Lecture50 : Application of surface integrals [Section 50.2]
In order to be able to do so, it becomes necessary to ensure that the function
is integrable over For example, this will be so if is continuous. For this, we can assume that is continuous. Thus, to be able to define , we should be able to say that our surface is such that at every point , three exist unit normal which varies continuously as very over . This motivates our next definition:
50.2.3
Definition :
A surface is said to be orientable if there exists a continuous vector-field
such that is the unit normal vector to at .
Orientability of a surface essentially means that there are two sides of the surface.
50.2.4
Examples :
(1)
Every simple closed surface is orientable, we can have a continuous inward or an outward normal to the surface.
For example, surfaces like sphere, ellipsoid, etc, are all orientable, with a continuous normal pointing in the region enclosed or pointing away from the region enclosed.
For example, this will be so if 
is continuous. For this, we can assume that 
is continuous. Thus, to be able to define 
, we should be able to say that our surface 
is such that at every point 
, three exist unit normal 
which varies continuously as 
very over 
. This motivates our next definition: 
is said to be orientable if there exists a continuous vector-field 
is the unit normal vector to 
at 
. 
