Maple 11 has included indicial notation capability to its new Physics package. This is a good start in performing tensorial calculations using Einsteinian summation convention. However, I wish that Maple further adds the following capabilities to make this package of practical use to folks involved in tensor analysis. This wish list is based upon my personal experience in using tensor analysis for continuum-mechanics and is in no way representative of the community involved in other fields like the theory of relativity.
1) Both covariant and contravariant indices are denoted in subscript. Including the convention of posting counter and covariant indices in super and subscripts, respectively, will help to improve the readability of the printed results.
2) The package needs multiple types of independent indices within a given spacetime. For example, in continuum mechanics, small letters are used to denote Eulerian framework and Capital letters are used to denote Lagrangian framework. The Physics package's "Simplify" and "Check" commands fail to understand that these are distinct types of indices. The "Setup" command only allows defining spacetime indices as lowercaselatin. However, for practical use , it needs to allow two or more types of spacetime indices. It also should allow building functional relationships between the variables in two coordinates systems. For example, if X denotes Eulerian coordinates, Y denotes Lagrangian coordinates, one should be able to define X=f(Y).
3) Lot more examples are needed to denote the use of X and Y spacetimes.
4) The metric tensor can be defined only in Euclidean and Pseudo-Euclidean spacetimes. More general definitions should be allowable.
Pawan Singh Takhar
Texas Tech University