PX275 - D4b - the vector surface area and element

suppose $d \vec{s}$ is the element of the vector surface area, $\vec{S}$ , such that:

d \vec{s} = \hat{n} d A

where, $\hat{n}$ is the normal unit vector, and $d A$ is the scalar area element

the total flux is:

total flux = \iint_{A} \vec{F} \cdot d \vec{s} = \iint_{A} \vec{F} \cdot \hat{n} d A

sometimes $\hat{n} d A = d \vec{A}$

cartesian flat surface

cylindrical coordinates

spherical coordiantes