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## Automatic handling of collision of tensor indices in products

Maple

Automatic handling of collision of tensor indices in products

The design of products of tensorial expressions that have contracted indices got enhanced. The idea: repeated indices in certain subexpressions are actually dummies. So suppose  and  are tensors, then in ,  is just a dummy, therefore  is a well defined object. The new design automatically maps input like  into . This significantly simplifies the manipulation of indices when working with products of tensorial expressions: each tensorial expression being multiplied has its repeated indices automatically transformed into different ones when they would collide with the free or repeated indices of the other expressions being multiplied.

This functionality is available within the Physics update distributed at the Maplesoft R&D Physics webpage (but for what you see under Preview that makes use of a new kernel feature of the Maple version under development).

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This shows the automatic handling of collision of indices

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Preview only in the upcomming version under development

Consider now the case of three tensors

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The product above has indeed the index  repeated more than once, therefore none of its occurrences got automatically transformed into contravariant in the output, and Check  detects the problem interrupting with an error  message

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However, it is now also possible to indicate, using parenthesis, that the product of two of these tensors actually form a subexpression, so that the following two tensorial expressions are well defined, and the colliding dummy is automatically replaced making that explicit

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This change in design makes concretely simpler the use of indices in that it eliminates the need for manually replacing dummies. For example, consider the tensorial expression for the angular momentum in terms of the coordinates and momentum vectors, in 3 dimensions

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Define  respectively representing angular and linear momentum

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Introduce the tensorial expression for

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The left-hand side has one free index, , while the right-hand side has two dummy indices  and

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If we want to computewe can now take the square of (11) directly, and the dummy indices on the right-hand side are automatically handled, there is now no need to manually substitute the repeated indices to avoid their collision

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The repeated indices on the right-hand side are now

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