@Earl Edgardo's posts are very good examples. For instance:

https://www.mapleprimes.com/posts/208437-ACA-2017--Computer-Algebra-In-Theoretical

An example of line elemnt is given in ?Physics,Setup

@John Fredsted Using the Setup's keyword Dgammarepresentation = chiral might be helpful:

@John Fredsted Well, that’s the answer as the question was first formulated. Please, note that an expression such as g_[mu,nu] is a tensor. This notation generalizes the matrix formalism. So mixing tensor and matrices doesn’t really make sense. As your problem seems quite complex, to give an overview, maybe could you publish a wider section of the calculus you’d like to perform…

@Joe Riel Yes, that's an interesting alternative, with the advantage of Oriented Object Programming (OOP) to create a multiple instance of the same object.

@Axel Vogt That's a good idea, but the reason to use a local variable is to avoid a collision with another variable in the worksheet that could have the same name. The example I provided is very simple, but this is intended to be included in a complex development.

@Carl Love Thanks for this detailed answer, that's exactly what need!

Hi Edgardo,

Very good improvement!

@ecterrab Thanks for this very interesting detailed answer! I understand that the problem is quite complex, and that this procedure could not be directly defined as a tensor. However, I find an interest in vv’s suggestion. Indeed, it is possible to proceed in two steps. First, formally define the differential operator using the new Setup/differentialoperators. As a tensor, this operator is indexed. The difficulty is to take into account this index value when making the operator definition explicit. Here is an example of application:

IndexedProcedure.mw

@vv Simple and elegant. After a few tests, it seems that's what I need. Thanks!

@Preben Alsholm Even after loading "ListOperations", you can still have the feature you describe by the use of the operator "~" for element wise operation:

@J F Ogilvie I do not have the same opinion. It is correct that Maple and the Physics Package is excellent for GR, but it is just incorrect to suggest that it is restricted to this field. As a professional physicist and researcher, I use it on a daily basis. It is a very powerful tool that can address many areas of Quantum Mechanics. Its capabilities spans from functional derivative to some very complex non-commutative algebraic operations, to mention only a few of its possibilities. Furthermore, it should be considered as a whole. Even if one is not directly involved in GR for instance, because of GR, the tensor engine of Maple is very powerful and simple to use, and an incredible number of fields requires tensor calculus.

@Thomas Richard Thanks, that’s exactly what I was looking for! It works perfectly.

@ecterrab 

Edgardo,

Thanks so much, the fix works perfectly!

Value@ThU The command value will restore the active mode.

@tontu: To enhance the display, enter first:

Physics:-Setup(mathematicalnotation = true)