Multivariable Mathematics - MATH 3500(H)–3510(H)
MWF 11:15–12:05, T 11:00–12:15
This is an integrated year-long course in multivariable calculus and linear algebra. It includes all the material in MATH 2270/2500 and MATH 3000, along with additional applications and theoretical material. There is greater emphasis on proofs, and the pace is quick. Typically the class consists of a blend of sophomores (some of whom have had MATH 2400(H)–2410(H), others of whom have had MATH 2260 or 2310H and MATH 3200) and freshmen who've earned a 5 on the AP Calculus BC exam. The text is my book, Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds.
Students who are unsure about what math class to take should contact me during the summer. Some students who would like to take MATH 3500(H) but aren't sure whether they will like it should give it a shot; if your schedule allows it, we can do a "section change" to MATH 2270 even after two or three weeks. Students who feel like they need more confidence in writing proofs should consider taking MATH 3200 concurrently in the fall semester. So far as grades are concerned, students who master the computational content of the course (the standard 3000 and 2270 material) ordinarily earn at least a B.
Students who would like some guidance in reading and writing proofs might want to look at a wonderful new book called How to Think Like a Mathematician: A Companion to Undergraduate Mathematics, by Kevin Houston, Cambridge University Press, 2009. You can get it used for under $25.
Differential Geometry - MATH 4250/6250
SPRING TR 9:30–10:45
This is an undergraduate introduction to curves and surfaces in R3, with prerequisites of either MATH 2270 (2500) and MATH 3000 or MATH 3510(H). The course is a study of curvature and its implications. The course begins with a study of curves, focusing on the local theory with the Frenet frame, and culminating in some global results on total curvature. We move on to the local theory of surfaces (including Gauss's amazing result that there's no way to map the earth faithfully on a piece of paper) and heading to the Gauss-Bonnet Theorem, which relates total curvature of a surface to its topology (Euler characteristic). As time permits, we'll discuss either hyperbolic geometry or calculus of variations at the end of the course.