Re: Is Mathematica capable of doing this?
- To: mathgroup at smc.vnet.net
- Subject: [mg37505] Re: [mg37469] Is Mathematica capable of doing this?
- From: "Hermann Schmitt" <schmitther at netcologne.de>
- Date: Sat, 2 Nov 2002 03:30:26 -0500 (EST)
- References: <NDBBJGNHKLMPLILOIPPOMEFKDEAA.djmp@earthlink.net> <200211010643.BAA11341@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Hello,
you may use entries with Head "Tensor", e.g. Tensor[xyz], where xyz
identifies an specific Tensor.
Then you can define functions Plus, Times, ... in the form:
Plus[x__Tensor] := .......
x__ Tensor means one ore more expressions with Head "Tensor".
Hermann Schmitt
----- Original Message -----
From: "Liguo Song" <Liguo.Song at vanderbilt.edu>
To: mathgroup at smc.vnet.net
Subject: [mg37505] [mg37493] Re: [mg37469] Is Mathematica capable of doing this?
> Thanks for the helpful input. I have been playing with the idea for a
while. And
> here are my answers for the questions that you brought out.
>
> First, about the super/sub-scripts. I can easily use a list of True/False
for
> sub/super-script. I can combine this list with the List that represent the
> tensor together to for a new object, Tensor.
>
> Second, output formatting can be done easily with
SubsuperscriptBox[string, sub,
> sup], where string represents the name of the Tensor, sub/sup are strings
to
> represent the scripts for the tensor. Spaces can be used to align the sub
and
> super-scripts.
>
> So, it boils down the my first question, how can I define a object,
Tensor,
> which will behave like Complex? So, I can redefine Times, Plus, Minus, and
other
> operators to handle Tensor.
>
> Also, how can I relate a symble to a Function as + relates Plus, * relates
to Times?
>
> Again, thanks for your thoughts on this topic.
>
>
> Liguo
>
>
> David Park wrote:
> > Liguo,
> >
> > I think it is a "dragon's egg" and certainly not the best way to learn
> > Mathematica. Basically you would have to Unprotect and add new
definitions
> > to Times and that would be only the start of it because how are you
going to
> > distinguish between superscripts and powers? How are you going to handle
> > mixed up and down indices? How are you going to get nice output
formatting?
> >
> > There are many nice tensor packages out there. The moderator of this
group
> > has the original powerful tensor package. As a way of learning some
tensor
> > calculus I have been working with Renan Cabrera on a package called
> > Tensorial. It can be obtained at my web site below. It is oriented
toward
> > learning the basic mechanics and reproducing textbook problems. You can
have
> > any symbols for tensor labels or indices. The index domain can be any
range
> > of numbers or a set of symbols. For example, {0,1,2,3} or {t,x,y,z} for
> > relativity problems. You can have colored indices to distinguish
different
> > coordinate frames.
> >
> > Here is how one would do your two problems in Tensorial.
> >
> > Needs["TensorCalculus`Tensorial`"]
> > SetMetric[{x, g}, IdentityMatrix[3]]
> >
> > DefineTensorShortcuts[{T, g}, 2]
> >
> > guu[u, v]Tdd[v, k]
> > % // MetricSimplify
> > (formatted output)
> > (formatted output, but Tud[u,k] in shortcut notation.)
> >
> > guu[u, v]Tdd[u, v]
> > % // IndexEinstein
> > (formatted output)
> > (formatted output but Tdd[1,1] + Tdd[2,2] + Tdd[3,3] in shortcut
notation.)
> >
> > The DefineTensorShortcuts statement defines T and g as labels of second
> > order tensors. The various up and down index configurations can be
specified
> > by appending "u"'s or "d"'s to the tensor label. So, for example,
gud[i,j]
> > is the shortcut for g with the first index i up, and the second index j
> > down. Isn't that easier than maneuvering between superscripts and
> > subscripts? MetricSimplify automatically carries our the raising or
lowering
> > of indices with the metric tensor. IndexEinstein automatically carries
out
> > summations on paired up and down indices.
> >
> > David Park
> > djmp at earthlink.net
> > http://home.earthlink.net/~djmp/
> >
> >
> > From: Liguo Song [mailto:Liguo.Song at vanderbilt.edu]
To: mathgroup at smc.vnet.net
> To: mathgroup at smc.vnet.net
> >
> > Dear MathGroup,
> >
> > I am in the process of learning to use Mathematica. Here are a couple of
> > questions that I want to ask the group.
> >
> > 1) Can I define a new object, Tensor, which will act like Complex? So,
two
> > Times[TensorA, TensorB] or TensorA*TensorB will invoke proper Times
function
> > to
> > handle it.
> >
> > 2) If the answer to the above question is yes, then can I use
> > super/sub-scripts
> > to represent the indices for the Tensor, and carry out the calculation
based
> > on
> > these indices? Such as, g^uv*T_vk will get T^u_k, which essentially
raises
> > the
> > first index of T_vk. Another example would be g^uv*T_uv will get a
scalor T.
> >
> > I know there are couple of Tensor analysis packages, comercial and free,
out
> > there. But, all the free packages I looked through won't be able to do
this.
> > And, figuring out how to do stuff is the best to learn how to use
> > Mathematica.
> >
> > Maybe, I am pursuing a dragon egg here. But, I'd still like to hear
about
> > how
> > well Mathematica can do to imitate this behavior.
> >
> > Thanks for any input on this.
> >
> >
> > Liguo
> >
>
>
>