Tuesday, July 21, 2009

Beyond Calculus II: Honors Vector Calculus or not...

A common question I often get from our very ambitious undergraduates focuses on a choice of vector calculus classes we offer. Vector Calculus, Calculus III, and Multivariable Calculus are all names for the same basic study of the properties of functions of more than one independent variable. This material is required for most engineering disciplines, as well as mathematics, and most of the natural science majors here at Hopkins. Since the techniques and material makes a lot more sense once students have studied most of the properties of functions of one independent variable, this course naturally follows the Calculus I and II sequence.

We have two flavors of vector calculus here at Hopkins:

110.202 Calculus III and 110.211 Honors Multivariable Calculus

The basic question is; Which should I take?

The basic answer is: depends....

Both of these courses fulfill the same requirements for all majors and minors that require multivariable calculus. Both can serve as prerequisite courses for any higher level course that requires multivariable calculus. Both cover the same basic material over the length of one semester, and run from the basic notions of vectors, matrices and the real space R^n through notions of continuous and differentiable functions of more than one independent variable, ending the basic material with the final major theorems tying together major aspects of the course: Green's, Stokes' and Gauss' Theorems.

The major difference between these two courses is one of focus. 110.202 Calculus III is more of a standard Calculus course, developing a blend of theoretical background on the nature of functions of more than one independent variable and the actual calculations involved in solving problems pertaining to this material. 110.211 Honors Multivariable Calculus, on the other hand, spend much more time on the theoretical nature of the material, digging deeper into the "why" aspects of calculus instead of "how things work". Students in the latter will develop a better understanding of content like the Inverse and the Implicit Function Theorems, and learn better how to analyze functions and problems that are not so straightforward. Furthermore, the honors version goes a bit beyond Gauss' Theorem and 110.202, with an introduction to differential forms, and a basic development of a generalized unified theory of the latter three theorems entitled "generalized Stokes'". Both courses are a challenge, but the latter is more so.

Students getting a BC score of 5 (or a 110.109 grade of B+ or better) can be encouraged to take this version if they are so inclined. Students with less strong scores should stay in 110.202, or at least should inquire further with the Math Department before registering for the honors version. In either case, while 110.211 is indeed a great course in vector calculus, taught the way mathematicians really want to teach a math course, it should be understood that the course will be quite a serious challenge.

Course sizes typically run over 100 easily for each lecture of 110.202, with about 4 recitation sections of 25 each. In contrast, 110.211 runs with about 40 students in 2 recitation sections.

Though always self-selected, students are usually quite enthusiastic about the honors version. it is also great fun to teach!

Hope this helps....

6 comments:

Tanmay Gokhale said...

Great post! I took the Honors math courses (Linear Algebra with Porod and Multivariable with Wilkin were the only ones offered at the time), and I loved them, but they're definitely not for everyone.

Richard Brown said...

Glad you like the courses, Tanmay. Haven't yet taught Honors Linear Algebra, but the other one was very enjoyable on my end also.

HD Kim said...

Thanks for the post! I took Honors Multivariable Calculus and Linear Algebra in 2000-2001 with Minicozzi and Morava. The material was incredibly stimulating and served me very well during my Ph.D. education. The best part about this course, however, was that there are 3 couples (out of only 15-20 people!) that originated from this class and are still together to this date (at least last I checked). Two of these couples (including me and my wife) are now married. And many of the classmates are still good friends, and we see each other periodically. This course definitely has a special place in my heart.

HD Kim said...

Oh, and good luck to you Dr. Brown. We tend to especially love our HMC and LA profs!

Richard Brown said...

Thank you, HD. And congratulations to you and yours. We typically do not think of our courses for their post-instance social value. But your endorsement does mean a lot to us. Good luck to you. And feel free to comment further to advise our current students. Your perspective as a former student is crucial to our central reason-to-be.

sabera said...

Mathematics of Linear Algebra