Friedrich Hagedorn wrote:
> On Mon, Oct 06, 2008 at 06:26:11PM -0400, Alan Bromborsky wrote:
>
>> Yes almost, but can you suppress the (x1,x2,x3) in the output so that
>> the output would be
>>
>> A*dB/dx1+B*dA/dx1 ?
>>
>
> I dont think that
>
> dA/dx1
>
> is a good idea for a CAS. This notation means the partial derivative of
> A(x1, x2, x3) in respect to x1. But x1 is only a 'local symbol' in the
> function A. The following terms are equivalent
>
> diff( A(x1, x2, x3), x1) == diff( A(b, c, e), b)
>
> == diff( A(x, y, z), x)
>
> Thats why there is an other notation
>
> diff(A, 1)
>
> means the partial derivative of A in respect to the first Argument of A.
> This notation it unique but not so convinient to read.
>
> By,
>
> Friedrich
>
> PS.: What is your original problem?
>
> >
>
>
I am calculating the geometric derivative of a multivector field. The
most general field for space time (4 basis vectors) can have 16 real
coefficients that are a function of the 4-position vector x =
(x0,x1,x2,x3) (note that x0=ct). There are one scalar, 4 vector, 6
bivector, 4 trivector, and one pseudoscalar coefficents. Lets just
consider the bivector coefficients B^{ij} which are functions of x. The
standard notation for the derivatives of B^{ij} would be
\partial_{k}B^{ij}. I am looking for a notation that duplicates the
standard notation as closely as possible both when printed out and in
the python program. I know from context that B^{ij} is a function of x
and do not want to have B^{ij}(x0,x1,x2,x3) repeated over and over in a
long expression especially if the B^{ij} are themselves expressed as
functions of implicit and explicit functions. For example if B^{12} =
x0*g^{12}(x0,x1,x2,x3)+x1*f^{12}(x0,x1,x2,x3) , having the
(x0,x1,x2,x3) appear in every implicit function and in every derivative
of a implicit function would be confusing, especially in long
expressions containing many terms.
For reference the most general multivector field is written in terms of
the real coefficients a, B^{ij}, T^{lmn}, and p using the summation
convention with j<k and l<m<n:
A = a
+v^{i}\gamma_{i}+B^{jk}\gamma_{k}\gamma_{k}+T^{lmn}\gamma_{l}\gamma_{m}\gamma_{n}+
p\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}
where the space time basis vectors are \gamma_{0}, \gamma_{1},
\gamma_{2},\and gamma_{3} and the geometric products of the basis
vectors are completely antisymmetric.
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