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

# IllustrationClosed 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: $$\boldsymbol{F}_{\perp} \cdot \text{d}\boldsymbol{l} = 0$$.