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.