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GrassmannCalculus`

ComplementPalette

ComplementPalette
	generates a palette of the basis elements of the Grassmann algebra together with their corresponding GrassmannComplement elements.

Details and Options



Examples

(1)

Basic Examples

(1)

In[1]:=

<<GrassmannCalculus`

The

ComplementPalette

for the default coordinate system:

In[2]:=

SetActiveAssociation

PublicGrassmannAtlas

"Grassmann Plane"

ComplementPalette

Out[2]=

Complement Palette

Basis	Complement
1	★⋀ e x ⋀ e y
★	e x ⋀ e y
e x	-(★⋀ e y )
e y	★⋀ e x
★⋀ e x	e y
★⋀ e y	- e x
e x ⋀ e y	★
★⋀ e x ⋀ e y	1

Th following is the palette for a coordinate with a non-Euclidean metric.

In[3]:=

SetActiveAssociation

PublicGrassmannAtlas

"Spherical"

ComplementPalette

Out[3]=

Complement Palette

Basis	Complement
1	Csc[θ] e r ⋀ e θ ⋀ e φ 2 r
e r	Csc[θ] e θ ⋀ e φ 2 r
e θ	-Csc[θ] e r ⋀ e φ
e φ	Sin[θ] e r ⋀ e θ
e r ⋀ e θ	e φ Csc[θ]
e r ⋀ e φ	- e θ Sin[θ]
e θ ⋀ e φ	2 r e r Sin[θ]
e r ⋀ e θ ⋀ e φ	2 r Sin[θ]

A "Form" basis uses the inverse metric.

In[4]:=

SwitchBasis

["Form"]

ComplementPalette

Out[4]=

Complement Palette

Basis	Complement
1	2 r Sin[θ]dr⋀dθ⋀dφ
dr	2 r Sin[θ]dθ⋀dφ
dθ	- 4 r Sin[θ]dr⋀dφ
dφ	4 r 3 Sin[θ] dr⋀dθ
dr⋀dθ	dφ 4 r Sin[θ]
dr⋀dφ	-dθ 4 r 3 Sin[θ]
dθ⋀dφ	drCsc[θ] 2 r
dr⋀dθ⋀dφ	Csc[θ] 2 r

The "Orthonormal" basis uses a Euclidean metric.

In[5]:=

SwitchBasis

["Orthonormal"]

ComplementPalette

Out[5]=

Complement Palette

Basis	Complement
1	 e r ⋀  e θ ⋀  e φ
 e r	 e θ ⋀  e φ
 e θ	-  e r ⋀  e φ 
 e φ	 e r ⋀  e θ
 e r ⋀  e θ	 e φ
 e r ⋀  e φ	-  e θ
 e θ ⋀  e φ	 e r
 e r ⋀  e θ ⋀  e φ	1

Set the default book preferences.

In[6]:=

★A

ComplementPalette

Out[6]=

Complement Palette

Basis	Complement
1	e 1 ⋀ e 2 ⋀ e 3
e 1	e 2 ⋀ e 3
e 2	-( e 1 ⋀ e 3 )
e 3	e 1 ⋀ e 2
e 1 ⋀ e 2	e 3
e 1 ⋀ e 3	- e 2
e 2 ⋀ e 3	e 1
e 1 ⋀ e 2 ⋀ e 3	1

SeeAlso

GrassmannComplement

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RelatedGuides

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Grassmann Complement

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