Here are some Mathematica packages that I have written.
VEST (Vector Einstein Summation Tools)
Carries out abstract vector calculus computations and simplifications. VEST is designed to be simple and intuitive to use, while also being powerful enough to handle very large numbers of terms and find complicated multiterm identities. Along with a tutorial, the package can be downloaded here.
NMStools (Nonmodal stability tools)
Contains a few useful functions for studying the nonmodal stability of linear systems of differential equations (in 1-D). The most useful function is EigenNDSolve, which works like the native Mathematica function NDSolve, but solves for the full spectrum of the specified eigenvalue boundary differential equation. NMStools can be downloaded here.