Brief Sketches of Post-Calculus Courses

algebra has many classical and new open problems. Much research now focuses on the interface between algebra and other fields. See http://www.openproblemgarden.org/category/algebra http://www.openproblemgarden.org/category/group theory and https://en.wikipedia.org/wiki/List of unsolved problems in mathematics. For a fuller, yet still brief, description, see https://en.wikipedia.org/wiki/Abstract algebra Prerequisites for Math 4121: Math 2971 and either Math 2184 or 2185. Prerequisites for Math 4122: Math 4121. MATH 4239 AND 4240, REAL ANALYSIS, I AND II The word analysis in mathematics refers to the study of functions and their limits. Hence real analysis is the study of functions of a real variable or of several real variables. The first collegelevel course in analysis is calculus 1 and 2, the familiar first-year course about differentiation and integration of functions of a single real variable. Math 4239, the first semester of the real analysis sequence, revisits the same ideas as are discussed in calculus 1 and 2: limits, continuity, differentiation, integration, sequences, and series. The course is sometimes called Advanced Calculus. (In this context, we refer to the calculus 1 and 2 sequence as Elementary Calculus.) Even though the topics in advanced calculus seem to be the same as in elementary calculus, the perspective is so different that it sometimes seems to students that they are entirely different subjects. In the elementary courses, almost every problem is about some formula, like x sinx or e−x 2 or arctan(ln(x + 1)), and the goal is to execute some computation with that formula. In Math 4239, hardly any problems ask specifically about one formula; instead, you learn what must be true about any formula. So in the elementary calculus sequence, a typical problem might begin “Suppose f(x) = x+tan(x).” In Math 4239, a typical problem might begin “Suppose f is a continuous function from R to R.” This is the abstract viewpoint, which we adopt because it is efficient and enlightening. Math 4239 investigates the foundations behind what is taught in calculus 1 and 2. We investigate various conditions that ensure the existence of limits, derivatives, and integrals. In calculus 1 and 2, we start with nice formulas that tend to automatically guarantee the existence of such things, and we simply compute them. The focus of Math 4239 is the assortment of theorems that assert such guarantees. So real analysis is the theory behind calculus. Here are a few questions that illustrate ideas found in Math 4239: • Can a function from R to R be continuous at all x but differentiable at no x? • Can a function from R to R be continuous at every irrational number but continuous at every rational number? • Does every continuous function on [0, 1] have a definite integral? • Under what circumstances does a Maclaurin series represent the function from which it came? To analyze such questions, certain concepts are introduced that typically are not seen in the elementary calculus sequence. These concepts include open and closed sets, completeness, compactness, liminf and limsup, bounded variation, uniform continuity, and uniform convergence. Part II of the course, Math 4240, covers the theory behind multivariable calculus. We develop an understanding of the derivative as a linear operator, with the familiar chain rule manifesting itself as a theorem about the product of matrices. We rework the definition of the Riemann integral in more than one dimension. We state, prove, and use the implicit and inverse function theorems. And if time permits, we may introduce the idea of a differential form in order to understand what is called the generalized Stokes theorem, which is a grand generalization and unification of the fundamental theorem of calculus, the fundamental theorem of line integration, Green’s theorem, Gauss’s theorem, and Stokes’s theorem. The real analysis courses sit centrally between pure and applied mathematics. They display the abstraction and rigor of pure mathematics, but the objects of study are at the center of all applied mathematics. Every student contemplating post-graduate study in mathematics (pure or applied), physics, or economics should complete this sequence, or at the very least the first half of it. Prerequisites for Math 4239: Math 1232 and Math 2971. Prerequisites for Math 4240: Math 2184, Math 2233, and Math 4239.