Illustration Surface integral with orthogonal surface vector + vector field

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.