In my opinion any serious mathematician should at least be aware of the computational tools available nowadays. The package Sage is a free multi-purpose package for doing math. It is similar functionality to Mathematica, Maple and Matlab but has several significant advantages. Perhaps the most important one is that it is free and runs in your browser without the need to install anything. Below are a few simple examples that you can run directly on this page. Just push the evaluate buttons and try to hack the code to suit your own desires. If you want to know more, here is a place to start. For heavier computations, saving, collaborating and sharing your work, consider signing up for the sage cloud (free).
Plotting vector fields. Push the evaluate button below to see a plot of the vector field. You may also replace the functions of $x$ and $y$ with your own. One thing to take into account is that you should write $2x$ as $2*x$.
Simple Plot Here is a plot of a smooth bump function (Gaussian scaled by tangent).
Differentiation Tired of differentiating by hand? Try this.
Integration What about integrals?
Programming One of the great things about sage is that it is fully integrated with the popular programming language Python. For example guess what this does: