Contour Integral/Examples
Examples of Contour Integrals
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:
- $\ds \int_\Gamma \mathbf E \cdot \d \mathbf l$