While the course catalogue says that the only formal pre-requisites for Math 32a are Math 23b and either Math 21b, or Math 22b (or permission of the instructor), to be more precise, in order to understand the lectures well and be comfortable with the concepts we will be dealing with, you should be reasonably familiar with inner-product spaces and also with the basic theory of ODE (ordinary differential equations). Most of you have probably seen these, but just to be sure, we will start the course with a quick review of these subjects, and at the same time we will develop some associated special machinery that we will need to use later in the course.

As an aside, let me remark concerning two features of this course that most students find give it a lot of interest and appeal. On the one hand, it deals with very familiar and intuitive objects (curves and surfaces), and on the other hand it brings together and uses almost all the concepts and techniques you have seen discussed in your earlier analysis courses to develop a good understanding of these objects.