Talk:Interior product: Difference between revisions
Cmloegcmluin (talk | contribs) adding some questions, observations, and suggestions |
Cmloegcmluin (talk | contribs) →Questions, observations, suggestions: add summary table |
||
| Line 100: | Line 100: | ||
If it's of any assistance, I've implemented all of these products in Wolfram Language. You can access the shared computable notebook here and make your own copy: https://www.wolframcloud.com/obj/5d4e22f3-af6b-4c4a-a7ce-af860362ae72 | If it's of any assistance, I've implemented all of these products in Wolfram Language. You can access the shared computable notebook here and make your own copy: https://www.wolframcloud.com/obj/5d4e22f3-af6b-4c4a-a7ce-af860362ae72 | ||
And here's a summary table that has helped me get my head around this situation: | |||
{| class="wikitable" | |||
|+ | |||
!operations | |||
!progressive product (AKA wedge product, exterior product) | |||
a ∧ b | |||
!regressive product (AKA vee product) | |||
a ∨ b | |||
<nowiki>*</nowiki>(*a ∧ *b) | |||
!right interior product | |||
a ⨽ b | |||
∗(∗a ∧ b) | |||
examples given where grade(a) ≥ grade(b) | |||
!(left) interior product | |||
a ⨼ b | |||
<nowiki>*</nowiki>(a ∧ *b) | |||
examples given where grade(a) < grade(b) | |||
!symmetrical interior product | |||
a • b = if grade(a) ≥ grade(b), a ⨽ b; else a ⨼ b | |||
|- | |||
!resultant grade, assuming empty intersections | |||
!grade(a) + grade(b) | |||
!grade(a) + grade(b) - dimensionality | |||
!grade(a) - grade(b) | |||
!grade(b) - grade(a) | |||
!if grade(a) ≥ grade(b), grade(a) - grade(b); else grade(b) - grade(a) | |||
|- | |||
!resultant variance | |||
!same as a (and b) | |||
!same as a (and b) | |||
!same as a | |||
!same as b | |||
!if grade(a) ≥ grade(b), same as a; else same as b | |||
|- | |||
!multicovector with multicovector | |||
⟨] ⟨] | |||
|⟨12 19 28 34] ∧ ⟨19 30 44 53] = ⟨⟨1 4 10 4 13 12]] | |||
|⟨⟨⟨1 2 -3 -2 1 -4 -5 12 9 -19]]] ∨ ⟨⟨⟨⟨1 2 1 2 3]]]] = ⟨⟨-6 7 2 -15 25 20 -3 -15 -59 -49]] | |||
|ND | |||
|ND | |||
|ND | |||
|- | |||
!multivector with multivector | |||
[⟩ [⟩ | |||
|[4 -4 1 0⟩ ∧ [13 -10 0 1⟩ = [[12 -13 4 10 -4 1⟩⟩ | |||
|[[44 -30 19⟩⟩ ∨ [[28 -19 12⟩⟩ = [4 -4 1⟩ | |||
|ND | |||
|ND | |||
|ND | |||
|- | |||
!multicovector with multivector | |||
⟨] [⟩ | |||
|ND | |||
|ND | |||
|⟨⟨⟨1 2 -3 -2 1 -4 -5 12 9 -19]]] ⨽ [-3 2 -1 2 -1⟩ = ⟨⟨6 -7 -2 15 -25 -20 3 15 59 49]] | |||
|⟨12 19 28] ⨼ [[44 -30 19⟩⟩ = [4 -4 1⟩ | |||
|(in terms of other two interior products) | |||
|- | |||
!multivector with multicovector | |||
[⟩ ⟨] | |||
|ND | |||
|ND | |||
|[[44 -30 19⟩⟩ ⨽ ⟨12 19 28] = [-4 4 -1⟩ | |||
|[-3 2 -1 2 -1⟩ ⨼ ⟨⟨⟨1 2 -3 -2 1 -4 -5 12 9 -19]]] = ⟨⟨-6 7 2 -15 25 20 -3 -15 -59 -49]] | |||
|(in terms of other two interior products) | |||
|} | |||