dual(u)
 
oriented volume
 
for a given vector space the highest grade element is unique. The outer product of n vectors is therefore a multiple of the unique oriented volume for Cln.                                
 
 Cl3 :
 e123 = e1 ^ e2 ^ e3    represent an oriented space.
 e123  commute with all elements, is called a pseudoscalar.
 e123² = -1
 e123  represent an oriented unit volume (e1 -> e2 ->  e3)
-e123  represent the opposite orientation
 e123  commutes with all elements of the algebra (as a scalar)    
 
we use also the notation j = e123
any bivector can be expressed as an imaginary vector, called a pseudovector .
ex :
e12 = e1e2(e1e2e3) / j = j e3
the center of Cl3 (the part that commutes with all elements) is spanned by {e0, j} and  is identified with the complex field.
Every elements of Cl3 is a linear combination of e0, e1, e2, e3, e12, e23, e13, j over the reals or, equivalently, of e0, e1,  e2, e3 over the complex field.
 
Cl1 --> e1
Cl2 --> e12
Cl3 --> e123
Cl4 --> e1234
Cl5 --> e12345
 
           
dual
 
dual1(u) = u / j
dual of a blade, representing a subspace, is a blade representing the orthogonal complement of that subspace
(space of all vectors perpendicular to it)  :
grade(dual(u)) = n - grade(u)     (Cln)
 
ex : (Cl3)
dual1 (e0)           = - e123
dual1 (e123)        =   e0
dual1 (e1)           = - e23                                 dual1 (e12) = e3
dual1 (e2)           = +e13    (-e31)                   dual1 (e23) = e1
dual1 (e3)           = -e12                                  dual1 (e13) = -e2    (-e31)
 
dual1 ( u ^ v ) = u . dual1 ( v )       dual1 ( u . v ) = u ^ dual1 ( v )
 
Gibbs cross product u x v = dual1 ( u ^ v )
 
 
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