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 ${\displaystyle S\subset \mathbb {R} ^{3}}$ ${\displaystyle p\in S}$ ${\displaystyle V\subset \mathbb {R} ^{3}}$ ${\displaystyle U\subset \mathbb {R} ^{2}}$ ${\displaystyle \phi \colon U\to V\cap S}$

• the mapping is a homeomorphism . So it is continuous, bijective and has a continuous inverse function . ${\displaystyle \phi }$
• the mapping is continuously differentiable. ${\displaystyle \phi }$
• for every point the differential has full rank , i.e. is injective . ${\displaystyle q\in U}$ ${\displaystyle {\tfrac {\mathrm {d} \phi }{\mathrm {d} x}}(q)\colon \mathbb {R} ^{2}\to \mathbb {R} ^{3}}$

The figure is called parameterization . The third requirement ensures that a tangential plane can be attached to every point on the surface . ${\displaystyle \phi }$

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 ${\displaystyle S\subset \mathbb {R} ^{3}}$ ${\displaystyle f(x,y)=z}$

${\displaystyle \phi (x,y)={\begin{pmatrix}x\\y\\f(x,y)\end{pmatrix}}}$

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. ${\displaystyle (x,y)}$

Bullet

By the figure , which by ${\displaystyle f\colon \left]-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right[\to \mathbb {R} ^{2}}$

${\displaystyle f(\phi )={\begin{pmatrix}r\cdot \cos(\phi )\\r\cdot \sin(\phi )\end{pmatrix}}}$

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. ${\displaystyle r}$ ${\displaystyle r=\left|f(\phi )\right|={\sqrt {r^{2}(\cos ^{2}(\phi )+\sin ^{2}(\phi ))}}}$

With the help of this curve parameterization, the parameterization of a sphere is obtained , which is determined by the function with ${\displaystyle g\colon \left]-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right[\times \left[0,2\pi \right[\to \mathbb {R} ^{3}}$

${\displaystyle g(\phi ,\psi )={\begin{pmatrix}r\cos(\phi )\cos(\psi )\\r\cos(\phi )\sin(\psi )\\r\sin(\phi )\end{pmatrix}}}$

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. ${\displaystyle g}$ ${\displaystyle (0,0,1)}$ ${\displaystyle (0,0,-1)}$

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 . ${\displaystyle r=|f(\phi ,\psi )|}$

Torus

Torus

Be . The parameterization of the circular line of a circle with radius and center is similar to the one above ${\displaystyle r<R}$ ${\displaystyle f\colon \left]-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right[\to \mathbb {R} ^{2}}$ ${\displaystyle r}$ ${\displaystyle (R,0)}$

${\displaystyle f(\phi )={\begin{pmatrix}R+r\cos(\phi )\\0+r\sin(\phi )\end{pmatrix}}.}$

With the help of this curve parameterization, the parameterization of a torus is given by ${\displaystyle g\colon \left[0,2\pi \right[\times \left[0,2\pi \right[\to \mathbb {R} ^{3}}$

${\displaystyle g(\phi ,\psi )={\begin{pmatrix}(R+r\cos(\phi ))\cos(\psi )\\(R+r\cos(\phi ))\sin(\psi )\\0+r\sin(\phi )\end{pmatrix}}}$

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. ${\displaystyle (R,0)}$ ${\displaystyle x_{3}}$

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

${\displaystyle (x,y)\mapsto f(x,y)}$

is parameterized by the figure

${\displaystyle (x,y)\mapsto (x,y,f(x,y)).}$

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 . ${\displaystyle S\subset \mathbb {R} ^{3}}$ ${\displaystyle p\in S}$ ${\displaystyle V\subset S}$ ${\displaystyle V}$ ${\displaystyle z=f(x,y)}$

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

1. ↑ Volkmar Wünsch: Differential Geometry. Curves and surfaces. BG Teuber Verlagsgesellschaft, Stuttgart / Leipzig 1997, ISBN 3-8154-2095-4 , p. 102.