Definition:Contour/Length

Definition
Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $C$ be a contour in $C$ defined by the (finite) sequence $\sequence {C_1, \ldots, C_n}$ of directed smooth curves in $\R^n$.

Let $C_i$ be parameterized by the smooth path $\rho_i: \closedint {a_i} {b_i} \to \R^n$ for all $i \in \set {1, \ldots, n}$.

The length of $C$ is defined as:


 * $\ds \map L C := \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \size {\map {\rho_i'} t} \rd t$

Complex Plane
The definition carries over to the complex plane, in which context it is usually applied:

Also see

 * Length of Contour is Well-Defined: $\map L C$ is defined and independent of the parameterizations of $C_1, \ldots, C_n$.