In this chapter we explore the concept of a mathematical function. Undoubtedly, functions have been prevalent in your previous mathematical experiences. You may have spent a lot of time working with functions given by an explicit formula, such as. In this chapter we develop a broader understanding of functions, presenting a commonly used and intuitive (but incomplete) definition in Section 5.1, followed by a variety of related terminology and concepts, and then a rigorous definition in Section 5.5.

What is a Function?

The notion of a function is broader than numerical functions that are given by explicit formulas.

DEFINITION 5.1 (Initial Version). Let A and B be sets. A function from A to B is a rule that assigns to each unique associated element.

There are two standard ways to represent “f is a function from A to B.” One is to write

and the other is

Both notations are supposed to indicate that the function f takes an input element and outputs an element, and we express this by writing f (a) = b. In the sections that follow, you will often read statements like “Let f : A → B.” This is a short but complete sentence stating that f is a function from A to B.

For example, consider the function g defined in a piecewise fashion for real numbers x:

The formulas and both give real numbers if is a real number, so we could use the notation g : because both A and B equal. As another example, you might have encountered parametric curves which trace out patterns in the plane. One of these gives a circle parametrized by

as illustrated in Figure 1. Here the input is a real number t ∈ R and the output is a point, so we write or.

The Hilbert Hotel, Count von Count, and Cookie Monster

Below are two stories. The first is well known within the mathematics community; the second is presented with two classic characters from children's television. In both cases, the stories highlight that our intuition about counting elements and determining the size of finite sets does not always transfer well to the setting of infinite sets.

The Hilbert Hotel. David Hilbert was a German mathematician who contributed transformative results in a number of areas of mathematics and provided valuable vision and leadership to the mathematics community. Because he introduced the following example in the 1920s, it is often described as the Hilbert Hotel. The Hilbert Hotel has many rooms. In fact, each room is numbered with a natural number, and there are just as many rooms as there are natural numbers. Thus there is a room #1, a room #2, …, a room #n, … and so on.

The Hilbert Hotel has acquired a nice reputation among tour group bus drivers, since it always seems to be able to accommodate very large tour groups. For instance, a bus drove up one evening that held infinitely many passengers: every passenger's shirt was labelled with a natural number, and every natural number appeared on just one shirt. This did not pose a problem for Hilbert, of course, because he made a single, precise announcement over his intercom system that told everyone where to go: “Go to the room whose number corresponds to the number on your shirt.” So passenger #1 went to room #1, passenger #2 went to room #2, and so on.

Later that evening, a single person drove up in a car and asked for a room. Hilbert couldn't send the person to the first vacant room, since there were no vacancies. Instead, Hilbert simply made the following announcement over the intercom to everyone who already had a room: “I apologize for the inconvenience, but in order for us to accommodate a new guest, we need you to change rooms. If you are in room #k, please move to room #(k + 1) for the night.

Differential forms constitutes an approach to multivariable calculus that simplifies the study of integration over surfaces of any dimension in Rp. This topic introduces algebraic techniques into the study of higher dimensional geometry and allows us to recapture with rigor the results obtained in the preceding chapter.

There are many approaches to defining and exploring differential forms, all leading to the same objective. One is to just define a form as a symbol with certain properties. That was the approach taken when I first encountered the subject. Though this appeals to many and ultimately leads to the same objective, it strikes me as inconsistent with the approach we have taken. So in this chapter I have decided to follow the approach in [7] where a form is defined in terms that are more in line with what I think is the background of readers of this book. The reader might also want to look at [1], [9], and [10] where there are different approaches.

Introduction

Here we will define differential forms and explore their algebraic properties and the process of differentiating them. The first definition extends that of a surface as given in the preceding chapter.

Definition. Let q ≥ 1. A q-surface domain or just a surface domain is a compact Jordan subset R of Rq such that int R is connected and R = cl (int R). A q-surface in Rp is a pair where R is a q-surface domain and is a function from R into Rp that is smooth on some neighborhood of R. The trace of is the set. If G is an open subset of Rp and, we say is a q-surface in G; let Sq(G) be the collection of all q-surfaces contained in G.