Electric flux

Flux describes the amount of a given quantity that passes through a given area. For an electric field $\vec{E}$ that is constant across an area $\vec{A}$, the electric flux is $\Phi_E = \vec{E} \cdot \vec{A}$, which is $EA\cos\theta$ with $\theta$ the angle between the field and the area vector. In general the AP Physics C: E&M sheet prints it as a surface integral:

Two conventions do most of the work. The area vector is perpendicular to the plane of the surface, and outward for a closed surface. The sign of the flux is then given by that dot product, so field leaving a closed surface counts positive and field entering counts negative.

What follows is worth holding on to. A field lying in the plane of a surface gives zero flux, since field and area vector are at right angles. A uniform field through a closed surface gives zero net flux, because whatever enters leaves. The unit is $\text{N} \cdot \text{m}^2/\text{C}$.

Flux is the input to Gauss's law, which sets the net flux through a closed surface equal to the enclosed charge divided by $\varepsilon_0$. The Gauss's law guide does the applying, over a Gaussian surface chosen for symmetry.

This is AP Physics C: Electricity and Magnetism material, Topic 8.5. The AP Physics 2 sheet prints magnetic flux but no electric flux.