Douglas Blumeyer's RTT How-To: Difference between revisions
Cmloegcmluin (talk | contribs) a wild 's' appeared |
Cmloegcmluin (talk | contribs) →multicommas: update table to reflect understanding that MLA tensors avoid variance changes too so they're more of a complement than a dual |
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!operator | !operator | ||
!example | !example | ||
!alternate example | !alternate example 1 | ||
!alternate example 2 | |||
!alternate example 3 | |||
!example (ASCII only) | !example (ASCII only) | ||
!alternate example (ASCII only) | !alternate example (ASCII only) | ||
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|¬[1 4 4⟩ = [4 -4 1⟩ | |¬[1 4 4⟩ = [4 -4 1⟩ | ||
|[̅1̅ ̅4̅ ̅4̅⟩ = [4 -4 1⟩ | |[̅1̅ ̅4̅ ̅4̅⟩ = [4 -4 1⟩ | ||
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|~[1 4 4> = [4 -4 1> | |~[1 4 4> = [4 -4 1> | ||
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|- | |- | ||
|2 | |2 | ||
|MLA dual | |MLA dual | ||
|is in exterior algebra form: compresses the antisymmetric/skew-symmetric matrix/tensor into a list of minors; this operation is the dual that RTT uses | |||
|yes | |yes | ||
|distinguish covariance from contravariance | |distinguish covariance from contravariance | ||
| | |diamond operator, (postfix) degree symbol | ||
| | |⋄⟨⟨1 4 4]] = [4 -4 1⟩ | ||
|⟨1 4 4]]° = [4 -4 1⟩ | |||
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| | | | ||
| | |<><<1 4 4]] = [4 -4 1> | ||
| | | | ||
|- | |- | ||
|3 | |3 | ||
|MLA | |MLA complement | ||
|is in tensor form: uses the full antisymmetric/skew-symmetric matrix/tensor itself; this operation is also known as the "Hodge dual", but "Hodge star" is preferred to avoid confusion with a variance-changing MLA dual | |||
| | |no | ||
| | |demonstrate agnosticism to and unchanging of variance | ||
|Hodge star | |Hodge star, asterisk operator | ||
|⋆⟨⟨0 1 4] ⟨-1 0 4] ⟨-4 -1 0]] = [4 -4 | |⋆⟨⟨0 1 4] ⟨-1 0 4] ⟨-4 -1 0]] = ⟨4 -4 1] | ||
| | |⋆[[0 1 4] [-1 0 4] [-4 -1 0]]⁰₂ = [4 -4 1]⁰₁ | ||
|*<<0 1 4] <-1 0 4] <-4 -1 0]] = | |∗⟨⟨0 1 4] ⟨-1 0 4] ⟨-4 -1 0]] = ⟨4 -4 1] | ||
|*[[0 1 4] [-1 0 4] [-4 -1 0]] type ( | |∗[[0 1 4] [-1 0 4] [-4 -1 0]]⁰₂ = [4 -4 1]⁰₁ | ||
|*<<0 1 4] <-1 0 4] <-4 -1 0]] = ⟨4 -4 1] | |||
|*[[0 1 4] [-1 0 4] [-4 -1 0]] type (0,2) = [4 -4 1] type (0,1)) | |||
|} | |} | ||