Definition:Contour Integral

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Definition

Let $OA$ be a curve in a vector field $\mathbf F$.

Let $P$ be a point on $OA$.

Let $\d \mathbf l$ be a small element of length of $OA$ at $P$.

Let $\mathbf v$ be the vector induced by $\mathbf F$ on $P$.

Let $\mathbf v$ make an angle $\theta$ with the tangent to $OA$ at $P$.


Line-integral.png


Hence:

$\mathbf v \cdot \d \mathbf l = v \cos \theta \rd l$

where:

$\cdot$ denotes dot product
$v$ and $\d l$ denote the magnitude of $\mathbf v$ and $\d \mathbf l$ respectively.


The contour integral of $\mathbf v$ along $OA$ is therefore defined as:

$\ds \int_O^A \mathbf v \cdot \d \mathbf l = \int_O^A v \cos \theta \rd l$


Complex Plane

Let $C$ be a contour defined by a finite sequence $C_1, \ldots, C_n$ of directed smooth curves in the complex plane $\C$.

Let $C_k$ be parameterized by the smooth path:

$\gamma_k: \closedint {a_k} {b_k} \to \C$

for all $k \in \set {1, \ldots, n}$.

Let $f: \Img C \to \C$ be a continuous complex function, where $\Img C$ denotes the image of $C$.


The contour integral of $f$ along $C$ is defined by:

$\ds \int_C \map f z \rd z = \sum_{k \mathop = 1}^n \int_{a_k}^{b_k} \map f {\map {\gamma_k} t} \map {\gamma_k'} t \rd t$


Also known as

A contour integral is called a line integral or a curve integral in many texts.


Examples

Work Done

Let $\mathbf F$ be a force acting as a point-function giving rise to a vector field $\mathbf V$.

Let $OA$ be a contour in $\mathbf V$ along which a particle $P$ is moved by $\mathbf F$.

Let $\d \mathbf l$ be a small element of length of $OA$ at $P$.

Then the work done by $\mathbf F$ moving $P$ from $O$ to $A$ is given by the contour integral:

$\ds \int_O^A \mathbf F \cdot \d \mathbf l$


Potential Difference

Let $\mathbf E$ be an electric field acting over a region of space $R$.

Let $OA$ be a contour in $R$.

Let $\d \mathbf l$ be a small element of length of $OA$ at a point $P$.

Then the potential difference between $O$ to $A$ is given by the contour integral:

$\ds \int_O^A \mathbf E \cdot \d \mathbf l$


Circulation of Fluid

Let $\mathbf v$ be the velocity within a body $B$ of fluid as a point-function.

Let $\Gamma$ be a closed contour in $B$.

Let $\d \mathbf l$ be a small element of length of $\Gamma$ at a point $P$.

Then the circulation of $B$ over $\Gamma$ is given by the contour integral:

$\ds \int_\Gamma \mathbf v \cdot \d \mathbf l$


Electromotive Force

Let $\mathbf E$ be an electromagnetic field acting over a region of space $R$.

Let $\Gamma$ be a closed contour in $R$.

Let $\d \mathbf l$ be a small element of length of $\Gamma$ at a point $P$.

Then the electromotive force in $\Gamma$ is given by the contour integral:

$\ds \int_\Gamma \mathbf E \cdot \d \mathbf l$


Sources