Differential Geometry
Dr. Sarah J. Greenwald

Where to Get Help

  • Office Hours 326 Walker Hall 262-2363. Sometimes, if no one comes to office hours, I go down the hall to the mailroom, photocopy machine, or to talk to another professor. If I am not in my office during office hours, you should walk down the hall to look for me, and interrupt to tell me that you are there. I am always around and happy to help you during office hours unless otherwise posted to the webpage. You do not need to make an appointment to use office hours - just drop by! If you can't make office hours, contact me on ASULearn, which I'll try to answer at least once a day.
  • Check the main web page often (every couple of days and at least 3 times a week) for work due. This lists and explains readings, hw, projects and tests (ie it is a part of my syllabus).
  • ASULearn is the easiest way to ask a math question outside of class and office hours. You are responsible for reading all posts from me. I prefer that you use office hours since it is easier to discuss material in person, but if you can not make them, then it is a great alternative. I usually check it every day including the weekends.

    Required Resources

  • Differential Geometry and Its Applications
    The Mathematical Association of America Edition in 2007
    Classroom Resource Materials
    John Oprea
    469 pp., hardbound, 2007
    ISBN: 978-0-88385-748-9
  • i-clicker
  • printouts of your Maple work
  • access to a web-browser and to Maple (on-campus access is sufficient as long as you have the time to work on campus while the labs are open)

    Course Goals

  • To develop geometric problem solving skills and 3-D spatial visualization skills.
  • To develop a greater appreciation for connections between various disciplines of mathematics, including geometry, linear algebra, complex analysis, and differential equations, along with an introduction to these subjects as they apply to differential geometry.
  • To understand the importance of differential geometry in various scientific fields, including physics.


    This is an introductory course in the differential geometry of curves and surfaces in space, presenting both theoretical and computational components, intrinsic and extrinsic viewpoints, and numerous applications. The geometry of space-time will also be considered. Prerequisite: MAT 2130. Corequisite: MAT 2240.
  • Geometry of curves in space, including Frenet formulas
  • Theory of surfaces, including curvature, geodesics, and metrics
  • Geometry of space-time and applications to general relativity

    Assignment Types and Grades

  • Participation in Classroom Activities 5% You are expected to contribute to discussions and i-clicker questions in a meaningful way and actively engage the material in class. You must be prepared for each class and check the main web page regularly for hw. These will include readings from the book, Maple applets, paper folding activities, as well as the following short readings:
    Curves and Surfaces, both by Dogan Comez, myself and Jill Thomley
    How Flies Fly: Kappatau Space Curves by Rudy Rucker
    How to Create Your Own Universe in Three Easy Steps by Lawrence Brenton
    Relativity by David Brink
    These kinds of baseline activities will result in a participation grade of 3.5/10. Other activities can increase or decrease this grade. Utilizing office hours and ASULearn, asking and answering thought provoking questions, coming up with creative ways of thinking about the material, and explaining the material to others are some examples of positive participation that will increase your grade. On the other hand, performing activities that detract from the professional classroom environment or distract Dr. Sarah (who is very easily distracted) will result in a lowered participation grade. Many activities and class discussions are designed to be completed during class. Thus, attendance is required at ALL classes, and will form a portion of your grade. Missing more than the equivalent of 4 weeks during the semester (8 class days) will result in an automatic F in the course. Save your absences for emergencies. If you must be late to a class, or must leave early, then do still attend.
  • Homework 30% Work will not be accepted without explanation and must also be turned in on or before the due date. May occur during the last week of classes. If there is some reason you must miss a class, then obtain the assignment from the web pages. The lowest graded homework will be dropped - save this for emergencies. If all of your homework is turned in AND you have received at least a grade of 70% for all work, then you will receive a work completing credit of +1 added on to your final average. No lates allowed*.
  • Exams 50% You should view exams primarily as a learning experience. This means that exams are not only an opportunity for you to demonstrate your mastery of the material, but are also an opportunity for you to be challenged with new material in order for you to make new connections. To encourage exams as a learning experience some extra points will be granted for complete and correct test corrections. No lates allowed*. May occur the last week of classes.
  • Final Project 15% You must participate to pass the class *

    Also see the University-wide syllabus and policy statements which we adhere to.
    * Work may occur during the last week of classes. Accommodations in the determination of your final grade will be made for extenuating circumstances that are documented to prevent you from completing work early/on time. The grading is a ten point scale: A ≥93; 90≤ A- < 93; 87 ≤ B+ <90...

    Receiving Graduate Credit

    In order to receive credit for 5530, graduate students who are enrolled will complete extra graduate problems and assignments as well as an additional component of the final project. See the individual assignments on the main webpage for details.

    Other Policies and Methodology

    As per the University-wide Statement on Student Engagement with Courses Plan to spend at least 2-3 hours outside of class for each credit hour in class, (on average). You are responsible for all material covered and all announcements and assignments made at each class, whether you are present or not. You are also responsible for announcements made on the web pages, so check them often.

    Asking questions, and explaining things to others, in or out of class, is one of the best ways to improve your understanding of the material. This course is to be an environment in which everyone feels comfortable asking questions, making mistakes, offering good guesses and ideas, and is respectful to one another. Turn in projects or prepare to present problems even if it they are not complete, even if only to say, "I do not understand such and such" or "I am stuck here." Be as specific as possible. When writing up work, be sure to give acknowledgment where it is due. Submitting someone else's work as your own (PLAGIARISM) is a serious violation of the University's Academic Integrity Code which defines:

    Plagiarism includes, but is not limited to, borrowing, downloading, cutting and pasting, and paraphrasing without acknowledgement, including from online sources, or allowing an individual's academic work to be submitted as another's work.

    The purpose of homework is to learn and practice computational strategies, concepts, and develop critical thinking and problem-solving skills, so you should try problems on your own. Feel free to talk to me or each other if you are stuck on this assignment, but be sure to acknowledge any sources - including each other, like "The insight for this solution came from a conversation with Joel." If you know how to do a problem and are asked for help, try to give hints rather than the solution: Give a man a fish and you feed him for a day. Teach a man to fish and you feed him for a lifetime (or at least this course)

    In this course, you will be challenged with problems that you have never seen before. I do not expect you to be able to solve all the issues immediately. Instead, I want to see what you can do on your own. Out in the real world, this is important, since no matter what job you have, you will be expected to seek out information and answers to new topics you have not seen before. This may feel uncomfortable and frustrating. I understand this and want to help you through the process. It helps to remember that there are no mathematical dead-ends! Each time we get stuck, it teaches us something about the problem we are working on, and leads us to a deeper understanding of the mathematics.

    In the real world though, you are not expected to face your work alone. In this class, you are also not expected to face your work alone. I am always happy to help you in class, during office hours (or by appointment), or on ASULearn, and will try to give you hints and direction. At times though, to encourage the exploration process, I may direct you to rethink a problem and to come back to discuss it with me again afterwards. This occurs when I believe that the struggle to understand is imperative for your deep understanding of the material.

    Instructor Bio

    I am a Professor of Mathematics and a Gender, Women and Sexuality Studies affiliate at Appalachian State University. I received my PhD from the University of Pennsylvania. My scholarship areas include Riemannian geometry of orbifolds (linear algebra is important here), popular culture as it pertains to mathematics, and women and minorities in mathematics. Recognition for my teaching includes a 2005 Mathematical Association of America Alder Award winner for distinguished teaching and the winner of the 2010 Appalachian State University Wayne D. Duncan Award for Excellence in Teaching in General Education. In 2010 I was also inducted into the Appalachian State University College of Arts and Sciences Academy of Outstanding Teachers and in 2011 I was named the College of Arts and Sciences Outstanding Teacher of the Year. I am the associate editor of the Association for Women in Mathematics Newsletter and a member of the editorial board of PRIMUS. Andrew Nestler and I co-created the educational website SimpsonsMath.com. My interactive mathematics lecture has been distributed on approximately one million DVDs worldwide as a 25-minute DVD extra for the 20th Century Fox Futurama movie Bender's Big Score and it is listed as "Mind-bending." Jill Thomley and I co-edited the 3-volume Encyclopedia of Mathematics & Society, which was named a "Best Reference 2011" by Library Journal. I've spoken about the impacts of scientific popular culture representations on NPR's Science Friday and all over the country.

    I am married to the bassist Joel Landsberg. We both happen to be on IMDb: Joel and me. In my spare time I like to travel, hike and conduct genealogy research (I also enjoy popular culture, as you can probably tell from some of my scholarly interests). In addition to my own personal genealogy, I like to give back to the broader community. I am the project coordinator for sites like the Bialobrzegi ShtetLink and the Book of Remembrance of the Community of Bialobrzeg. These projects strive to research and preserve information about small Jewish communities that were destroyed in World War II. My great-grandparents lived there (it was the Russian empire back then!) in the late 1800s. Some of what I really like about mathematics is also what I enjoy about genealogy - the sense of exploration, discovery and aha moments that come with lots of patience and effort.