  ## Encyclopedia > Rectifiable curve

Article Content

# Rectifiable curve

A rectifiable curve is a curve which has a well-defined finite length. Rectifiable curves are mainly important in complex analysis because they are needed to define the path integral.

Suppose γ : [a, b] -> C is a continuous function from an interval into the complex plane. This curve γ is called rectifiable if the following supremum is finite:

sup {∑i=1n |γ(ti)-γ(ti-1)| : n in N and at0<t1<...<tnb}
The value of this supremum is called the length of the curve γ.

In an analogous manner (by replacing the absolute value with the Euclidean distance or a norm), one can define rectifiable curves γ : [a, b] -> Rn and, more generally, γ : [a, b] -> V where V is a normed vector space.

Every continuous and piecewise continuously differentiable[?] curve γ is rectifiable, and its length can be computed as the ordinary Riemann integral

L = ∫ab |γ'(t)| dt

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 DB ... Decibel; see Bel DB is a French automobile maker; see DB (car) DB is the abbreviation for Deutsche Bahn, the major German railway company DB is the abbreviation of ...  