Thinking about multivariable functions

The only thing separating multivariable calculus from ordinary calculus is this newfangled word "multivariable". It means we will deal with functions whose inputs or outputs live in two or more dimensions. Here we lay the foundations for thinking about and visualizing multivariable functions.

Welcome to multivariable calculus! Soon you will learn how to apply to tools of calculus to multivariable functions, but to start things off let's start getting a feel for what these multivariable functions can actually look like.

The main two tactics for understanding scalar-valued functions are graphs and contour maps.

To understand vector-valued functions, it's common to either think parametrically, in which you think of the function as drawing a curve or surface in the output space, or with a vector field, in which you plop a vector on various points in space.

Thinking about a function as a transformation means thinking about how it moves points from the input space to the output space. A nice way to visualize this is with animations that actually move space.

Having an image to hold in your head as you think about a multivariable function can be crucial for understanding that function. Graphs, contour maps, parametric functions, vector fields and transformations all provide different ways to visualize functions in higher dimensions.