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

# IllustrationSurface integral with orthogonal surface vector + vector field

Share — copy and redistribute the material in any medium or format

Adapt — remix, transform, and build upon the material for any purpose, even commercially.

Sharing and adapting of the illustration is allowed with indication of the link to the illustration.

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.