Real Mathematics!

I am gearing up for the fall semester here at Hopkins.  This fall, I am teaching our version of vector calculus (aka multivariable calulus), 110.202 Calculus III.  It is a great course, beautifully visual and quite subtle in many ways.  Good stuff! 

I sent out a "hello" email to my 300+ students, inviting them to check out the webpage and generally welcoming them to the course.  In this email, I say near the end:

Even though one may think of calculus as simply a math course where one learns some techniques for solving physics and statistics problems, it actually is much more than this.  Instead of simply learning techniques, we will be learning how and why the techniques even exist, what they say about the structure of mathematics like calculus, and how to think analytically and reason deductively and abstractly.  THIS is the real mathematics.  The techniques will come along for the ride.  You will learn those also. 

Perhaps the best way to drive this point home is the following:  It does not matter what your current and/or future major is or will be.  You are here at Hopkins to train to be a scholar at something.  Part of that training includes proficient and efficient understanding of the abstract logical structural framework found in all complex ideas and constructions.  This is really what mathematics is.  We typically use numbers and operations on those numbers to study and exhibit mathematical ideas because they provide the self-consistent framework needed for the study.  I will say a lot more about this on the first day of class. 

Perhaps this is one of my personal definitions of mathematics.  But I like it.  Make sense?

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....

Chicken or Egg or ...: Which comes first? CalcIII, LinAlg, or DiffEq?

If I can point to a set of questions that are asked most often of the Math Department, one definitely on the list is the following:

Is it better to take Calculus III or Linear Algebra first?

Throw Differential Equations into the mix, and you get a branching of one's math career into three distinct paths. All three of these courses, at least here at Hopkins, have a full year of single variable calculus as a prerequisite; necessary for technique as well as theory in the case of multivariable calculus (Calculus III) and Differential Equations, and necessary for a sufficient level of "mathematical maturity" in the case of Linear Algebra.

But for many majors, and interests, one must take courses in and well understand two, if not all three, of these topics. So what order makes the most sense?

Its a good question. It turns out, it is not really important....

I am starting a new series about these and other courses under the tag and title "Beyond Calculus II". In this series, I will explain better the idea and focus of these three (and other) courses taken after a full year of calculus is achieved. Here at Hopkins, a large population of our students start their tenure here at this level.

For now, though, let's stick with the topic above. To start:

There is neither multivariable calculus nor differential equations in linear algebra, yet there is a bit of linear algebra in both of the others. In contrast, linear algebra is a more mature course, sometimes requiring more in the way of expanding one's frame of reference mathematically than the other two.

That said, we actually took a look at performance among students who took the two courses 110.202 Calculus III and 110.201 Linear Algebra back to back over a two year period (there are quite a few of them). I will pass on the details of this study, but we found that there was no real preferred order to these two, at least as far as ultimate grades go.

Couple this with the fact that any linear algebra found in either calculus III or differential equations is essentially covered within the courses, and any of the three may be taken in any real order. Hence preference for time slots, professors, and/or friends in the course may be of higher priority in your choice than content.

And one last note, our course 110.302 Differential Equations, fairly standard in content with most sophomore-level courses at American universities, is a course in ordinary differential equations (involving functions of one independent variable, in contrast with partial differential equations). One can easily describe this course as Calculus II.5 (weird notation, hih?). It can be viewed as the proper successor to Calculus II, rather than Calculus III. Just sayin....