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IllustrationSurface integral with orthogonal surface vector + vector field

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Visualizing the surface integral of a vector field $$\boldsymbol{F}$$ within a surface $$A$$:$\int_A \boldsymbol{F} \cdot \text{d}\boldsymbol{a}$where $$\text{d}\boldsymbol{a}$$ is an infinitesimal surface element of the surface $$A$$.

The vector field $$\boldsymbol{F}$$ can be divided into component $$\text{d}\boldsymbol{F}_{||}$$ parallel to $$\text{d}\boldsymbol{a}$$ and into component $$\boldsymbol{F}_{\perp}$$ perpendicular to $$\text{d}\boldsymbol{a}$$. The scalar product $$\boldsymbol{F}_{\perp} \cdot \text{d}\boldsymbol{a} = 0$$ with the vertical component does not contribute to the surface integral.