To see some of the abilities of the new Mathematica, see
Demonstrations.Wolfram.COMFrom the last website I mentioned, let me recommend you e.g. this article about image processing - a new set of capabilities besides built-in parallel computing, geodesy & GIS, and other new things:
Reference.Wolfram.COM (standard help pages)
- Search the previous web e.g. for surface and browse for a while
Screencasts (short educational videos, try this 2-minute one)
The Learning Center (screencasts, seminars, tutorials)
MathWorld (a Wikipedia-like math encyclopedia with lots of technical references and many Mathematica notebooks)
Truth to be told, I had to convert my images to BMP to be able to simply drag them to the Mathematica windows. But it was worth the hassle. (Update: the importing has been fixed after a very helpful, hard-working employee of Wolfram Research told me how to change a default directory so that the value doesn't contain "š" from "Luboš". Now, I can import JPG both locally and from the web easily.) The following screenshot (click to zoom in) shows a Nobel prize winner, including a spousal unit, partitioned into some matrices (besides some gamma matrix and quasinormal windows):
If you have any doubts that my copy of the program can evaluate the rest, here is the Dynkin diagram of David Gross's 10 x 10 matrix. Be sure he's neither simple nor compact as a group but he can remove a sword from your throat if you happen to eat it.
But it is still Mathematica, with its amazing abilities to formally and numerically manipulate with expressions, functions, matrices, factorize, simplify, integrate, differentiate, solve equations, draw functions etc. If your living standards are above the average of your country and you are into maths, you should seriously buy your own copy unless you have another access to it!
You may also search for Mathematica on this blog. You will find articles about Mathematica 6 (just 18 months ago), STRINGVACUA, LHC olympics, Wolfram Publicon, spheres turned inside out, and many other things. Yes, I could now verify that Larry Summers' triple integral is 128/45 Pi = 0.905415, close to his estimate 1.