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?
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Some people do write it with a double integral sign ($\iint$). But many people (including myself) find it more elegant to use the single integral sign ($\int$). But you're right - whenever we evaluate a surface integral, we do need to perform a double integral. – Kenny Wong May 11 '17 at 00:52
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In some texts, it is the convention to use a double integral sign for surface integrals, as in $$ \iint_S F \cdot n , dS $$ – Ben Grossmann May 11 '17 at 00:53
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In a sense, we can think of every integral as really being a "single integral" with respect to some measure over some space. – Ben Grossmann May 11 '17 at 00:54