Duality in coordinates
Let us introduce some coordinates 
on 
.A generic 
-dimensional submanifold can be defined by the following equation,
depending on an odd parameter 
and an even parameter 
:
-dimensional submanifold can be defined by the following equation,
depending on an odd parameter and an even parameter :
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We conclude that 
may be
coordinatized with and .
may be
coordinatized with and .Since 
is a functor, a vector field on should automatically define
some vector field on 
.
Theorem 2: The following vector field 
on :
is a functor, a vector field on should automatically define
some vector field on .
Theorem 2: The following vector field on :
defines the following vector field on 
:
:
where 
is defined as follows:
is defined as follows:
Proof:
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