SamplePublisher`GrassmannCalculus`
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
In[2]:=
 | ★ℬ |
5
A multi-vector represents a geometric object of indefinite shape. We can separately alter the lengths of each edge without changing the overall measure of the object. Here, using the "Augmented" method we specify the relative lengths of each edge and compensate with an overall coefficient. If we simplify the expression we obtain the original expression.
In[3]:=
2p⋀q⋀r%//
[{1,2,3},CoefficientMethod"Augmented"]
[%]
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 | ★ |
Out[3]=
2p⋀q⋀r
Out[3]=
1
3
Out[3]=
2p⋀q⋀r
If we want to make the leading coefficient 1 we use the "Unity" method. This will generally introduce radical expressions. It might be convenient to convert to use approximate numbers.
In[4]:=
2p⋀q⋀rstep1=%//
[{1,2,3},CoefficientMethod"Unity"]
[step1]step2=N[step1]
[step2]//Rationalize
 | DistributeProductWeights |
| ★ |
| ★ |
Out[4]=
2p⋀q⋀r
Out[4]=
p
1/3
3
2q
1/3
3
2/3
3
Out[4]=
2p⋀q⋀r
Out[4]=
(0.693361p)⋀(1.38672q)⋀(2.08008r)
Out[4]=
2p⋀q⋀r
If we wish to keep the leading coefficient unchanged we can use the "Constant" method. This will also generally introduce radical expressions.
In[5]:=
ap⋀q⋀rstep1=%//
[{1,2,3},CoefficientMethod"Constant"]
[step1]step2=N[step1]
[step2]//Rationalize
| DistributeProductWeights |
| ★ |
| ★ |
Out[5]=
ap⋀q⋀r
Out[5]=
a⋀q⋀r
p
1/3
6
2/3
2
1/3
3
2/3
3
1/3
2
Out[5]=
ap⋀q⋀r
Out[5]=
a(0.550321p)⋀(1.10064q)⋀(1.65096r)
Out[5]=
ap⋀q⋀r
We can insert a set of symbolic weights in the factors and compensate with the leading coefficient.
In[6]:=
p⋀q⋀r%//
[{a,b,c}]
[%]
| DistributeProductWeights |
| ★ |
Out[6]=
p⋀q⋀r
Out[6]=
(ap)⋀(bq)⋀(cr)
abc
Out[6]=
p⋀q⋀r
A leading coefficient can be moved into a specific factor with:
In[7]:=
ap⋀q⋀r%//
[{1,a,1}]
[%]
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| ★ |
Out[7]=
ap⋀q⋀r
Out[7]=
p⋀(aq)⋀r
Out[7]=
ap⋀q⋀r
A coefficient can be moved from one factor to another with:
In[8]:=
p⋀(bq)⋀r%//
[{1,1/b,b}]
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Out[8]=
p⋀(bq)⋀r
Out[8]=
p⋀q⋀(br)
A 'squeeze' transformation (or mapping) is one that shrinks one side of a rectangle and expands the other side by the same amount so as to maintain the area.
In[9]:=
p⋀q%//
[{1/h,h}]
[%]
| DistributeProductWeights |
| ★ |
Out[9]=
p⋀q
Out[9]=
p
h
Out[9]=
p⋀q
The Dirac delta function is the following limit.
In[10]:=
Limit
[{1/h,h}][p⋀q],hInfinity
 | DistributeProductWeights |
Out[10]=
h∞
p
h
In[11]:=
| ★ℬ |
5
e
1
3
e
4
8
e
5
2
e
2
3
e
4
8
e
5
14
e
3
e
5
7
| DistributeProductWeights |
Out[11]=
10++⋀++⋀-
e
1
3
e
4
8
e
5
2
e
2
3
e
4
8
e
5
14
e
3
e
5
7
Out[11]=
10a++⋀b++⋀c-
e
1
3a
e
4
8
a
e
5
2
e
2
3b
e
4
8
b
e
5
14
e
3
c
e
5
7
abc
And it works with some other products.
In[12]:=
mProduct/.WedgeCircleTimes%//
[{2,-3,5}]
| DistributeProductWeights |
Out[12]=
10++⊗++⊗-
e
1
3
e
4
8
e
5
2
e
2
3
e
4
8
e
5
14
e
3
e
5
7
Out[12]=
-2++⊗-3--⊗5-
1
3
e
1
3
e
4
4
e
5
e
2
9
e
4
8
3
e
5
14
e
3
5
e
5
7
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""