Karthik C has a very nice video entitled Emacs-Calc: The Poor Man’s Mathematica. It’s a tutorial on some of the features of Emacs calc. He begins by covering the basic stack machine nature of calc and how to perform simple calculations. He quickly covers some—but by no means all—of the vector calculations, and then moves on to symbolic computations.
He shows to to symbolically solve various types of equations. This is tremendously powerful and mirrors the capabilities of Mathematica. He also demonstrates how to do symbolic differentiation and integration. Finally, he shows how to solve numerically when a symbolic solution is not available.
The video is just short of 49 minutes and covers a lot of ground but it still only scratches the surface. It’s a really great introduction to what calc can do and even if you don’t remember everything he shows you, the help system will remind you of the commands once you tell it what you want to do.
I use calc all the time but mostly for simple calculations so it’s nice to have a video that reminds me of some of its more complicated capabilities. Of course, the manual is very complete if you need in-depth information. If you’re just starting, you’ll want Sue D Nymme’s quick reference guide. It far and away the best quick reference for calc.