For example, in calculus the definitions of limit and continuity contain expressions such as |x - c| . A function f is continuous at c if for every e > 0 there is a d > 0 such that if |x - c| < d then |f(x) - f(c)| < e.
How can we define continuity for higher-dimensional spaces, such as R2 or any Rn where there is no notion of absolute value? What's really behind the definition is that |x - c| and |f(x) - f(c)| can be understood as distances along the real number line. The definition makes sense in any context in which we have a notion of distance. This includes the familiar example of Rn but also less familiar ones, such as "spaces" whose elements are themselves functions. A set with a suitable notion of distance is called a "metric space"; this is the primary context for 302, rather than specific cases such as R or R2 .
This process of abstraction and generalization has several benefits. One virtue is efficiency: instead of giving new proofs of theorems for each new context, we can give one proof that covers many contexts. Another is understanding: we are able to shed light on what is really essential. For example, it is the notion of distance and not the notion of absolute value that is really at the heart of continuity.
One difference between 302 and courses below the 300-level is the degree of abstraction. Another is that the balance of computation vs. theoretical work shifts dramatically. In calculus, almost all of a student's work is on computations. In Math 206 students spend time on both proofs and computations. In Math 302 most time is spent on proofs, and relatively little on computations. In 302 and other advanced courses you will get a taste of what mathematicians do for a living.
Both 305 and 302 are required for the mathematics major, and one option for the mathematics minor requires either 302 or 305. Which of the two should one take as a first 300-level math course? Both are challenging; which is easier? Students often find their first 300-level course the hardest, whichever it is, since it is there that they first really learn how to construct proofs. Apart from that, some enjoy the algebraic manipulations of 305, while some prefer the visualization that plays a key role in 302. For someone planning a mathematics major it is good to take one or the other soon after 206 since they are prerequisites for other 300-level courses.
Prerequisite: 205; and at least one from 206, 208, 212, 214, 223, 225
Distribution: Mathematical Modeling
