luis wrote:
> Hello Alan,
>
> Thanks for your answer. Apparently all I need from the exterior
> algebra
> can be done with '^'. If so, I can still use, as you suggest, a
> diferent metric in the setup. (I will probably use a euclidien
> metric)
> and then use the geometric product ('*') to compute the dual: "-I*a".
> I will look tomorrow into the Doran and Lasenby.
>
> Regards,
>
> Luis
>
> -----
>
> On Jan 18, 9:32 pm, Alan Bromborsky <[email protected]> wrote:
>
>> luis wrote:
>>
>>> Hi,
>>>
>>> Thanks,
>>>
>>> Non, I don't want a Minkowsky metric. I am simply doing exterior
>>> algebra in R^4 (or R^n) and in such case I need a null metric
>>> to have the wedge product in Lambda(R^n).
>>>
>>> As you can see
>>> inhttp://en.wikipedia.org/wiki/Clifford_algebra#As_quantization_of_exte...
>>> the exterior algebra Lamda(R^n) is the same as Cl(R^n,Q) with Q a null
>>> metric.
>>>
>>> Do you know another way of 'simulating' the wedge (alternating)
>>> product of the
>>> exterior algebra?
>>>
>>> Luis
>>>
>>> ------
>>>
>>> On Jan 18, 8:19 pm, Alan Bromborsky <[email protected]> wrote:
>>>
>>>> luis wrote:
>>>>
>>>>> Hi,
>>>>>
>>>>> Thanks. Since I want to do exterior algebra, I am using a null
>>>>> metric.
>>>>> Also, I am not in 3-dim and so I don't have a cross product.
>>>>>
>>>>> Let's suppose we have the following code:
>>>>>
>>>>> #######################
>>>>> #!/bin/env python
>>>>> from sympy.galgebra.GAsympy import *
>>>>> set_main(sys.modules[__name__])
>>>>> MV.setup('e0 e1 e2 e3', '0 0 0 0, 0 0 0 0, 0 0 0 0, 0 0 0 0')
>>>>> a = e0*e1*e2*e3
>>>>> ######################
>>>>>
>>>>> We know that in this case
>>>>>
>>>>> a = e0^e1^e2^e3 and
>>>>> *a = 1 (one)
>>>>>
>>>>> but I don't know how to obtain this value (one) with sympy,
>>>>> starting from 'a'. I suspect it should be trivial, but I
>>>>> don't see how to do.
>>>>>
>>>>> Luis
>>>>>
>>>>> ---------------------------
>>>>>
>>>>> On Jan 18, 7:24 pm, Alan Bromborsky <[email protected]> wrote:
>>>>>
>>>>>> luis wrote:
>>>>>>
>>>>>>> Hi,
>>>>>>>
>>>>>>> Is there some way in 'galgebra.GAsympy'
>>>>>>> to apply the Hodge-* operator?
>>>>>>>
>>>>>>> Thanks,
>>>>>>>
>>>>>>> Luis
>>>>>>>
>>>>>>> PS. I am using MV.setup(basis,metric) with a null metric.
>>>>>>>
>>>>>> Seehttp://en.wikipedia.org/wiki/Geometric_algebra. I think the Hodge
>>>>>> dual of a vector, v, is just the geometrics product of the vector and
>>>>>> the pseudoscalar I that is
>>>>>>
>>>>>> v* = v*I
>>>>>>
>>>>>> for 3-dimensions (signature(+++)) we have the relation that the cross
>>>>>> product of two vectors a and b is
>>>>>>
>>>>>> a x b = -I a ^ b
>>>>>>
>>>>>> where I is the pseudo-scalar and ^ is the outer (exterior) product
>>>>>>
>>>>>> seehttp://wiki.sympy.org/wiki/Geometric_Algebra_Moduleforhowto set
>>>>>> up your metric.
>>>>>>
>>>> I think you want (for minkowski metric)
>>>>
>>>> #!/bin/env python
>>>> from sympy.galgebra.GAsympy import *
>>>> set_main(sys.modules[__name__])
>>>> MV.setup('e0 e1 e2 e3', '1 0 0 0, 0 -1 0 0, 0 0 -1 0, 0 0 0 -1')
>>>>
>>>> since metric is orthogonal
>>>>
>>>> a = I = e0*e1*e2*e3
>>>>
>>>> or
>>>>
>>>> a = I = e0^e1^e2^e3
>>>>
>>>> a* = a*I = I*I = +1 or -1 depending on the metric for minkowski metric a*I
>>>> = -1,
>>>>
>>>> for 4-dimensional euclidian a*I = 1 (MV.setup('e0 e1 e2 e3', '1 0 0 0, 0 1
>>>> 0 0, 0 0 1 0, 0 0 0 1'))
>>>>
>>>> remember that for vectors a,b
>>>>
>>>> a*b = a.b + a^b
>>>>
>>>> geometric algebra is a clifford algebra
>>>>
>>>> for higher dimension rotations let U = a^b where U**2 = -1 (always true in
>>>> a euclidian space) define the rotation
>>>> plane and theta the ammount of rotation. The the rotor is
>>>>
>>>> R = cos(theta/2)+sin(theta/2)U
>>>>
>>>> and the rotation of a vector x->x' is given by
>>>>
>>>> x' = R*x*rev(R) (all geometric products)
>>>>
>>>> where rev(R) = cos(theta/2)-sin(theta/2)U
>>>>
>>>> if U**2 = 1 replace cos and sin with cosh and sinh
>>>>
>> The ^ product of geometric algebra is alternating since a^b = -b^a and
>> a^a= 0 for vectors regardless of the metric. Note that the ^
>> (wedge,outer,exterior) product and the | (dot,inner) product are built
>> into GAsympy along with the * (geometric) product. All are defined for
>> arbitrary multivectors. If you use ^ or | be sure to use parenthesis
>> since the python operator priority for ^ and | is not what we would use
>> in a geometric algebra expression. If you have a good library available
>> look at "Geometric Algebra for Physicists" by Doran and Lasenby, page
>> 218. They show the relationship between geometric calculus and
>> differential forms. I haven't dealt with differential forms in 30 years
>> so my knowledge about the correspondence with geometric calculus is
>> probably negative.
>>
> >
>
>
This link may be usefule to you.
http://www.mrao.cam.ac.uk/~clifford/introduction/intro/node12.html
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