640:432:01 Differential Geometry
Class
meets: TTh5(3:20  4:40pm), SEC206.
Instructor:
Dr. ZhengChao Han
Office Hours:
M12pm, T11:00noon. Hill230.
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), PrenticeHall, 1976 (503 pp.);
(ISBN# 0132125897; 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), ISBN10: 0130652466.
 Elementary Differential Geometry (Paperback), by Andrew Pressley,
published by Springer, Corr. 2nd printing edition (September 18, 2002), ISBN10: 1852331526.
 Elementary Differential Geometry, Revised 2nd Edition, Second Edition (Hardcover), by Barrett O'neill,
published by Academic Press; 2 edition (March 27, 2006), ISBN10: 0120887355.
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 withwe 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 Makeup 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 makeup 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 22 and
Homework Assignment 3 are posted, please do prereading on Section 22 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:

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.
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:

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 10point
problems will be graded in detail, with partial credit given for incomplete solutions. The
2point 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.
Course Grading Policy
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% (noon3pm, Friday, May 9, location to be announced).
For comments regarding this page,
please send email to zchan at math dot rutgers dot edu.