There will be an emphasis on the geometric properties of the objects we are studying, rather than their more formal algrebraic properties. To this end, we will be using the computer to generate lots of pictures and examples, in hopes that this will aid your intuition about how surfaces and curves work in 3-space (and beyond). Be prepared to do some drawings yourself; there is no substitute for the understanding you can get by actually working carefully with the surfaces by hand.
An underlying theme of the course will be an attempt to tie a number of seemingly independent ideas together into a unified whole; in particular, we will develop a new definition of what a derivative is that encompasses a wide range of situations, and includes the single-variable derivative as a special case.
The basic idea for the course will be to study what a "rate of change" means for multivariable functions, and the intuition will be that it should mean "the expected change in the function for a unit change in the parameters of the function". The content of the course will be to interpret this statement for several important categories of multivariable functions. Along the way we will develop a number of tools from linear algebra that will be crucial aids in studying functions of several variables.
See also the course outline for the timing of these topics.