Mr. Integral
Math notes/Calculus III/Vector Calculus

Oriented Surface Flux — Cheat sheet

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Core rule

∬SF⋅n dS=∬DF(r)⋅(ru×rv)du dv.\iint_S\mathbf F\cdot\mathbf n\,dS=\iint_D\mathbf F(\mathbf r)\cdot(\mathbf r_u\times\mathbf r_v)du\,dv.

Watch for

Check orientation before integrating. The cross product already includes the area scale.