It's been suggested by various wits that the most widely used application of Mathematica is cheating on calculus homework. Actually, Mathematica does most of the heavy lifting in physics, applied math, and many areas of economics. Version 4.1 extends a list of traditional strengths, and makes significant steps toward integration with the Web.
Building on Basics
Mathematica is not only capable of crunching any calculus problem but also can correct the surprisingly frequent errors in standard handbook tables of integrals. The application's steady improvement in the range of integration problems now allows it to solve piecewise functions and integration over regions defined by inequalities.
An important benefit of symbolic-integration research is increased power for solving ordinary differential equations, and version 4.1 can now handle many classes of nonlinear first- and second-order differential equations, as well as all the equations that yield the special functions of mathematical physics. It now makes sense to apply the function DSolve to any differential equation that's even slightly unfamiliar -- you may find some surprising results. Speed and Size
Mathematica has always included a fairly capable statistics package, but it got bogged down with large data sets forcing users to invest in a dedicated statistics program. Regression calculations, one of the most useful applications, were particularly slow. The speed improvement on regression from 4.0 to 4.1 is truly startling -- the first time I tried it on a standard test problem, I thought I had accidentally called up a previously calculated result. Mathematica is now a real alternative to a dedicated statistics program, although a program such as StatView still has better graphics.
What Wolfram did for equations by introducing a complex set of live, functioning notation, they have now done for the Web. Using MathML, an XML dialect that's largely been developed under the active guidance of Wolfram Research, you can simply cut and paste equations as valid input. The commands Save As . . . HTML+MathML and Copy As MathML let you put Mathematica expressions into Web pages, and IBM's free techexplorer Web browser plug-in (for Netscape Navigator or Internet Explorer) lets other viewers view and manipulate those expressions. The combination of these technologies creates a Web math blackboard that's accessible to everyone.
Another element of Web integration with far-reaching potential is the J/Link tool kit. J/Link lets you write Java programs inside a Mathematica notebook, using the full range of Java classes. It also lets Java programs or applets call Mathematica functions. J/Link functionality enables everything from simple online loan calculators to Web-based math textbooks that adapt to a student's individual requirements, to an MBA suite of finance courses. It's good news for the Mac in higher education that Wolfram faithfully supports Mac development, including offering a Linux PPC version of 4.1 and a Mac OS X version in a few months.Integral Logic Mathematica 4.1 can integrate functions over complex regions pieced together with logical operations.