Definition:Contour Integral
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 $P$.
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:
- $\ds \int_\Gamma \mathbf E \cdot \d \mathbf l$
Also see
- Results about contour integrals can be found here.
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {II}$: The Products of Vectors: $3$. Line and Surface Integrals: $(2.11)$