Module 1 : A Crash Course in Vectors
Lecture 1 : Scalar And Vector Fields
Area as a Vector Quantity
  The magnitude of the vector also happens to be the area of the parallelogram formed by the vectors $\vec A$ and $\vec B$. The fact that a direction could be uniquely associated with a cross product whose magnitude is equal to an area enables us to associate a vector with an area element. The direction of the area element is taken to be the outward normal to the area. (This assumes that we are dealing with one sided surfaces and not two sided ones like a Möbius strip.
  For an arbitrary area one has to split the area into small area elements and sum (integrate) over such elemental area vectors
 
\begin{displaymath}\vec S = \int d\vec S\end{displaymath}
  A closed surface has zero surface area because corresponding to an area element $d\vec S$, there is an area element $-d\vec S$ which is oppositely directed.
 
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