Section 1 Introduction
Calculus I and II are the study of functions of a single real variable, using the techniques of limits, derivatives, intergrals and series. In Calculus I and II, we built a set of very powerful tools to understand functions, their increase and descrease, their asymptotic growth, their behaviour near undefined points, their maxima and minima, their slopes, and the areas under their graphs. Qualitatively, we can describe and visualize single-variable functions very precisely using the tools of calculus.
However, all the functions studied so far have been functions of one real variable with one real output. This course starts with the obvious observation that a function can depend on numerous variables and have outputs in several variables. Vector calculus asks this question: What can the techniques of calculus tell us about these multi-variable functions? How do the major ideas of derivatives, integrals and series extend to situations with more than one variable?
The short answer is that they all extend; moreover, they extend in deeply fascinating, mysterious and intricate ways.