# Math 32a, Spring 2006

Course Description: This course is about the differential geometry of curves and surfaces in the plane and Euclidean 3-space. The basic concept we will try to master is the measurement of curvature, using the tools of multivariable calculus. We will use the book Elementary Differential Geometry, by Andrew Pressley, and cover the first 6 chapters plus additional topics.

Grading: The homework will count for 35%, the midterm for 25%, and the final will be 40%.

Office Hours: For the first two weeks, office hours will be Tuesday 11-12, Wednesday 2-3, and Thursday 11-12, and by appointment.

Homework: The homework will have two parts. The first part will be problems from the book, which will be corrected but not graded (the book has solutions!). Here's the catch: Every student will be required to present in class (at least once, probably twice) a solution to one of these problems. You will know in advance if you are presenting, and I will reserve the right to ask you to justify statements or calculations you make. The second part will consist of other problems that I will supply, which will be graded. Assignments as well as review sheets for exams can be found on WebCT.

Contact information: The easiest way to reach me is via email: ruberman @brandeis.edu. My office is 310 Goldsmith, and my phone is x3074.

### Course Outline

Curves in the Plane and in Space: Parameterized curves, curves defined by equations. Arc length; curvature of space curves and plane curves; Serret-Frenet Formulas. Global properties of plane curves: Isoperimetric Inequality and Four Vertex Theorem.

Surfaces in R3: Smooth surfaces, tangents, normals, orientability. Inverse function theorem and parameterized versus implicitly defined surfaces. Measuring distances on surfaces: the First Fundamental Form, isometries, and surface area.

Curvature of Surfaces: Second Fundamental Form, Normal and Principal curvatures, Gaussian curvature. Gauss' Theorema Egregium.

Additional Topics, depending on time: Geodesics, Minimal Surfaces and Soap Bubbles, Surfaces of Constant curvature, Gauss-Bonnet Theorem.

### General Course Policies

You may discuss the homework problems with other students in the class; however, if you do, you should write on your homework submission the students with whom you discussed the assignment. (You do not need to mention any help you received from the TA's or instructor.) You may not copy the written work of another student or allow another student to copy your written work. What you submit should be your own work.

If you are a student with a documented disability on record at Brandeis University and wish to have a reasonable accommodation made for you in this class, please see me immediately. Please keep in mind that reasonable accomodations are not provided retroactively.

You are expected to be honest in all of your academic work. The University policy on academic honesty is distributed annually as section 5 of the Rights and Responsibilities Handbook. Instances of alleged dishonesty will be forwarded to the Office of Campus Life for possible referral to the Student Judicial System. Potential sanctions include failure in the course and suspension from the University. If you have any questions about my expectations, please ask.

Last updated: January 18, 2006