fordulatszámok

A fordulatszámok, or rotation numbers, are a concept in mathematics, particularly in the study of vector fields and differential geometry. They quantify the number of times a curve winds around a given point or object. Essentially, a fordulatszám measures the net number of full turns a tangent vector to a curve makes as you traverse the curve.

The simplest case is a closed curve in a plane. If the curve encloses a point, the fordulatszám with respect to that point is an integer. If the curve winds counterclockwise around the point, the fordulatszám is positive; if it winds clockwise, it is negative. A curve that does not enclose the point has a fordulatszám of zero.

This concept extends to higher dimensions and more complex scenarios. For instance, in the context of vector fields, the fordulatszám can describe the behavior of the field lines around singularities. It's a powerful tool for understanding topological properties and the global behavior of mathematical objects. The calculation often involves integration along the curve, and the result is always an integer, reflecting the discrete nature of "full turns." The theory of fordulatszámok plays a role in areas like dynamical systems and the study of differential equations.