# Definition:Contour/Closed

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## Definition

Let $\R^n$ be a real cartesian space of $n$ dimensions.

Let $C$ be the **contour** in $\R^n$ defined by the (finite) sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ of directed smooth curves in $\R^n$.

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

$C$ is a **closed contour** if and only if the start point of $C$ is equal to the end point of $C$:

- $\rho_1 \left({a_1}\right) = \rho_n \left({b_n}\right)$

### Complex Plane

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

$C$ is a **closed contour** if and only if the start point of $C$ is equal to the end point of $C$:

- $\gamma_1 \left({a_1}\right) = \gamma_n \left({b_n}\right)$

## Also see

- Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints, from which it follows that this definition is independent of the parameterizations of $C_1$ and $C_n$.