Multivariate functions

First of all, multivariate calculus is a generalization of univariate calculus. I will rely on your understanding of univariate calculus. The stronger that foundation, the easier your life in multivariate.
We will be generalizing
- functions
- graphs
- domains and boundaries
- limits
- derivatives
- extrema
- integrals and numerical integration techniques
- Taylor Series
- The Fundamental Theorem of Calculus
etc.!
A fair number of theorems which you have already encountered along your calculus journay will also be generalized.
As we begin, let's start with functions: they're the key to the study we're about to undertake.
Take a few minutes to consider these questions, and discuss with a classmate:
1. What is a function? Give a definition.
2. What classes of functions are important in univariate calculus, and why?
3. Give examples of some applications of different types of univariate functions from your calculus experience.
4. Give an example of a function on which we can't do calculus.
5. Give an example of a function whose range elements are not numbers, but on which we might be able to do calculus.
Why study multivariate calculus?
It is applicable in any subject area, especially in any area making use of statistics and modeling. Life is full of multivariate functions! While univariate functions are the easiest to work with (being simpler), they're a bad place to stop (being unrealistic, or unsophisticated).
Multivariate calculus is essentially a generalization of univariate calculus: there's not a lot that's truly new, but it will be more complex and take unfamiliar forms.
We can think of univariate calculus as shadows from the multivariate world: you're now about to see the beasts that were casting those shadows.
One of the most serious issues is that it's hard to represent multivariate functions of two variables on the plane. Attempted solutions:
- Use 3-space! Models
- projections -- e.g. July was likely Earth's hottest month in what was Earth's hottest year (since we started keeping track).
- contours or "slices" -- e.g. Arctic Sea Ice
- tables (see Wind Chill, p. 903)
When function have more than two independent variables, we're really out of luck, from a visualization standpoint. That's when a computer may come in really handy.... Movies, for example, may help.
We'll be using technology to help us understand multivariate functions (including Mathematica). I encourage you to download a free copy.

Website maintained by Andy Long. Comments appreciated.