Regular area
A regular surface or differentiable surface or short surface is a mathematical object from the differential geometry . With the help of this term, the commonly used term area is precisely defined in a mathematical context. The following definition clearly means that pieces of a plane are deformed and stapled together in such a way that no corners or edges are created, so that a tangential plane can be created at every point of the resulting structure . In contrast to the topological surface , a derivation of a mapping can be explained on the regular surface - due to the existence of a tangential plane .
definition
There are different but equivalent methods of defining a regular surface. In elementary differential geometry , a regular surface is defined by a parameterization. In differential topology , a more abstract sub-area of differential geometry, the regular surfaces are two-dimensional special cases of n-dimensional differentiable manifolds.
Through parameterizations
A subset is called a regular area if for each there is a neighborhood , an open set and a mapping such that
- the mapping is a homeomorphism . So it is continuous, bijective and has a continuous inverse function .
- the mapping is continuously differentiable.
- for every point the differential has full rank , i.e. is injective .
The figure is called parameterization . The third requirement ensures that a tangential plane can be attached to every point on the surface .
As a two-dimensional manifold
Alternatively, a regular surface can also be understood as a topological surface with a differentiable structure . In particular, a regular surface is a two-dimensional, differentiable submanifold .
Examples
Regular surfaces
Examples of regular surfaces are the 2- sphere , the ellipsoid , the hyperboloid and the torus . The torus and the 2-sphere (spherical surface) will be discussed in more detail in a moment. The proof that these objects are regular surfaces can often be simply given with the theorem of regular values from differential geometry. In particular, every two-dimensional, differentiable manifold is a regular surface.
Concrete parameterizations
Parameterizations play an important role in relation to surface integrals . If an area can also be described by a differentiable function , one receives with
a parameterization and the area is regular. However, in this way you can only parameterize areas for which you do not have to assign more than one z-value to a pair . The following two examples, which are often used, cannot be represented in this way if you only want to use one parameterization mapping.
Bullet
By the figure , which by
is given, a curve parameterization of the circular line of a semicircle in the right half-plane with radius and center zero is obtained, as the equation shows.
With the help of this curve parameterization, the parameterization of a sphere is obtained , which is determined by the function with
is described. The fact that the properties required from the definition apply can be found under spherical coordinates . However, one must note that this parameterization "forgets" the points and . It is not possible to describe a complete sphere with a global parameterization. At least two images are required for this.
This parameterization is clearly obtained by starting at any point on the sphere and revolving it on a semicircle and at each point that is reached, revolving the sphere once completely in the direction perpendicular to it. You can also show equality here .
Torus
Be . The parameterization of the circular line of a circle with radius and center is similar to the one above
With the help of this curve parameterization, the parameterization of a torus is given by
can be described. This clearly means that a torus is created when you take a circle with a center and turn it around the axis around the zero point.
Graphs of differentiable functions
As already mentioned in the examples, the graph of a differentiable function is always a regular surface. The graph of the function
is parameterized by the figure
The example of the spherical shell shows that the reverse does not apply. However, locally there is a reversal of the statement. Let be a regular surface and a point. Then there exists a neighborhood of p such that it is the graph of a differentiable function which has the form .
literature
- Konrad Königsberger : Analysis. Volume 2. 3rd revised edition. Springer, Berlin et al. 2000, ISBN 3-540-66902-7 .
- Manfredo P. do Carmo: Differential geometry of curves and surfaces. Prentice-Hall, Englewood Cliffs NJ 1976, ISBN 0-13-212589-7 .
Individual evidence
- ↑ Volkmar Wünsch: Differential Geometry. Curves and surfaces. BG Teuber Verlagsgesellschaft, Stuttgart / Leipzig 1997, ISBN 3-8154-2095-4 , p. 102.