Notation for surface integrals over vector fields

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Why do every question I see for a surface integral over a vector field $F$ under the surface $S$ be denoted by $$\int_{S} F \cdot n dS$$? As in, why does it only have a single integral? Where as once we expand the $dS$ to be $dudv$ (our parametric variables), we have a double integral? Shouldn't we have a double integral in the first place?