Mathematica and differential topology

[Phrak]

Solving equations in differential topology by pen and paper, and in Microsoft word has been tedious and error prone. Can Mathematica help?

Mathematica would be required to deal with tensor equations on a pseudo Riemann manifold of four dimensions with complex matrix entries. It should be capable of repartitioning into subspaces (space and time) or it would be of little help to me. As well, I require the capability to deal with real valued entries in 8 dimensions having metric signature (2-, 6+) --though ten dimensions would be preferable.

Can Mathematica, Version 7 handle all this?

 

[Dale]

I'm sure it can handle it, but I am not sure how much you could do using the built in capabilities and how much you would need to set things up on your own.

 

[Phrak]

Im not sure what you mean by setting thing up.

As one example,

[tex]G = G_{\mu\nu} = G_{[\mu \nu]}[/itex]

The brakets indicate that G is antisymmetric. G, in this case is the dual of the electromagnetic field tensor, F. *dG is easy enough to partition into spacelike and timelike parts.

$$(*dG)_i = -\epsilon_{i \lambda \mu\nu}\partial_{[\lambda}G_{\mu\nu]}}$$
$$(*dG)_0 = -\epsilon_{0ijk}\partial_{[i}G_{jk]}}$$

(give or take a factor of 2) 'd' is the exterior derivative, '*' is the Hodge dual (antisymmetric tensor and a scalar factor). It doesn't matter much if your not familiar with the notion, really--just the general idea.

Eventually I should recover equations such as these wave equations

$$\frac{1}{2}(*d*dF+d*dG)_{0i} = \partial _t(\nabla x B)_{i} + \partial_i (\nabla \cdot E) - \nabla^2 E_i + ...$$
and
½(*d*dF+d*dG)_jk = ...

This is the sort of partitioning I require. The first is time-space. The second is space-space.

I can make the conversions to vector calculus. It's the interim steps that I have to check and rechech and check again that bother me. Some go on for a few pages--and this is in four dimensions. In eight dimensions I don't trust my math skills.

 

[Dale]

When you write a matrix or tensor equation it is nothing more than a shorthand representation for a whole system of equations. You can certainly use Mathematica to work with the system of equations, and you can use matrix notation as shorthand, but I don't know if the tensor notation is built in. If not, then you would need to do a little bit of preparation to tell the system how to interpret a tensor equation. There is also a pretty decent amount of user-created notebooks available, and if the built-in features do not include tensor manipulation then I am sure someone has developed such a package that you could use.

 

[Phrak]

Thanks, Dale. From what you say, Mathematica can manifupate arrays of at least 8 indices, and if I don't find built-in operations I can build my own? I neeed to find a Mathematica primer.

 

[Dale]

Yes, that is correct. If you need help let me know, I have something like 13 or 14 years of experience with Mathematica.

 

[Phrak]

I appreciate the offer. In the course of researching I found a list of add-ons to mathematica that you might be interested in too. http://en.wikipedia.org/wiki/Tensor#Software_for_use_with_Mathematica"

I'm still reseaching to see if price and performance will suit.