GrassmannCalculus`
Measure  |
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Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
We will calculate some simple measures in the Grassmann Plane.
In[2]:=
 | SetActiveSpacePreferences |
 | PublicGrassmannAtlas |
The distance between a point on the negative x axis and the Origin.
In[3]:=
| Measure |
e
x
Out[3]=
3
Notice that we had to subtract the two points to obtain the distance. That's because the Origin itself has a Measure.
In[4]:=
| Measure |
Out[4]=
1
So the Measure of the point itself is:
In[5]:=
| Measure |
e
x
Out[5]=
10
The Measure of an unlocated vector in the plane:
In[6]:=
| Measure |
e
x
e
y
Out[6]=
5
The area of a right triangle in the plane with vertices at .
{★,★+3,★+4}
e
x
e
y
In[7]:=
vertexA=★;vertexB=★+3;vertexC=★+4;1/2
[(vertexB-vertexA)⋀(vertexC-vertexA)]
e
x
e
y
| Measure |
Out[7]=
6
That used the conventional baseheight, with a right angle at but we could use any two sides of the triangle even though they are not at right angles.
1
2
vertexA
In[8]:=
1/2
[(vertexC-vertexB)⋀(vertexA-vertexB)]
| Measure |
Out[8]=
6
The Measure of the basis n-element is with the Euclidean matrix.
1
In[9]:=
| Measure |
e
x
e
y
Out[9]=
1
Here we add two symbolic vectors and calculate the measure of their bivector.
In[10]:=
| ★★V |
| Measure |
Out[10]=
-+(⊖)(⊖)
2
(⊖)
Next we add scalar symbols for a general metric and recalculate the measure of the basis n-element.
In[11]:=
| ★★S |
g
_,_
| DeclareMetric |
g 1,1 | g 1,2 |
g 1,2 | g 2,2 |
| Measure |
e
x
e
y
Out[11]=
-+
2
g
1,2
g
1,1
g
2,2
The metric requires the use of the basis elements and so it is not reflected in purely symbolic expressions.
In[12]:=
| Measure |
Out[12]=
-+(⊖)(⊖)
2
(⊖)
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