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Is Mathematica capable of doing this?


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



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