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Geometric Algebra
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geometric-algebra.md 
0 # zero order identity element
N_1 = O -> N
I_1 = O -> I # first order identity element
N_2 = I_1 -> N_1
E_2 = I_1 -> E_1 # second order identity element
N_3 = E_2 -> N_2

composition

A_2 . B_2 = C_2
  • analogous to addition, multiplication, etc.
  • works on tensors of the same order
  • produces tensor of same order

application

A_2(B_1) = C_1
  • analogous to addition, multiplication, etc.
  • works on argument tensors of order minus 1
  • produces tensor of order minus 1

mapping

B_1 -> C_1 = A_2
  • direct inverse of application
  • analogous to subtraction, division, etc.
  • produces tensor of order plus 1

inversion

A_2(B_1) = C_1
A_2 = B_1 -> C_1

B_1 -> C_1 = A_2
C_1 = A_2(B_1)

lowering order by appling to identity element

N_2(I_1) = N_1

raising order by mapping from identity element

I_1 -> N_1 = N_2

negation rules

I_1 -> (N_2(I_1)) = N_2
(I_1 -> N_1)(I_1) = N_1

composition to application conversion

A_2 . B_2 =
A_2(B_2(I_1))

application to composition conversion

A_2(B_1) =
A_2 . (I_1 -> B_1)

commutative application rule

A_2(B_1) =
A_2 . (I_1 -> B_1) =
(I_1 -> B_1) . A_2 =
(I_1 -> B_1)(A_2(I_1))

distributive property

C_2(A_1 . B_1) = C_2(A_1) . C_2(B_1)

distribution over twice lower order?

C_3(I_1 -> (A_1 . B_1)) = D_2


C_3 . (E_2 -> (I_1 -> (A_1 . B_1))) =

what does it mean to apply a higher order tensor to a minus 2 lower one?

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