Definition:Contour

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Definition

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

Let $C_1, \ldots, C_n$ be directed smooth curves in $\R^n$.

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

For each $i \in \left\{ {1, \ldots, n-1}\right\}$, let the end point of $\rho_i$ equal the start point of $\rho_{i + 1}$:

$\rho_i \left({b_i}\right) = \rho_{i + 1} \left({a_{i + 1} }\right)$


Then the finite sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ is called a contour (in $\R^n$).


If $C_1, \ldots, C_n$ are defined only by their parameterizations $\rho_1, \ldots, \rho_n$, then the contour can be denoted by the same symbol $\rho$.


Parameterization

Let $C_1, \ldots, C_n$ be 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 $i \in \set {1, \ldots, n}$.

Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$.


The parameterization of $C$ is defined as the function $\rho: \closedint {a_1} {c_n} \to \R^n$ with:

$\map {\rho \restriction_{\closedint {c_i} {c_{i + 1} } } } t = \map {\rho_i} t$

where:

$\ds c_i = a_1 + \sum_{j \mathop = 1}^i b_j - \sum_{j \mathop = 1}^i a_j$ for $i \in \set {0, \ldots, n}$
$\rho \restriction_{\closedint {c_i} {c_{i + 1} } }$ denotes the restriction of $\rho$ to $\closedint {c_i} {c_{i + 1} }$.


Closed Contour

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 $C$:

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


Simple Contour

Let $C_1, \ldots, C_n$ be 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 $i \in \set {1, 2, \ldots, n}$.

Let $C$ be the contour in $\R^n$ defined by the finite sequence $C_1, \ldots, C_n$.


$C$ is a simple contour if and only if:

$(1): \quad$ For all $i, j \in \set {1, \ldots, n}, t_1 \in \hointr {a_i} {b_i}, t_2 \in \hointr {a_j} {b_j}$ with $t_1 \ne t_2$, we have $\map {\rho_i} {t_1} \ne \map {\rho_j} {t_2}$
$(2): \quad$ For all $k \in \set {1, \ldots, n}, t \in \hointr {a_k} {b_k}$ where either $k \ne 1$ or $t \ne a_1$, we have $\map {\rho_k} t \ne \map {\rho_n} {b_n}$.


Length

Let $C$ be a contour in $\R^n$ 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:

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


Image

Let $C$ be a contour in $\R^n$ 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 image of $C$ is defined as:

$\ds \Img C := \bigcup_{i \mathop = 1}^n \Img {\rho_i}$

where $\Img {\rho_i}$ denotes the image of $\rho_i$.


Endpoints

Let $C_1, \ldots, C_n$ be 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 $i \in \left\{ {1, \ldots, n}\right\}$.

Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$.


The start point of $C$ is $\rho_1 \left({a_1}\right)$.

The end point of $C$ is $\rho_n \left({b_n}\right)$.


Collectively, $\rho_1 \left({a_1}\right)$ and $\rho_n \left({b_n}\right)$ are referred to as the endpoints of $C$.


Complex Plane

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


Let $C_1, \ldots, C_n$ be directed smooth curves in the complex plane $\C$.

For each $k \in \set{ 1, \ldots, n}$, let $C_k$ be parameterized by the smooth path $\gamma_k: \closedint {a_k}{b_k} \to \C$.

For each $k \in \set{ 1, \ldots, n-1}$, let the endpoint of $\gamma_k$ equal the start point of $\gamma_{k + 1}$:

$\map {\gamma_k}{b_k} = \map {\gamma_{k + 1} }{a_{k + 1} }$


Then the finite sequence $\sequence{C_1, \ldots, C_n}$ is a contour.


Also see