Regularitätskurven

Regularitätskurven, also known as regularity curves, are a concept in mathematics, particularly in differential geometry and calculus of variations. They are curves that possess a certain degree of smoothness or differentiability. In simpler terms, a regularitätskurve is a curve where the tangent vector exists and is non-zero at every point. This ensures that the curve does not have any sharp corners, cusps, or self-intersections locally.

The formal definition of a regularitätskurve is a differentiable curve $\gamma: I \to \mathbb{R}^n$, where $I$ is an interval, such that its derivative $\gamma'(t)$ is non-zero for all $t \in I$. The derivative $\gamma'(t)$ represents the tangent vector to the curve at the point $\gamma(t)$. The condition that $\gamma'(t) \neq 0$ is crucial. If $\gamma'(t) = 0$ at some point, the curve might be "stopping" or changing direction abruptly, leading to a lack of regularity.

Regularitätskurven are fundamental in many areas of mathematics. For instance, in the calculus of variations, problems often involve finding curves that minimize or maximize certain functionals. The assumption that the curves in question are regularitätskurven simplifies the analysis and ensures the existence of necessary conditions for extremality, such as the Euler-Lagrange equations. In differential geometry, the concept of curvature and torsion, which describe the bending and twisting of a curve, are well-defined for regularitätskurven. Furthermore, arc length parametrization, a convenient way to describe curves, can be achieved for regularitätskurven.