\begin{definition}[Definition:Line Integral]
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 $P$.
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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:
:$\ds \int_O^A \mathbf v \cdot \d \mathbf l = \int_O^A v \cos \theta \rd l$
\end{definition}