products : gp(u, v)  outp(u, v)  inp(u, v)
                        Clifford algebra products :
                     
En a vector space, dimension n,.with vectors  a, b, c, ...
The associative vector algebra Cln, dimension 2^n, consists of all sums of products a b c..., named A, B, C ..., subject to the following axioms :
 
1 - A + B = B + A
(A + B) + C = A + (B + C)
A + 0 = 0 + A = A                                                                                              0 null vector
a(A + B) = aA + aB        (a + b)A = aA + bA     a (bA) = (ab)A                                a, b  reals
1A = A1 = A
 
2 - product AB is also an element of Cln and multiplication is associative and distributive :
A(B + C) = AB + AC      (B + C)A = BA + CA
(AB)C = A(BC)
 
3 - real scalars a wich are part of Cln commutes with all elements :  aA = Aa
 
4 - for a vector a,     a a = = |a|²    is a scalar .
 
A, B, C, ... are called Clifford numbers, or multivectors. If we represent a vector in Euclidian space (n = 3) by a matrix, vector product is defined by matrix product, and all axioms above are verified.
 
an arbitrary multivector  can be decomposed into a sum of pure grade terms :
A = <A>0 + <A>1 + <A>2 + <A>3  + ...    
A+ = <A>0 + <A>2  + ...    is the set of even Clifford numbers
A- =  <A>1 + <A>3  + ...    is the set of odd  Clifford numbers
A+ A- = A-     A- A- = A+
A+ A+ = A+                (product of two even number is even number)  ==> closed under multiplication : the subspace A+ is a subalgebra, called the even subalgebra of Cln
                      Geometric product :
 
a, b two vectors.  (a + b)² =+ ab + ba +  -->  ab + ba = (a + b)² --  is a scalar, by axiom 4.
we define the inner product for vectors as :          
a . b = 1/2(ab + ba)
the remaning, antisymetric part of the geometric product is defined as the exterior product, and return a bi-vector.
a ^ b = 1/2(ab - ba)          
these definition combines to give :
ab = a . b + b ^ a          
                                               
ex : A = 3e0 + 2 e123       B = e1 + e2            gp(A,B) =  3e1 + 3e3 + 2e12 + 2e23          grade(1,2)
       a = e1 + 2e2 - e3                                      gp(a,a)  =  6e0                                             grade(0)
 
                      Inner product
 
A, B two blades,  j and k grade :                             A . B = <AB>|j-k|
 
vectors a, b :  k = j =1      k-j =0                  a.b = <ab>0         ( = scalar product)
multivectors A B :                                        A.B =  sum of blades inner product
 
ex :  a = e1 + e2      b = e2 - e3                  inp(a,b) = e0                                     grade(0)
A = 2e23 - e13   B = e2 +e3                        inp(A,B) = -e1 + 2e2 - 2e3                   grade(1)
A = e0 + e1 + e23  B = e2 + e13                   inp(A,B) = e2 + e13                             grade(1,2)
 
                       Outer product
 
A, B two blades,  j and k grade :                   A ^ B = <AB>j+k
vectors a, b :  k = j =1      j+k =2                 a ^ b = <ab>2     ( = bivector)
multivectors A B :                                        A ^ B =  sum of blades outer product
outer product associativity :                          A ^ (B ^ C) = (A ^ B) ^ C
 
ex :  a = e1 + e2      b = e2 - e3                  outp(a,b) = e12 - e23 - e13                 grade(2)
A = 2e23 - e13   B = e2 +e3                        outp(A,B) = e123                                grade(3)
A = e0 + e1 + e23  B = 2e2 + e13                  outp(A,B) = 2e2 + 2e12 + e13               grade(1,2)
 
                     scalar product  
 
A*B = <A rev(B)>0
 
ex : A = 3e0 + 2e12
      B = e1 + e12
 grade(gp(A,rev(B)),0) = 2e0
 

                     left contraction,  right contraction , Hestenes dot product
 
lc(u,v)                                                                             u, v multivectors
rc(u,v)                                                                             u, v multivectors
doth(u,v) = u.v - <u>0 v - u <v>0 + <u>0 <v>0                     u, v multivectors
 


                         
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