Peanos surface

Model of the Peanos surface in the Dresden model collection

In mathematics, the Peano area is the graph of the function

${\displaystyle f(x,y)=(2x^{2}-y)(y-x^{2})\ .}$

It was given by Giuseppe Peano in 1899 as a counterexample to a conjecture for the existence of a local maximum / minimum of a function of two variables.

This surface was referred to by Georg Scheffers in his textbook on descriptive geometry in 1920 as the surface of Peano . It is also called the Peano saddle .

The presumption to be rebutted

If the sections of the graph of a function with planes through the axis all have a local maximum at that point , then the function also has a local maximum at that point . ${\displaystyle f(x,y)}$ ${\displaystyle z}$ ${\displaystyle (0,0)}$ ${\displaystyle f}$ ${\displaystyle (0,0)}$

The area of ​​Peano shows: This assumption is wrong. It is sufficient to show that the following applies to the function : ${\displaystyle f(x,y)=(2x^{2}-y)(y-x^{2})}$

1. Each intersection curve of the surface with a plane through the axis has a local maximum at the point . ${\displaystyle z}$ ${\displaystyle (0,0,0)}$
2. In any environment of has both positive and negative values. ${\displaystyle (0,0)}$ ${\displaystyle f}$

Peano surface

properties of ${\displaystyle f}$

The function has the following properties: ${\displaystyle f(x,y)=(2x^{2}-y)(y-x^{2})}$

1. ${\displaystyle f(x,y)=0\;}$on the parables and . ${\displaystyle y=2x^{2}}$ ${\displaystyle y=x^{2}}$
2. ${\displaystyle f(x,y)>0\;}$between these parabolas, i.e. for (pink in the picture) and ${\displaystyle x^{2}<y<2x^{2}}$
3. ${\displaystyle f(x,y)<0\;}$ otherwise (light blue in the picture).
4. If one restricts through with , it is easy to check that every such restriction has a local maximum at the zero point . ${\displaystyle f}$ ${\displaystyle ax+by=0}$ ${\displaystyle a^{2}+b^{2}>0}$ ${\displaystyle f}$ ${\displaystyle (0,0,0)}$
5. The function values ​​along the parabola (red in the picture) are positive  (!) Outside the zero point . so has a saddle point at the zero point . (Any curve running between the parabolas can be used for this consideration.) ${\displaystyle y={\sqrt {2}}x^{2}}$ ${\displaystyle f}$

The usual saddle point test with the determinant of the Hessian matrix does not give a result because the determinant is 0.

Further models of the Peanos surface

• University of Dresden
• University collections

literature

1. ↑ Peano's surface. In: math.tu-dresden.de. Retrieved August 2, 2020 . 
2. ^ Arnold Emch: A model for the Peano Surface. In: American Mathematical Monthly. 29, No. 10, 1922, pp. 388-391.
3. ↑ Angelo Genocchi, Giuseppe Peano (ed.): Differential calculus and basic features of integral calculus. BG Teubner, 1899, p. 332.
4. ↑ Kuno Fladt: Analytical Geometry of Special Surfaces and Space Curves , Springer-Verlag, 2013, p. 197.
5. ↑ Georg Scheffers: Textbook of the descriptive geometry. Volume II, 1920, pp. 261-263.
6. ↑ SN Krivoshapko, VN Ivanov: Encyclopedia of Analytical Surfaces. Springer, 2015. See in particular the section Peano Saddle, pp. 562–563.
7. ↑ George K. Francis: A Topological Picturebook. Springer-Verlag, New York 1987, ISBN 0-387-96426-6 , p. 88.
8. ^ Kurt Meyberg, Peter Vachenauer: Höhere Mathematik 1. Springer-Verlag, 1995, ISBN 3-540-59188-5 , p. 403.