English: Diagram of a vector field ${\displaystyle \mathbf {F} (\mathbf {x} )}$ passing through a small rectangular portion of a surface ${\displaystyle dS}$. The surface integral or "flux" of the field through the surface is equal to the normal (perpendicular) component of the vector field ${\displaystyle F_{\text{n}}(\mathbf {x} )=F\cos \theta }$ through the surface, multiplied by the area ${\displaystyle dS}$. The normal component can be calculated by taking the dot product of the field with the unit normal vector at that point ${\displaystyle \mathbf {n} (\mathbf {x} )}$, so the flux through the surface is equal to

${\displaystyle \mathbf {F} (\mathbf {x} )\cdot \mathbf {n} dS}$

To find the surface integral of the field through a surface ${\displaystyle S}$, this is integrated over the surface

${\displaystyle \int _{S}\mathbf {F} \cdot \mathbf {n} dS}$