You already have studied some aspects of curves and surfaces in your multivariable calculus course (Math 115), but that was from a very computational standpoint. This course will address those concepts from a much more theoretical viewpoint, and will extend them in a number of ways. The motivating ideas will come from trying to understand the local geometry of curves and surfaces: for example, how do you measure how a curve is turning or how a surface is warping? How do you compare one curve or surface to another in these respects?
Students traditionally find this material to be difficult. The ideas are abstract, and the computations are detailed and often long and error prone. You need to be prepared to spend a lot of time thinking about the underlying ideas and how they work together, and you need to be ready to go over the computations several times to catch mistakes. There are no quick shortcuts to understanding this material; it simply takes a lot of patience, practice, and perseverance.
As is the case with so many things, mathematics is not something you can learn by watching others do it. That is particularly true of differential geometry. Working out the details (not just seeing the details) is a crucial part of the process. For this reason, you will be doing the problem sets individually, not in groups (see the problem set policy for more details). I know that many of you find this difficult to do, but it is an essential part of the learning process. You may work in groups on the homework, but not on the problem sets.

