Smooth curve

smooth curve

piecewise smooth curve

A smooth curve is a continuously differentiable parameterized curve (here path ) with a non-vanishing derivative . This clearly means that the path does not stop at any point when running through the parameter or that the direction changes abruptly.

A curve in general is smooth if at least one smooth path has the curve to the image.

In contrast to these definitions, a smooth mapping must be infinitely differentiable.

Formal definition

Let be a curve in the parametric representation ${\displaystyle K}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle f:t\mapsto (f_{1}(t),\ldots ,f_{n}(t))}$

• ${\displaystyle K}$is called smooth if for are continuously differentiable and holds for all . ${\displaystyle f_{i}}$ ${\displaystyle i=1,\ldots ,n}$ ${\displaystyle [a,b]}$ ${\displaystyle (f_{1}^{\prime }(t),\ldots ,f_{n}^{\prime }(t))\neq (0,\ldots ,0)}$ ${\displaystyle t\in [a,b]}$
• ${\displaystyle K}$is called piecewise smooth if there is a partition of such that on every interval for is smooth. ${\displaystyle P={t_{0},\ldots ,t_{m}}}$ ${\displaystyle [a,b]}$ ${\displaystyle K}$ ${\displaystyle [t_{k-1},t_{k}]}$ ${\displaystyle k=1,\ldots ,m}$

literature

• Kurt Endl, Wolfgang Luh : Analysis. An integrated representation; Study book for students of mathematics, physics and other natural sciences from the 1st semester. Volume 2. 7th revised edition. Aula-Verlag, Wiesbaden 1989, ISBN 3-89104-455-0 , p. 343.
• Harro Heuser : Textbook of Analysis. Part 2. 5th revised edition. Teubner, Stuttgart 1990, ISBN 3-519-42222-0 , p. 365.

Web links

• piecewise smooth at PlanetMath