Let be a closed bounded region in whose boundary is an orientable surface . Let
be a continuously differentiable vector-field in an open set containing the region . Then
where is the outward normal to the surface .
Proo f :
(For Simple regions )
We shall assume that the region has the property that any straight line parallel to any one of the coordinate axes intersects at most in one line segment or a single point. For such a region , we have to show that
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Let the outward normal at any point on have direction cosines and i.e., let
be a closed bounded region in 
whose boundary is an orientable surface 
. Let 
. Then 
is the outward normal to the surface 
. 
)
has the property that any straight line parallel to any one of the coordinate axes intersects 
at most in one line segment or a single point. For such a region 
, we have to show that 
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at any point on 
have direction cosines 
and 
i.e., let 
is same as proving: 
