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 $P$ by $\mathbf F$.

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 line integral of $\mathbf v$ along $OA$ is therefore defined as:
 * $\ds \int_O^A \mathbf v \cdot \d \mathbf l = \int_O^A v \cos \theta \rd l$