Integral permukaan
caption: Definisi permukaan integral bergantung pada pemisahan permukaan menjadi elemen permukaan kecil.
caption: Ilustrasi elemen permukaan tunggal. Elemen-elemen ini dibuat sangat kecil, dengan proses pembatas, sehingga mendekati permukaan.
caption: A curved surface S


{\displaystyle S} with a vector field F



{\displaystyle \mathbf {F} } passing through it. The red arrows (vectors) represent the magnitude and direction of the field at various points on the surface
caption: Surface divided into small patches d
S
=
d
u
d
v


{\displaystyle dS=dudv} by a parameterization of the surface
caption: The flux through each patch is equal to the normal (perpendicular) component of the field F

n


(

x

)
=
F
(

x

)
cos
⁡
θ


{\displaystyle F_{n}(\mathbf {x} )=F(\mathbf {x} )\cos \theta } at the patch's location x



{\displaystyle \mathbf {x} } multiplied by the area d
S


{\displaystyle dS} . The normal component is equal to the dot product of F

(

x

)


{\displaystyle \mathbf {F} (\mathbf {x} )} with the unit normal vector n

(

x

)


{\displaystyle \mathbf {n} (\mathbf {x} )} (blue arrows)
caption: The total flux through the surface is found by adding up F

⋅

n


d
S


{\displaystyle \mathbf {F} \cdot \mathbf {n} \;dS} for each patch.  In the limit as the patches become infinitesimally small, this is the surface integral ∫

S



F
⋅
n


d
S


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