GrassmannCalculus | Wolfram Resource Object

GrassmannCalculus`

ComposeSimpleBivector

ComposeSimpleBivector[c]
	composes a simple bivector as the exterior product of two vectors constructed from the currently declared basis.

ComposeSimpleBivector[c,s]
	begins indexing the scalars at s .

Details and Options



Examples

(1)

Basic Examples

(1)

In[1]:=

<<GrassmannCalculus`

You can use the full form of

ComposeSimpleBivector[c]

, or its alias

★

c

.

In[2]:=

★A;

ComposeSimpleBivector

[c]

,

★

c

//Column

Out[2]=

(

c

1

e

1

+

c

2

e

2

+

c

3

e

3

)⋀(

c

4

e

1

+

c

5

e

2

+

c

6

e

3

)

(

c

1

e

1

+

c

2

e

2

+

c

3

e

3

)⋀(

c

4

e

1

+

c

5

e

2

+

c

6

e

3

)

Contrast that with composing a general bivector:

In[3]:=

ComposeBivector

[c]

Out[3]=

c

1

e

1

⋀

e

2

+

c

2

e

1

⋀

e

3

+

c

3

e

2

⋀

e

3

You will get the same result whether you are in a point space or a vector space.

In[4]:=



★ℬ

3

;

★

a

,

★

3

;

★

a

//Column

Out[4]=

(

a

1

e

1

+

a

2

e

2

+

a

3

e

3

)⋀(

a

4

e

1

+

a

5

e

2

+

a

6

e

3

)

(

a

1

e

1

+

a

2

e

2

+

a

3

e

3

)⋀(

a

4

e

1

+

a

5

e

2

+

a

6

e

3

)

You can compose simple bivectors in any dimension greater than 1.

In[5]:=

Table

★ℬ

i

;

★

c

,{i,2,4}//Column

Out[5]=

(

c

1

e

1

+

c

2

e

2

)⋀(

c

3

e

1

+

c

4

e

2

)

(

c

1

e

1

+

c

2

e

2

+

c

3

e

3

)⋀(

c

4

e

1

+

c

5

e

2

+

c

6

e

3

)

(

c

1

e

1

+

c

2

e

2

+

c

3

e

3

+

c

4

e

4

)⋀(

c

5

e

1

+

c

6

e

2

+

c

7

e

3

+

c

8

e

4

)

You can compose lists of simple bivectors:

In[6]:=

★ℬ

3

;

★

{a,b,c}

//Column

Out[6]=

(

a

1

e

1

+

a

2

e

2

+

a

3

e

3

)⋀(

a

4

e

1

+

a

5

e

2

+

a

6

e

3

)

(

b

1

e

1

+

b

2

e

2

+

b

3

e

3

)⋀(

b

4

e

1

+

b

5

e

2

+

b

6

e

3

)

(

c

1

e

1

+

c

2

e

2

+

c

3

e

3

)⋀(

c

4

e

1

+

c

5

e

2

+

c

6

e

3

)

You can make the starting index of the coefficients whatever you want.

In[7]:=

★

{a,b},0

//Column

Out[7]=

(

a

0

e

1

+

a

1

e

2

+

a

2

e

3

)⋀(

a

3

e

1

+

a

4

e

2

+

a

5

e

3

)

(

b

0

e

1

+

b

1

e

2

+

b

2

e

3

)⋀(

b

3

e

1

+

b

4

e

2

+

b

5

e

3

)

In[8]:=

★

{a,b},{-3,α}

//Column

Out[8]=

(

a

-3

e

1

+

a

-2

e

2

+

a

-1

e

3

)⋀(

a

0

e

1

+

a

1

e

2

+

a

2

e

3

)

(

b

α

e

1

+

b

1+α

e

2

+

b

2+α

e

3

)⋀(

b

3+α

e

1

+

b

4+α

e

2

+

b

5+α

e

3

)

Note that all the scalar symbols generated as coefficients of the basis elements have been automatically declared as scalar symbols.

In[9]:=

ScalarSymbols

Out[9]=

{a,b,c,d,e,f,g,h,

a

-3

,

a

-2

,

a

-1

,

a

0

,

a

1

,

a

2

,

a

3

,

a

4

,

a

5

,

a

6

,

b

0

,

b

1

,

b

2

,

b

3

,

b

4

,

b

5

,

b

6

,

b

α

,

b

1+α

,

b

2+α

,

b

3+α

,

b

4+α

,

b

5+α

,

c

1

,

c

2

,

c

3

,

c

4

,

c

5

,

c

6

}

You can also generate templates for simple bivectors using the placeholder symbol. This allows you to tab through the composed result and enter your own values.

In[10]:=

★ℬ

4

;

★



Out[10]=

(

e

1

+

e

2

+

e

3

+

e

4

)⋀(

e

1

+

e

2

+

e

3

+

e

4

)

One could also enter a specific simple bivector by dotting lists of coefficients with the Basis vectors.

In[11]:=

{a,b,c,d}.Basis⋀{1,2,-3,-4}.Basis

Out[11]=

(a

e

1

+b

e

2

+c

e

3

+d

e

4

)⋀(

e

1

+2

e

2

-3

e

3

-4

e

4

)

SeeAlso

ComposeBivector

▪

ComposeSimpleForm

▪

ComposeSimpleMVector

RelatedGuides

▪

Composers

▪

Grassmann Calculus

▪

Grassmann Calculus

""