Covariant derivatives of tensors?

• To: mathgroup at smc.vnet.net
• Subject: [mg107153] Covariant derivatives of tensors?
• From: Erik Max Francis <max at alcyone.com>
• Date: Thu, 4 Feb 2010 06:26:34 -0500 (EST)

```Working on my tensor library, I'm trying to implement the covariant
derivative for an arbitrary-rank tensor.  I'm keeping track of which
indices are contravariant/upper and covariant/lower, so the problem
isn't managing what each term would be, but rather I'm having difficulty
seeing how to take an arbitrary tensor and "add" a new index to it.
This in effect requires running Table with an arbitrary number of
indices, and then adding one.  Given the arbitrariness of the
multidimensional array, I'm not seeing how to do it.  The naive approach
would be something like:

Table[
<complex function involving many terms of a[[i1]][[i2]]...>
{j, n}, {i1, n}, {i2, n}, ... {ir, n}]

where the variable si1 .. ir (r of them) range over the value 1 through
n for each of the indices of the tensor (of rank r), j is the additional
index added by the covariant derivative, and n is the dimensionality of
the space.

I'm not seeing how to do this dynamically, since I don't know in advance
what the rank of the tensor is, and I'm still relatively new to
Mathematica.  Any ideas?

--
Erik Max Francis && max at alcyone.com && http://www.alcyone.com/max/
San Jose, CA, USA && 37 18 N 121 57 W && AIM/Y!M/Skype erikmaxfrancis
All the people in my neighborhood turn around and get mad and sing
-- Public Enemy

```

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