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Covariant derivatives of tensors?

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  • Subject: [mg107153] Covariant derivatives of tensors?
  • From: Erik Max Francis <max at>
  • 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:

	    <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 &&
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