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Level 3
Level 3 requires the basics of vector calculus, differential and integral calculus. Suitable for undergraduates and high school students.

Illustration 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: \(\boldsymbol{F}_{\perp} \cdot \text{d}\boldsymbol{l} = 0 \).

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  • License: CC BY 4.0This illustration may be used with indication of the copyright!
  • Copyright: © 2020
  • This illustration was uploaded by FufaeV on .
  • This illustration was updated by FufaeV on .