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When performing surface integrals, the equations for the surface integrals have the normal vector encoded into the formula, as either the field F dotted with the normal vector (in vector surface integrals) or the field F multiplied by the magnitude of the normal vector (in scalar surface integrals).

Where and how do these normal vectors show in in Stoke's theorem (and by association, Green's and Gauss' theorem). In other words in Stoke's theorem, what is the d$\Sigma$? I understand it is the differential element of the surface, but how does the normal vector that we see in the general surface integral formula get included in there? And when it goes to the right side of the equation, why is it changing to dr?


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