Why this course: multivariable calculus?

In calc 1 and 2, one learns about limits, derivatives and integrals of a function [default name f(x)] of a single independent variable [default name x]. To visualize them we introduce a dependent variable y and study the graph, which is the curve y = f(x) in the x-y plane. Entering calc 3, it is assumed that you can differentiate and integrate explicit expressions, and you have graphing calculators and Maple computer software on your laptop to easily make sure you get these mechanical operations correct. There is no excuse for getting them wrong, since a few seconds of keyboarding in Maple will check them for you. These mechanics that deal with the actual formula for a function are the least important part of calculus. Rather it is understanding how to use the results of those mechanics and what they mean in the context of a generic function of a single variable that is important.

Now we want to increase the number of dependent variables, then the number of independent variables, then both together, and understand how the single variable calculus ideas fit into the context of more than one variable and how we can visualize them geometrically. Increasing the number of dependent variables keeping only one independent variable leads to the vector valued functions of a single variable which can be visualized as curves in the plane, in space, in higher dimensional spaces. Increasing the number of independent variables keeping one dependent variable leads to single real valued functions on the plane, in space, on higher dimensional spaces, which can be visualized as surfaces in space or in higher dimensional spaces through their graphs, or in terms of curves in the plane or surfaces in space, etc., if we look at the sets of points with the same function value (contour plots). Increasing both the number of independent and dependent variables simultaneously leads to the idea of vector fields in the plane, in space, etc.

We already know how to take limits, differentiate and integrate. In multivariable calculus we need to learn what these operations mean in the context of more than one variable, and what geometry we can use to visualize them and the associated concepts. In particular, a key concept in dealing with more than one variable is the concept of parametrizing sets of points in order to describe subsets of a space, like curves in the plane or in space, or surfaces in space, or regions bounded by curves in the plane, or by surfaces in space, etc. Coming up with a parametrization (a quantitative description) of the set of points is required for integrating along a curve, or between curves, or between surfaces. Using these descriptions, one reduces integration over curves, surfaces and regions between them to iterated integrals, namely nested (successive) integrals each with respect to a single independent variable. The goal of integration theory is to learn how to arrive at these parametrizations in order to reduce integration to operations we already know how to do. The goal of differentiation here is similarly to elevate the calc1 derivative to various derivative operations in the multivariable calculus context and understand visually what they mean in the geometry of the curves, surfaces, and regions that we must deal with.

In calc2 we have already gotten a little experience with describing regions of space over which we want to integrate using the "method of disks/washers" to find the volume enclosed by a surface of revolution, which only requires one independent variable to describe because of the rotational symmetry. [This is a really just a first step exercise in dealing with calculus in 3 dimensions, since not many of us are interested in volumes of revolution for their own sake!]

The fundamental theorem of calculus shows how differentiation and integration are related to each other and how antidifferentiation can be used to evaluate integrals. In multivariable calculus, similar more complicated relationships exist between multivariable differentiation and integration that are often encountered in physics class without proper mathematical preparation. At the end of the semester we will arrive at these relationships, although stop short of studying all of them [Gauss's law and Stoke's theorem in 3 dimensions that you might have encountered in simpler forms in physics are both aspects of Green's Theorem in the plane.]