GrassmannCalculus | Wolfram Resource Object

GrassmannCalculus`

ComposeLineElement

ComposeLineElement[c]
	composes a line element as an exterior product of a point and a vector, each expressed as a linear combination of basis elements in the currently declared basis. The coefficients of the components are symbolic scalars of the form c i .

ComposeLineElement[c,s]
	begins the indexing of the scalar symbols at s .

Details and Options



Examples

(1)

Basic Examples

(1)

In[1]:=

<<GrassmannCalculus`

Set the book Point 3-dimensional space.

In[2]:=

★A;

★

3

;

Compare a point, vector, simple bivector and a line element in 3-space.

In[3]:=

MapThreadRow[{#1,#2[c]}]&,{"Point: ","Vector: ","Simple Bivector: ","Line Element: "},

ComposePoint

,

ComposeVector

,

ComposeSimpleBivector

,

ComposeLineElement

//Column

Out[3]=

Point: ★+

c

1

e

1

+

c

2

e

2

+

c

3

e

3

Vector:

c

1

e

1

+

c

2

e

2

+

c

3

e

3

Simple Bivector: (

c

1

e

1

+

c

2

e

2

+

c

3

e

3

)⋀(

c

4

e

1

+

c

5

e

2

+

c

6

e

3

)

Line Element: (★+

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 use the full form of

ComposeLineElement[c]

, or its alias

★

c

.

In[4]:=

★A;

★

3

;

ComposeLineElement

[c]

,

★

c

//Column

Out[4]=

(★+

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

)

The current space needs to be a point space. If it is not, no composition will occur.

In[5]:=



★ℬ

3

;

★

a

,

★

3

;

★

a

//Column

Out[5]=

ComposeLineElement[a]

(★+

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 line elements in any dimension.

In[6]:=

Table

★

i

;

★

c

,{i,1,4}//Column

Out[6]=

(★+c

e

1

)⋀(

c

2

e

1

)

(★+

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 line elements.

In[7]:=

★

2

;

★

{a,b,c}

//Column

Out[7]=

(★+

a

1

e

1

+

a

2

e

2

)⋀(

a

3

e

1

+

a

4

e

2

)

(★+

b

1

e

1

+

b

2

e

2

)⋀(

b

3

e

1

+

b

4

e

2

)

(★+

c

1

e

1

+

c

2

e

2

)⋀(

c

3

e

1

+

c

4

e

2

)

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

In[8]:=

★

{a,b,c},0

//Column

Out[8]=

(★+

a

0

e

1

+

a

1

e

2

)⋀(

a

2

e

1

+

a

3

e

2

)

(★+

b

0

e

1

+

b

1

e

2

)⋀(

b

2

e

1

+

b

3

e

2

)

(★+

c

0

e

1

+

c

1

e

2

)⋀(

c

2

e

1

+

c

3

e

2

)

In[9]:=

★

{a,b,c},{-2,0,α}

//Column

Out[9]=

(★+

a

-2

e

1

+

a

-1

e

2

)⋀(

a

0

e

1

+

a

1

e

2

)

(★+

b

0

e

1

+

b

1

e

2

)⋀(

b

2

e

1

+

b

3

e

2

)

(★+

c

α

e

1

+

c

1+α

e

2

)⋀(

c

2+α

e

1

+

c

3+α

e

2

)

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

In[10]:=

ScalarSymbols

Out[10]=

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

a

-2

,

a

-1

,

a

0

,

a

1

,

a

2

,

a

3

,

a

4

,

b

0

,

b

1

,

b

2

,

b

3

,

b

4

,

c

0

,

c

1

,

c

2

,

c

3

,

c

4

,

c

α

,

c

1+α

,

c

2+α

,

c

3+α

}

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

In[11]:=

★

4

;

★



Out[11]=

(★+

e

1

+

e

2

+

e

3

+

e

4

)⋀(

e

1

+

e

2

+

e

3

+

e

4

)

Then that could be copied, filled and assigned to a bivector.

In[12]:=

line1=(★+2

e

1

+6

e

2

+-1

e

3

+0

e

4

)⋀(a

e

1

+b

e

2

+c

e

3

+d

e

4

)

Out[12]=

(★+2

e

1

+6

e

2

-

e

3

)⋀(a

e

1

+b

e

2

+c

e

3

+d

e

4

)

Subscripted scalars are convenient for generic derivations. For specific cases it will often be easier simply to dot

Basis

with Lists of coefficients in an exterior product.

In[13]:=

line1=({1,2,6,-1,0}.Basis)⋀({0,a,b,c,d}.Basis)

Out[13]=

(★+2

e

1

+6

e

2

-

e

3

)⋀(a

e

1

+b

e

2

+c

e

3

+d

e

4

)

In[14]:=

Clear[line1]

SeeAlso

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RelatedGuides

▪

Composers

""