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Illustration Closed line integral of a vector field

Closed line integral of a vector field
Closed line integral of a vector field
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A vector field \(\boldsymbol{F}\) is integrated along a line:\[ \oint_L \boldsymbol{F} \cdot \text{d}\boldsymbol{l}\]where \(\text{d}\boldsymbol{l}\) is an infinitesimal line element along the considered line \(L\), whose beginning and end are connected (closed line integral).

In order to illustrate the scalar product in the integral, the vector field \(\boldsymbol{F}\) is divided into component \(\boldsymbol{F}_{||}\) parallel to \(\text{d}\boldsymbol{l}\) and into component \(\boldsymbol{F}_{\perp}\) orthogonal to \(\text{d}\boldsymbol{l}\). The scalar product with the orthogonal component does not contribute to the line integral: \(\boldsymbol{F}_{\perp} \cdot \text{d}\boldsymbol{l} = 0 \).