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4: Functions of Two Variables

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    Pre-Calculus Idea – Topological Maps

    If you’ve ever hiked, you have probably seen a topographical map. Here is part of a topographic map of Stowe, Vermont.

    topographical map
    (Figure courtesy of United States Geological Survey and

    Points with the same elevation are connected with curves, so you can read not only your east-west and your north-south location, but also your elevation. You may have also seen weather maps that use the same principle – points with the same temperature are connected with curves (isotherms), or points with the same atmospheric pressure are connected with curves (isobars). These maps let you read not only a places location but also its temperature or atmospheric pressure.

    In this chapter, we will use that same idea to make graphs of functions of two variables.

    • 4.1: Functions of Two Variables
    • 4.2: Calculus of Functions of Two Variables
      Now that you have some familiarity with functions of two variables, it's time to start applying calculus to help us solve problems with them. In Chapter 2, we learned about the derivative for functions of two variables. Derivatives told us about the shape of the function, and let us find local max and min – we want to be able to do the same thing with a function of two variables.
    • 4.3: Optimization
      The partial derivatives tell us something about where a surface has local maxima and minima. Remember that even in the one-variable cases, there were critical points which were neither maxima nor minima – this is also true for functions of many variables. In fact, as you might expect, the situation is even more complicated.
    • 4.E: Functions of Two Variables (Exercises)

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