Maths - Multivector inverse program

There are a number of open source programs that can work with Grassmann and Clifford Algebra. I have used Axiom, how to install Axiom here.

First we setup a completely general 3D basis:

a	b	c
d	e	f
g	h	i

(I have put user input in red):

(1) -> )library GRAS
   GrassmannAlgebra is now explicitly exposed in frame frame1
   GrassmannAlgebra will be automatically loaded when needed from
      /home/martin/GRAS.NRLIB/GRAS
(1) -> Eu := GrassmannAlgebra(3,Expression(Fraction(Integer)),[[a,b,c],[d,e,f],[g,h,i]])

   (1)
   GrassmannAlgebra(3,Expression(Fraction(Integer)),[[a,b,c],[d,e,f],[g,h,i]])
                                                                 Type: Domain

First we square a general vector :

(2) -> m:Eu := multivector[0,x,y,0,z,0,0,0]

   (2)  x e  + y e  + z e
           1      2      3
Type: GrassmannAlgebra(3,Expression(Fraction(Integer)),[[a,b,c],[d,e,f],[g,h,i]])
(3) -> m*m

           2                               2                   2
   (3)  i z  + ((h + f)y + (g + c)x)z + e y  + (d + b)x y + a x
Type: GrassmannAlgebra(3,Expression(Fraction(Integer)),[[a,b,c],[d,e,f],[g,h,i]])

This gives a scalar value equal to:

a	b	c
d	e	f
g	h	i

now we will try squaring bivectors:

4) -> bivector:Eu := multivector[0,0,0,x,0,y,z,0]

   (4)  x e e  + y e e  + z e e
           1 2      1 3      2 3
Type: GrassmannAlgebra(3,Expression(Fraction(Integer)),[[a,b,c],[d,e,f],[g,h,i]])
(5) -> bivector*bivector

   (5)
                   2
     (- e i + f h)z  + (((- e - a)i + f h + c g)y + (- e i + f h - a e + b d)x)z
   +
                   2                                                2
     (- a i + c g)y  + (- a i + c g - a e + b d)x y + (- a e + b d)x
   +
                                        2
     ((d - b)x z + (d - b)x y + (d - b)x )e e
                                           1 2
   +
                           2
     ((g - c)y z + (g - c)y  + (g - c)x y)e e
                                           1 3
   +
              2
     ((h - f)z  + ((h - f)y + (h - f)x)z)e e
                                          2 3
Type: GrassmannAlgebra(3,Expression(Fraction(Integer)),[[a,b,c],[d,e,f],[g,h,i]])
(6) ->

- a e + b d	- a e + b d	- a e + b d
- a i + c g	- a i + c g	- a i + c g
- e i + f h	- e i + f h	- e i + f h

now we will try multipy even terms by cojugate:

(6) -> bivector:Eu := multivector[w,0,0,x,0,y,z,0]

   (6)  w + x e e  + y e e  + z e e
               1 2      1 3      2 3
Type: GrassmannAlgebra(3,Expression(Fraction(Integer)),[[a,b,c],[d,e,f],[g,h,i]])
(7) -> conj:Eu := multivector[w,0,0,-x,0,-y,-z,0]

   (7)  w - x e e  - y e e  - z e e
               1 2      1 3      2 3
Type: GrassmannAlgebra(3,Expression(Fraction(Integer)),[[a,b,c],[d,e,f],[g,h,i]])
(8) -> bivector*conj

   (8)
                 2
     (e i - f h)z  + (((e + a)i - f h - c g)y + (e i - f h + a e - b d)x)z
   +
                 2                                            2    2
     (a i - c g)y  + (a i - c g + a e - b d)x y + (a e - b d)x  + w
   +
                                              2
     ((- d + b)x z + (- d + b)x y + (- d + b)x )e e
                                                 1 2
   +
                               2
     ((- g + c)y z + (- g + c)y  + (- g + c)x y)e e
                                                 1 3
   +
                2
     ((- h + f)z  + ((- h + f)y + (- h + f)x)z)e e
                                                2 3
Type: GrassmannAlgebra(3,Expression(Fraction(Integer)),[[a,b,c],[d,e,f],[g,h,i]])
(9) ->

Maths - Inverse Program

EuclideanSpace