball (so-called " outer normal ").

Example 2: Explicit representation

Is the area specified in the form, the sheet is pressed by the differentials of the co-ordinates of.

Thus surface element and vectorial surface element are equal:

Projection on surface with known surface element

We hereafter assume that a surface with its surface element and the associated normal vector is known. For example,

• Xy plane:

• Lateral surface of a circular cylinder with radius:

• Sphere with radius:

For another surface normal vector with the surface element to be determined. The area is given approximately by, and thus the normal vector of the same.

We now project on along. Then the surface elements can be linked by means of:

It should each cut straight along the normal vectors of the surface only once. Otherwise you have to divide the area into smaller areas whose projection is then clearly, or choose a different base.

The vectorial surface element is:

Example 1

Is given a surface of the mold, and thus the following applies:

This area is now with projected into the plane, and; is

Example 2

Wanted is the surface element of a body of revolution about the axis, ie.

By projecting onto the generated surface of a circular cylinder with radius is obtained, the planar element:

The integrals

With the parameterizations and the surface elements can now define the surface integrals. This multi-dimensional integrals are Lebesgue integrals, but can be calculated in most applications as multiple Riemann integrals.

The scalar surface integral

The scalar surface integral of a scalar function on a surface with regular parametrization with is defined as

If, for example, the scalar surface integral is simply the area of ​​the surface.

The vectorial surface integral

The vectorial surface integral of a vector-valued function on a surface with regular parametrization with is defined as

An intuitive representation of this integral is done over the flow of a vector field through the surface: The size indicates, provides the contribution to the total flow of infinitesimally - small surface vector; flows namely how much of the surface by piece. The flow is at a maximum when the field vector is parallel to the surface normal, and zero when is perpendicular to, that is tangential to the surface - then " flowing " along the surface, but not through them.