The aim of this book, intended as a companion to a traditional text, is to explore the notions of multivariable calculus using a computer as a tool to help with computations and with visualization of graphs, transformations, etc. The software tool we have chosen is Maple; one could as easily have chosen Mathematica or Matlab. In some cases the computer is merely a convenience which slightly speeds up the work and allows one to accurately treat more examples.
In others it is an essential tool since the necessary computations would take many minutes, if not hours or days. We will, for example, use Maple to study the temperature distribution in a thin flat plate by reducing the problem to the solution of a system of, say, one hundred equations in one hundred unknowns, then using the resulting numerical data to construct a contour plot which shows lines of equal temperature. All this could be done by hand, but it would be laborious work indeed. Such problems would be out of reach without tools for computation and visualization.
Difficult computations and fancy pictures are, of course, not ends in themselves. We must understand the underlying mathematics if we are to know which computations to do and which pictures to draw. Likewise we must develop our own intellectual tools sufficiently well in order to understand, interpret, and make use of the data and images that we “compute.” Thus our focus will always be on the mathematical ideas and their applications. The role of Maple is to more vividly illustrate them and to widen the range of problems that we can successfully solve.

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