Douglas Blumeyer's RTT How-To: Difference between revisions

Line 1,328:

# Just as a vector is the dual of a covector, we also have a '''multivector''' which is the dual of a multicovector. Analogously, we call the thing the multivector represents a '''multicomma'''.

# We can calculate a multicomma from a comma basis much in the same way we can calculate a multimap from a mapping-row basis

# We can convert between multimaps and multicommas using an operation called “taking '''[[the dual]]'''”<ref>This operation uses the same process as is used for finding the complement in exterior algebra, however, whereas exterior algebra does not convert between vectors and covectors (it can be used on either one, staying within that category), with RTT's dual you switch which type it is as the last step.

# We can convert between multimaps and multicommas using an operation called “taking '''[[the dual]]'''”<ref>This operation uses the same process as is used for finding the complement in exterior algebra, however, whereas exterior algebra does not convert between vectors and covectors (it can be used on either one, staying within that category), with RTT's dual you switch which type it is as the last step. More details can be found below. The dual we use in RTT is #2 in the table. It essentially combines elements from both #1 and #3.

{| class="wikitable"

|+'''Table 6(footnote).''' Three similar duals

!~~<math>d</math>~~

!#

!~~<math>r</math>~~

!dual type

!~~<math>d~~ - ~~r</math>~~

!notes

!~~<math>{d \choose r}</math>~~

!variance changing

!~~<math>{d \choose {d - r}}</math>~~

!using RTT's extended bra-ket notation to

!~~count~~

!operator

!example

!alternate example

!example (ASCII only)

!alternate example (ASCII only)

!

|-

|3

|2

|1

|~~<math>(2~~,~~3) (2~~,~~5) (3~~,~~5)</math>~~

|Grassman/EA/orthogonal complement

|~~<math~~>~~(2) (3) (5)</math~~>

|also called "dual" within EA, but "complement" is preferred to avoid confusion with the variance-changing MLA dual

|3

|no

|demonstrate agnosticism to and unchanging of variance

|negation, overbar, tilde

|¬[1 4 4⟩ = [4 -4 1⟩

|[̅1̅ ̅4̅ ̅4̅⟩ = [4 -4 1⟩

|~[1 4 4> = [4 -4 1>

|

|-

|4

|2

|3

|MLA dual, in EA form

|1

|compresses the antisymmetric/skew-symmetric matrix/tensor into a list of minors; this is the dual that RTT uses

|~~<math>(2,3,5) (2,3,7) (2,5,7) (3,5,7)</math>~~

|yes

|<~~math>(2) (3) (5) (7)~~<~~/math~~>

|distinguish covariance from contravariance

|4

|asterisk

|∗⟨⟨1 4 4]] = [4 -4 1⟩

|

|*<<1 4 4]] = [4 -4 1>

|

|-

|5

|3

|2

|MLA dual, in tensor form

|<~~math>(2,3,5) (2,3,7) (2,3,11) (2,5,7) (2,5,11) (2,7,11) (3,5,7) (3,5,11) (3,7,11) (5,7,11)~~<~~/math~~>

|uses the full antisymmetric/skew-symmetric matrix/tensor itself; here this operation is known as the Hodge dual

|~~<math>~~(2,3) (2,5) ~~(2,7~~) ~~(2,11) (3,5), (3,7) (3,11) (5,7) (5,11) (7,11)</math>~~

|yes

~~|10~~

|distinguish covariance from contravariance

|Hodge star

|⋆⟨⟨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]¹₀

|*<<0 1 4] <-1 0 4] <-4 -1 0]] = [4 -4 1>

|*[[0 1 4] [-1 0 4] [-4 -1 0]] type (1,1) = [4 -4 1] type (1,0))

|}

@@ Line 1,328: / Line 1,328: @@
 # Just as a vector is the dual of a covector, we also have a '''multivector''' which is the dual of a multicovector. Analogously, we call the thing the multivector represents a '''multicomma'''.
 # We can calculate a multicomma from a comma basis much in the same way we can calculate a multimap from a mapping-row basis
-# We can convert between multimaps and multicommas using an operation called “taking '''[[the dual]]'''”<ref>This operation uses the same process as is used for finding the complement in exterior algebra, however, whereas exterior algebra does not convert between vectors and covectors (it can be used on either one, staying within that category), with RTT's dual you switch which type it is as the last step.
+# We can convert between multimaps and multicommas using an operation called “taking '''[[the dual]]'''”<ref>This operation uses the same process as is used for finding the complement in exterior algebra, however, whereas exterior algebra does not convert between vectors and covectors (it can be used on either one, staying within that category), with RTT's dual you switch which type it is as the last step. More details can be found below. The dual we use in RTT is #2 in the table. It essentially combines elements from both #1 and #3.
 {| class="wikitable"
 |+'''Table 6(footnote).''' Three similar duals
-!<math>d</math>
+!#
-!<math>r</math>
+!dual type
-!<math>d - r</math>
+!notes
-!<math>{d \choose r}</math>
+!variance changing
-!<math>{d \choose {d - r}}</math>
+!using RTT's extended bra-ket notation to
-!count
+!operator
+!example
+!alternate example
+!example (ASCII only)
+!alternate example (ASCII only)
+!
 |-
-|3
-|2
 |1
-|<math>(2,3) (2,5) (3,5)</math>
+|Grassman/EA/orthogonal complement
-|<math>(2) (3) (5)</math>
+|also called "dual" within EA, but "complement" is preferred to avoid confusion with the variance-changing MLA dual
-|3
+|no
+|demonstrate agnosticism to and unchanging of variance
+|negation, overbar, tilde
+|¬[1 4 4⟩ = [4 -4 1⟩
+|[̅1̅ ̅4̅ ̅4̅⟩ = [4 -4 1⟩
+|~[1 4 4> = [4 -4 1>
+|
 |-
-|4
+|2
-|3
+|MLA dual, in EA form
-|1
+|compresses the antisymmetric/skew-symmetric matrix/tensor into a list of minors; this is the dual that RTT uses
-|<math>(2,3,5) (2,3,7) (2,5,7) (3,5,7)</math>
+|yes
-|<math>(2) (3) (5) (7)</math>
+|distinguish covariance from contravariance
-|4
+|asterisk
+|∗⟨⟨1 4 4]] = [4 -4 1⟩
+|
+|*<<1 4 4]] = [4 -4 1>
+|
 |-
-|5
 |3
-|2
+|MLA dual, in tensor form
-|<math>(2,3,5) (2,3,7) (2,3,11) (2,5,7) (2,5,11) (2,7,11) (3,5,7) (3,5,11) (3,7,11) (5,7,11)</math>
+|uses the full antisymmetric/skew-symmetric matrix/tensor itself; here this operation is known as the Hodge dual
-|<math>(2,3) (2,5) (2,7) (2,11) (3,5), (3,7) (3,11) (5,7) (5,11) (7,11)</math>
+|yes
-|10
+|distinguish covariance from contravariance
+|Hodge star
+|⋆⟨⟨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]¹₀
+|*<<0 1 4] <-1 0 4] <-4 -1 0]] = [4 -4 1>
+|*[[0 1 4] [-1 0 4] [-4 -1 0]] type (1,1) = [4 -4 1] type (1,0))
 |}