University of Illinois at Urbana-Champaign, Department of Mathematics

**What this course is about:**

This is a first introduction to manifolds and to calculus on manifolds. Manifolds are spaces that generalize the familiar Euclidean plane and space. Some examples: surfaces, spacetime in relativity, space of configurations of robotic arms, space of rotation matrices. The intended audience of this course are math or physics undergrads and engineering or physics grad students. The content is similar to Math 518 which is a MUCH faster paced graduate course.

Math 481 (and Math 423) remedy a serious omission in the education of most math and physics undergrads. These two courses introduce students to exterior calculus of differential forms, preparing them for further courses in in geometry and topology, theoretical physics, and theoretical aspects of some engineering fields. This course will give you the background needed to take courses like: differentiable manifolds (Math 518), general relativity, and differential geometry (Math 519). The ideal sequence for learning differentiable manifolds and differential geometry is Math 423, 481, 518, 519 even though some students can skip around in this sequence without problems.

**Instructor** Anil Hirani

**Office Hours** See Moodle course page for details

**Homeworks** There will be 9-10 graded homeworks. These must be handed in on paper in class. You are encouraged to discuss the homeworks with others and course staff, but you should write up solution on your own. Your lowest homework score will be dropped.

**Textbook**
*An Introduction to Manifolds*, Second Edition, by Loring Tu. PDF available free from our library catalog. A physical copy may be available in Math library reserves in Altgeld Hall.

**Exams** There will be two evening midterm exams and a final exam. Conflict exams will be available if arrangements have been made with me at least one week in advance of the scheduled exam.

**Grading** Attendance 8%, Homeworks 15%, Two Midterm Exams: your lower score worth 17%, higher score worth 25%, Final Exam 35%.

**Attendance:** You can miss 2 lectures without penalty and without giving reason. After that, with each missed lecture your lost points will go up exponentially. Points lost = 2^(number of missed lectures - 1) upto a maximum of 8 points. So you lose 1, 2, 4, 8 percentage points from your semester final score if you miss 1, 2, 3, 4 lectures respectively. I will start taking attendance in week 2.

Page maintained by Anil N. Hirani

Last modified: Fri Jan 20 13:18:24 CST 2023