English: Diagram illustrating how a surface integral of a vector field over a surface is defined. It shows an arbitrary surface S with a vector field F , (red arrows) passing through it. The surface is divided into small (infinitesimal) regions dS . The surface integral is the sum of the perpendicular component of the field passing through each region multiplied by the area dS . The perpendicular component of the field is equal to the dot product of the field F(x) with the unit normal vector n(x) at the point dS :

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