2026 October 12,

Math 344: Differential Geometry

Carleton College, Fall 2026, Dr. Joshua R. Davis, , CMC 324, x4095

Introduction

Differential geometry is the study of curved spaces using calculus and linear algebra. It relates to several other areas of math — particularly topology, complex analysis, and differential equations. It is used in computer graphics, robotic control, DNA folding, and other scientific fields. Its most famous application is general relativity, where the space in question is our universe, and curvature is gravity arising from the presence of matter and energy.

This course is an introduction to differential geometry. We begin by studying curves, which are important in understanding higher-dimensional spaces. Then we study surfaces, which are richer and more nuanced than curves. Toward the end of the course, we spend some time with higher-dimensional spaces. (We glimpse topological aspects of differential geometry, such as the Gauss-Bonnet theorem. But we don't thoroughly study them, because Math 354: Topology is not a prerequisite.)

The official prerequisite for this course is Math 236: Mathematical Structures. I am happy to admit students who've taken CS 202: Math of CS instead. We prove a lot of theorems. Compared to other upper-level "pure" math courses, we also do a lot of calculations, which are error-prone and sometimes tedious. To expedite the calculations, and to visualize weird spaces, we use the software Mathematica. It is available in our computer labs, and Carleton students can install it for free on their own computers. I supply Mathematica tutorials and instruction. No prior experience is expected, and I do not expect you to write programs, but I do expect you to tweak given programs to explore new examples. Talk to me, if you are concerned about your background. :)

Responsibilities

The College's accreditation says that a 6-credit course is 150 hours of work. That's about 15 hours per week or 5 hours per class meeting. Those 5 hours break down into about 1 hour for class itself and 4 hours for homework, reading, studying, etc. If you find yourself spending much more time, then talk to me.

Participation

Our course meets in CMC 319 during period 2A. That's Monday 9:50-11:00, Wednesday 9:50-11:00, Friday 9:40-10:40. You are expected to attend every class meeting promptly, take notes on paper or a tablet, participate in discussion and group work, and ask and answer questions. Additionally, you are required to visit my office hours sometime during the first two weeks.

Laptops, phones, photos, and recordings are prohibited (except by special arrangement). Why? This course's material can't be typed easily on a keyboard. Using a laptop or phone measurably distracts the students around you. Photographing a chalkboard is not as educational as taking notes. I want our class to be a safe space, where students don't feel that they're on stage. My presentation is my copyrighted property.

It is important that our course be welcoming to all students, regardless of their identities, backgrounds, and experiences. We all sometimes say and do things that make life worse for others, and we should all strive not to. Please let me know if the class feels hostile to you, because of something that I or someone else has done.

Homework

Although class meetings may seem like the core of the course, homework assignments are actually where you learn the material. It is essential that you attempt each homework promptly, before the next class meeting. For then you better understand that next class meeting!

On homework, you are encouraged to figure out the problems with other students. However, you should always write/type your solutions individually, in your own words. You may not copy someone else's work or allow them to copy yours. You may not use AIs such as ChatGPT. Presenting someone else's work as your own is a violation of Carleton's Academic Integrity standards.

Writing is not just for literature and history majors. Written and oral communication skills are essential to every academic discipline and are highly prized by employers. In this course, your written work is evaluated both for correctness and for presentation. Compose your solutions as if the intended audience is your fellow students. By doing so, you show enough detail that the grader can ascertain whether you yourself understand the material.

Homework is usually due two class meetings after it was assigned. This schedule tries to keep you on track, so that work doesn't pile up. But we all have bad weeks, where we can't get everything done, right? If you need to submit an assignment late, then write "LATE" at the top of the front page and submit it as soon as you can. If the grader hasn't graded the assignment yet, then they can grade your paper with the others for full credit. If the grader has graded the assignment already, then you might not get credit. There are limitations to how much delay and complication a grader can handle.

When handing in an assignment, please mark it with the day that it was assigned (e.g., "Day 11") and staple your pages into a single packet, in the correct order. Is there a stapler in the classroom? Often not, so staple ahead of time. Is a paper clip just as good? Sorry, no.

Depending on time constraints in any given week, perhaps not all of your homework is graded. In order to ensure full credit, do all of the assigned problems by their due date. As far as your grade is concerned, homework serves primarily as the first step in studying for exams.

Exams

We have three exams. The second exam has a take-home component; otherwise, the exams are given in class. They are given roughly 1/3, 2/3, and 3/3 of the way through the term. The last exam is given during our official final exam period, in our usual room.

Are the exams cumulative? No and yes. No, in that each exam is focused on the material that has not been tested yet. Yes, in that much of the material is inherently cumulative. Also, the last exam might have some questions that explicitly address material from earlier in the course.

Grading

For better or worse, we are required to measure your learning using grades. Your numerical grade is based on the responsibilities above: participation 5%, homework 10%, Exam A 25%, Exam B 25%, Exam C 35%.

Numerical grades are converted to letter grades only at the end of the term. Grades are not curved, so students are not in competition with each other. Grades are also not based on predetermined percentages (90%, 80%, 70%, etc.), because Math 344 problems are difficult to tune so precisely. (I could accidentally write a difficult exam and wreck everyone's percentages.) Rather, I assign grades by comparing the students to the course goals. For example:

• A student who earns an A is not perfect, but they can speak fluently about nearly all of the course content. They exceed expectations.
• A student who earns a B learns most of the big concepts pretty well, but they are missing some concepts. They lack the depth of understanding needed to appreciate subtleties in proofs, etc.
• A student who earns a C gets some of the material but has significant gaps in their understanding of major concepts or their ability to prove important theorems.

Talk to me, if you are concerned about your grade. :)

Resources

I want all of my students to work hard and learn a lot. I try to give them all of the resources that they need. Here are the basics:

• Our textbook is A First Course in Differential Geometry: Surfaces in Euclidean Space by Woodward and Bolton. This book is often, but not always, the textbook of Math 344. It gives a thorough overview of curves and surfaces, with more material than we need.
• When we transition to higher-dimensional spaces, I give you additional notes or readings, to be determined later.
• Here are the exams from this term, with quartiles (75th, 50th, 25th percentiles):
  • Exam A, Solutions
• Here are the exams from my fall 2020 edition of Math 344. That edition was pretty different from what we're doing this term, and I used a different textbook, and all of the exams were take-home, and some of the exams are damaged. Anyway, maybe they can help you study.
  • Exam A, Solutions, Practice
  • Exam B, Solutions
  • Exam C, Solutions

Here are some Mathematica resources. The first two entries are Mathematica notebook files; right-click to save them to your computer, and then open them in Mathematica.

• Mathematica Nuts and Bolts tells you the basic logistics of using a Mathematica notebook.
• introduction.nb teaches the extreme basics of Mathematica for a general audience.
• memory.nb tells you how to store results in variables. (This is a surprisingly large pitfall in computer programming.)
• Mathematica is installed on our lab machines (CMC 102 and 304). Some of the Math/Stats Skills Center tutors can help with Mathematica.
• Install Mathematica using Carleton's license pool on your own computer, if you like. Getting a license might take a day, so get started early.
• Official Mathematica Documentation from Wolfram is gigantic.
• Mathematica Tutorial Video from Wolfram might help.

Remember that I encourage you to solve problems with classmates (even if the work that you submit must be your own). If you want help in finding a study partner, then e-mail me, perhaps describing some of your habits: working in the middle of the night, not waiting until deadlines, etc.

I hold several office hours per week; see my weekly schedule. No appointment is needed; just drop in! You can also ask questions immediately after class. If you can't make office hours, then consult my schedule and e-mail me with several possible meeting times.

If a health condition or other personal matter affects your participation in class, homework, exams, etc., then please let me know as soon as possible. Depending on the situation, we might want to confer with Accessibility Resources, Assistive Technology, Student Health and Counseling, or Title IX. When you ask me to help, I do my best to help. :)

Schedule

This schedule is tentative. It is tweaked and filled in as the course progresses.

To help you decode the schedule below, here is an example. Monday September 14 is Day 01 of the 28-day course. During class, I give an overview of the course, review some prerequisite material, and start discussing regular curves. You have a homework assignment called "Day 01", which you should attempt immediately, but which is due at the start of class on Day 03. You are expected to skim Section 1.1 of the textbook after (or before) class. We use the Mathematica notebook whatIsDiffGeom.nb during class, and you can peruse it again if you wish.