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MathGroup Archive 2006

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Re: vector derivatives

  • To: mathgroup at smc.vnet.net
  • Subject: [mg70327] Re: [mg70289] vector derivatives
  • From: Christopher Arthur <caa0012 at unt.edu>
  • Date: Fri, 13 Oct 2006 01:30:16 -0400 (EDT)
  • References: <200610120936.FAA04558@smc.vnet.net>

The Tensorial package is probably what you want.  I've played with it a 
little bit, and for example, it lets you use einstein summations in 
your notation--meaning that  x_i y^i represents a sum over i of the 
components of x and y.  But it's not an easy package to use, IMO, and 
if you want to do some computation rather than just notation and 
algebra, probably it isn't so useful. Quoting rych <rychphd at gmail.com>:

> I have (n) particles {ri} (ri- radius-vectors) with pairwise
> interaction potential (u) depending on the distance only
> (|rij|=|ri-rj|) (in d=3 Euclidean space). I'd like Mathematica to find
> the force, f_i and higher partial derivatives of the potential energy
> U:
> U = 0.5 \sum_i,j u(|rij|)
> -fk= \nabla_k U= \sum_j u`(|rkj|) rkj/|rkj|
> \nabla . \nablaU = \sum_i,j u``(|rij|)+(d-1)u`(|rij|)/|rij|
> and so on.
>
> And I'd like to have the results in that compact form. I start writing
> an exersise like this in Mathematica, - not pretty at all and the
> output is far from what I want.
> r1 = {a1, b1, c1}; r2 = {a2, b2, c2};
> l[r12_List] := Sqrt[r12 . r12]
> u[r1_List, r2_List] := l[r1 - r2]^n
> D[l[r1 - r2], r1]
> D[u[r1, r2], r1]
>
> What is the proper way of doing such a task in Mathematica (and in
> mathematics)? If the Tensorial package is the way to go, I don't mind
> learning it, metric tensors, differential forms and such.
> Thanks
> Igor
>
>



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