GrassmannCalculus | Wolfram Resource Object

GrassmannCalculus`

ComposeMPlaneElement

ComposeMPlaneElement[m,c]
	composes a multiplane element as the exterior product of a point and m vectors. The coefficients of the components are symbolic scalars of the form c i .

ComposeMPlaneElement[m,c,s]
	begins the indexing of the scalars at s .

Details and Options



Examples

(1)

Basic Examples

(1)

In[1]:=

<<GrassmannCalculus`

In[2]:=

savePreferences=

AllPreferences

;

Set book 4-dimensional Point space.

In[3]:=

★A

★

4

;

Compose a 3-Plane.

In[4]:=

ComposeMPlaneElement

[3,c]

Out[4]=

(★+

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

)⋀(

c

9

e

1

+

c

10

e

2

+

c

11

e

3

+

c

12

e

4

)⋀(

c

13

e

1

+

c

14

e

2

+

c

15

e

3

+

c

16

e

4

)

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

In[5]:=

★A;

★ℬ

3

;

ComposeMPlaneElement

[3,a],

★

3

;

ComposeMPlaneElement

[3,a]//Column

Out[5]=

ComposeMPlaneElement[3,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

)⋀(

a

7

e

1

+

a

8

e

2

+

a

9

e

3

)⋀(

a

10

e

1

+

a

11

e

2

+

a

12

e

3

)

You can compose m-plane elements in any dimension (although some may simplify to zero). For example, a 3-plane in a space of 1 or 2 dimensions is zero.

In[6]:=

Table

★

i

;

ComposeMPlaneElement

[3,c],{i,1,4}//Column

★

[Part[%,1,{1,2}]]

Out[6]=

(★+

c

1

e

1

)⋀(

c

2

e

1

)⋀(

c

3

e

1

)⋀(

c

4

e

1

)

(★+

c

1

e

1

+

c

2

e

2

)⋀(

c

3

e

1

+

c

4

e

2

)⋀(

c

5

e

1

+

c

6

e

2

)⋀(

c

7

e

1

+

c

8

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

7

e

1

+

c

8

e

2

+

c

9

e

3

)⋀(

c

10

e

1

+

c

11

e

2

+

c

12

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

)⋀(

c

9

e

1

+

c

10

e

2

+

c

11

e

3

+

c

12

e

4

)⋀(

c

13

e

1

+

c

14

e

2

+

c

15

e

3

+

c

16

e

4

)

Out[6]=

{0,0}

You can compose lists of m-plane elements.

In[7]:=

★

3

;

ComposeMPlaneElement

[{1,2},c]//Column

Out[7]=

(★+

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

)⋀(

c

7

e

1

+

c

8

e

2

+

c

9

e

3

)

In[8]:=

ComposeMPlaneElement

[{2,2},{α,β}]//Column

Out[8]=

(★+

e

1

α

1

+

e

2

α

2

+

e

3

α

3

)⋀(

e

1

α

4

+

e

2

α

5

+

e

3

α

6

)⋀(

e

1

α

7

+

e

2

α

8

+

e

3

α

9

)

(★+

e

1

β

1

+

e

2

β

2

+

e

3

β

3

)⋀(

e

1

β

4

+

e

2

β

5

+

e

3

β

6

)⋀(

e

1

β

7

+

e

2

β

8

+

e

3

β

9

)

In[9]:=

ComposeMPlaneElement

[{1,2},{α,β}]//Column

Out[9]=

(★+

e

1

α

1

+

e

2

α

2

+

e

3

α

3

)⋀(

e

1

α

4

+

e

2

α

5

+

e

3

α

6

)

(★+

e

1

β

1

+

e

2

β

2

+

e

3

β

3

)⋀(

e

1

β

4

+

e

2

β

5

+

e

3

β

6

)⋀(

e

1

β

7

+

e

2

β

8

+

e

3

β

9

)

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

In[10]:=

★

4

;

ComposeMPlaneElement

[{2,3},{a,c},{0,k}]//Column

Out[10]=

(★+

a

0

e

1

+

a

1

e

2

+

a

2

e

3

+

a

3

e

4

)⋀(

a

4

e

1

+

a

5

e

2

+

a

6

e

3

+

a

7

e

4

)⋀(

a

8

e

1

+

a

9

e

2

+

a

10

e

3

+

a

11

e

4

)

(★+

c

k

e

1

+

c

1+k

e

2

+

c

2+k

e

3

+

c

3+k

e

4

)⋀(

c

4+k

e

1

+

c

5+k

e

2

+

c

6+k

e

3

+

c

7+k

e

4

)⋀(

c

8+k

e

1

+

c

9+k

e

2

+

c

10+k

e

3

+

c

11+k

e

4

)⋀(

c

12+k

e

1

+

c

13+k

e

2

+

c

14+k

e

3

+

c

15+k

e

4

)

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

In[11]:=

ScalarSymbols

Out[11]=

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

a

0

,

a

1

,

a

2

,

a

3

,

a

4

,

a

5

,

a

6

,

a

7

,

a

8

,

a

9

,

a

10

,

a

11

,

a

12

,

c

1

,

c

2

,

c

3

,

c

4

,

c

5

,

c

6

,

c

7

,

c

8

,

c

9

,

c

10

,

c

11

,

c

12

,

c

13

,

c

14

,

c

15

,

c

16

,

c

k

,

c

1+k

,

c

2+k

,

c

3+k

,

c

4+k

,

c

5+k

,

c

6+k

,

c

7+k

,

c

8+k

,

c

9+k

,

c

10+k

,

c

11+k

,

c

12+k

,

c

13+k

,

c

14+k

,

c

15+k

,

α

1

,

α

2

,

α

3

,

α

4

,

α

5

,

α

6

,

α

7

,

α

8

,

α

9

,

β

1

,

β

2

,

β

3

,

β

4

,

β

5

,

β

6

,

β

7

,

β

8

,

β

9

}

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

In[12]:=

ComposeMPlaneElement

2,





Out[12]=

(★+

e

1

+

e

2

+

e

3

+

e

4

)⋀(

e

1

+

e

2

+

e

3

+

e

4

)⋀(

e

1

+

e

2

+

e

3

+

e

4

)

You can also compose specific m-planes by dotting lists of coefficients with the Basis inside of the exterior product.

In[13]:=

{1,2,3,4,5}.Basis⋀{0,a,b,c,d}.Basis⋀{0,6,7,8,9}.Basis

Out[13]=

(★+2

e

1

+3

e

2

+4

e

3

+5

e

4

)⋀(a

e

1

+b

e

2

+c

e

3

+d

e

4

)⋀(6

e

1

+7

e

2

+8

e

3

+9

e

4

)

One could also use this construction to compose a specific m-plane by a combination of points and vectors.

In[14]:=

★



⋀

★



⋀

★



Out[14]=

(★+

e

1

+

e

2

+

e

3

+

e

4

)⋀(★+

e

1

+

e

2

+

e

3

+

e

4

)⋀(

e

1

+

e

2

+

e

3

+

e

4

)

In[15]:=

RestorePreferences

[savePreferences]

SeeAlso

ComposeBoundSimpleMVector

▪

▪

▪

▪

▪

""