640:432:01 Differential Geometry

Class meets: TTh5(3:20 -- 4:40pm), SEC-206.
Instructor: Dr. Zheng-Chao Han
Office Hours: M1-2pm, T11:00-noon. Hill-230.
Email: zchan at math dot rutgers dot edu (I prefer to answer math questions in person during office hours, not through emails; I try to process my emails once per day ).
Text: The following is the required text for this course:

Manfredo DoCarmo, Differential Geometry of Curves and Surfaces (first edition), Prentice-Hall, 1976 (503 pp.); (ISBN# 0-13-212589-7; ISBN13: 9780132125895).

Additional supplementary material may be provided/recommended as the course progresses. I have requested to put the following references on reserve in the math library:

Note: Do not forget to "reload" the assignments pages - if you visited them before, your browser may be showing you only the old cached page.

General Comments on the Course

Differential geometry is the study of geometric properties of curves, surfaces, and their higher dimensional analogues using the methods of calculus. It has a long and rich history, and, in addition to its intrinsic mathematical value and important connections with various other branches of mathematics, it has many applications in various physical sciences, e.g., solid mechanics, computer tomography, or general relativity. Differential geometry is a vast subject. A comprehensive introduction would require prerequisites in several related subjects, and would take at least two or three semesters of courses. In this elementary introductory course we develop much of the language and many of the basic concepts of differential geometry in the simpler context of curves and surfaces in ordinary 3 dimensional Euclidean space. Our aim is to build both a solid mathematical understanding of the fundamental notions of differential geometry and sufficient visual and geometric intuition of the subject. We hope that this course is of interest to students from a variety of math, science and engineering backgrounds, and that after completing this course, the students will be in a position to (i) apply their knowledge and skills in this course to their related subjects, (ii) be ready to study more advanced topics such as global properties of curves and surfaces, geometry of abstract manifolds, tensor analysis, and general relativity.

Prerequisites:: Although the officially listed prerequisite is 01:640:311, what are not explicitly stated but are at least equally essential prerequisites are Multivariable Calculus and Linear Algebra. Most notions of differential geometry are formulated with the help of Multivariable Calculus and Linear Algebra. 01:640:311, which itself requires Multivariable Calculus and Linear Algebra as prerequisites, is an important prerequisite because it helps students build mathematical maturity and gain the ability to understand, formulate and present precise mathematical concepts and proofs. The following is a list of specific topics that you should review and familiarize yourself with---we acknowledge that the list of topics to follow and our homework policies have been adapted from the course page of Professor John Lee of the University of Washington. If two or three items on the list are unfamiliar, you are probably still OK as long as you’re willing to spend some extra time reviewing. If there are half a dozen unfamiliar items on the list, you’re not ready for this class. Of course many people will be rusty on various topics. Some of these topics will be reviewed briefly in class as they arise, but be prepared to do some review on your own.

Attendance and Make-up Policy: Class attendance is expected. Poor attendance will be used to decide borderline grade situations. Any changes to the syllabus, homework assignment and any update announcement for the midterms and final exam will be made in the lectures. No late work will be accepted. A make-up midterm will be given only if you have a valid reason such as serious illness (not a slight cold) or a family emergency, and provide an acceptable, written excuse (not an email message), or you will receive a grade of zero. If possible (particularly if you want to be sure that your excuse is an acceptable one), contact me before missing an exam.

Course Material

You may find a copy of our course's syllabus and homework assignment posted on line. Both are subject to adjustment. Any updated information should be posted on this web page. However, the most accurate information will be from the lectures.

Some Notes on Section 2-2 and Homework Assignment 3 are posted, please do pre-reading on Section 2-2 and these notes before our Feb. 5 lecture.

Homework Policy

Homework Assignment: Working through the homework will be essential for learning the material in this course. Plan to spend a lot of time on homework—six hours or more outside of class during most weeks. A typical homework assignment will consist of the following:

I strongly encourage you to form study groups and work together on the homework problems. By this I mean you can discuss, develop, and test an outline for proofs with others, but not the complete details; you can also study from other sources, but not to copy solutions, and when you write up your solutions to hand in, you must write your own solutions in your own words. If your solution's key idea was the result of discussion with others, or from other sources, you must give reference and explain how you came to understand it.

Homework Grading Guidelines The grading of the assignments will be based on correctness of concepts and content, soundness of logical structure in your arguments, and exposition. Homework should be neatly written using proper English grammar, i.e., a list of calculations never forms a complete answer. You should present your work just as you would do for a writing assignment in any other subject, giving the necessary background information, definition of terms you are about to explain, logical arguments, and your conclusions. More specifically, if you are presenting the solution to a problem, explain first what the problem is; if you are going to use some terms or concepts, try to give as clear a definition as possible(because mathematical concepts in a typical student's mind are often vague and may change from context to context, but in scientific discussions preciseness of concepts is needed); if you are giving an argument, try to explain the main points before you launch into the details. This may seem hard in the beginning. But you can improve quickly if you keep a journal to record what you have been thinking and doing in your work, and then try to organize the ideas in a coherent way as if you wanted to explain your ideas to a friend or to convince him/her of your arguments. Here are some other important instructions regarding homework:

Course Grading Policy

Your course grade will be determined on the following basis:

For comments regarding this page, please send email to zchan at math dot rutgers dot edu.