The total flux through the surface is found by adding up F

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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}