640:432:01 Differential Geometry

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:

• Differential Geometry and Its Applications (2nd Edition) (Hardcover), by John Oprea, published by Prentice Hall; 2 edition (December 9, 2003), ISBN-10: 0130652466.
• Elementary Differential Geometry (Paperback), by Andrew Pressley, published by Springer, Corr. 2nd printing edition (September 18, 2002), ISBN-10: 1852331526.
• Elementary Differential Geometry, Revised 2nd Edition, Second Edition (Hardcover), by Barrett O'neill, published by Academic Press; 2 edition (March 27, 2006), ISBN-10: 0120887355.

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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.

• Vector calculus: partial derivatives, the chain rule, dot products, cross products, tangent lines/planes, line integrals, surface integrals, gradients, vector fields.
• Point sets and continuous functions in the real line and the plane: open and closed sets, boundaries, limit points, closures, continuous and differentiable maps.
• Linear algebra: vector spaces, bases and dimension, linear transformations and their representation by matrices, determinants, matrix algebra, eigenvectors and eigenvalues (esp. of 2 x 2 or 3 x 3 matrices), inner products.

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.

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 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:

• Reading: Typically, you will be given approximately two to three sections of the textbook to read each week. This will usually correspond to the material that is being discussed in lecture that week. I will expect you to read through the chapter quickly before the relevant lectures, and then to reread it carefully after the lectures.
• Practice Problems: Most assignments will include a number of “practice problems.” These are not to be handed in for a grade, but I expect you to do (or at least figure out how to do) all of them for your own good. Understanding these problems will be important for solidifying your understanding of the text and lectures, and for preparing to do the required problems. Some of these problems are likely to show up on exams.
• Required Problems: The problems listed as “Required Problems” are for you to write up and hand in for a grade. These problems, consisting almost entirely of proofs, are the heart of the course, and they will constitute a significant part of your course grade.

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:

• Please staple your homework papers together, with the required problems in numerical order, and with your name on every page. Write legibly, in complete sentences, and leave ample margins in which the grader can write comments.
• Each of the “Required Problems” will be graded. Selected problems will be graded on a scale of 0 to 10 points, and the remaining ones will be graded on a scale of 0 to 2. The 10-point problems will be graded in detail, with partial credit given for incomplete solutions. The 2-point problems will be given 0 points if there is no reasonable attempt at a solution, 1 point if you make a reasonable attempt, and 2 points if there are no blatant mistakes. Since you won’t know in advance which problems will be graded for 10 points, it pays to try to answer all problems as thoroughly as possible.
• In computing your final grade, your raw homework scores will be converted to percentages, so that each assignment carries equal weight. Then your lowest homework score will be dropped, and the remaining assignments will be averaged.

Your course grade will be determined on the following basis:

• Regular assignments: 30%
• Midterms: 20% each (first one tentatively scheduled on Thursday, Feb. 21, and second one tentatively scheduled on Tuesday, April 8).
• Final Exam: 30% (noon--3pm, Friday, May 9, location to be announced).