RE: Is Mathematica capable of doing this?
- To: mathgroup at smc.vnet.net
- Subject: [mg37481] RE: [mg37469] Is Mathematica capable of doing this?
- From: "David Park" <djmp at earthlink.net>
- Date: Fri, 1 Nov 2002 01:42:57 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
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
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