I have a couple abstract indexed quantities, both differential elements
$dx = dx^\mu e_\mu + x^\mu de_\mu$
$du = du^\mu e_\mu + u^\mu de_\mu$
I can compute the expression $(dx + du) \cdot (dx + du) - dx \cdot dx$ manually on paper, contracting the products appropriately and taking the differences. However, evaluating that end result for specific parameterizations and basis representations becomes messy (I've done only the 3D Euclidian cartesian and cylindrical coordinate cases).
This seems to be a perfect opportunity for a symbolic computation engine, but I'm having trouble starting. The mathematica book hints that tensor expressions can be represented by lists, but an expression of the above form does not require any specific dimensionality, and can still be symbolically manipulated (on paper).
If I use a list, does that list not have to have specific dimension? For example, I'm guessing that I could use an explicit 3D representation for the upper index coordinates like so:
x = { x1, x2, x3 }
u = { u1, u2, u3 }
then define a metric tensor for the lower index coordinates. I don't know how I'd represent the basis vectors $e_\mu$ in Mathematica though, even if I restricted myself to 3D Euclidean spaces.
Is this sort of computation within the scope of Mathematica, and if so, how does one setup the variables?
Answer
On this wikipedia page you find a collection of Tensor software and Mathematica has the biggest section.
The package Ricci, which username acl pointed out in his answer is there, and I personally have used xAct. It looks like this
http://img692.imageshack.us/img692/211/pic1ni.png
And yes, as you suggest in your question, for smaller computation in specific dimensions you can also work in components directly. For me, this usually looks something like this (only have a screenshot atm.).
http://img835.imageshack.us/img835/5023/bild4go.png
Although the expression $de_\mu$ makes me think the package variant is best suited for you.
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