Line Integral/Examples

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Examples of Line 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 curve 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 line 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 curve 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 line 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 curve 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 line 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 curve 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 line integral:

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