Day 2 in Mat329

Last time: Welcome

Next time: Limits and Continuity

Today:

Announcements
- Reminder: stow those small, cellular rectangles....
- No quiz tomorrow. We'll start next week.
- Roll
Last time we began with a brief overview of Multivariate Functions. Some comments:
- Definition of a function (p. 10):
  
  A function $f$ is a rule that assigns to each element x in a set (domain) $D$ exactly one element (called $f(x)$) in the set (range) $E$.
  The difference between multivariate and univariate functions is all in that domain $D$.
  While this definition seems to correspond to univariate functions, the variable $x$ could be a number, an ordered-pair of numbers, a person, a function, etc. -- and $f(x)$ could be any of these (or even a brand of soda). So this is really a very general definition.
- While we are moving from univariate to multivariate functions, we will tend to focus on functions of two variables. These are obviously the simplest non-univariates cases. KISS!
We didn't get to the examples in that file, so I'll just revisit it for those, then on to section 14.1.
Just a reminder: with each section, I'll produce a section summary, which highlights some of the things I think most important. In the old days, I'd hand those out. In these times of tight budgets, I'll just be putting them on-line.
Sometimes I'll refer to them, and sometimes I won't!
Drawing surfaces
- Axes in 3 dimensions: here is a summary from section 13.1. Let's take a look.
- Let's start with planes! I was doing some construction this past summer, and usually we were making planes (or rather portions thereof).
  Try representing this plane, p. 913, #25, using several different (physically realistic) projections.
  Our author comments (and I concur strongly) that ``...in much the same way that linear functions of one variable are important in single-variable calculus, we will see that linear functions of two variables play a central role in multivariable calculus.'' (p. 905)
Similarly we can generalize quadratics, which are so important for practical problems. And we'll see that things get more complicated. In the univariate case there are just bowls and umbrellas (as well as "degenerate" quadratics, which are straight lines).
In a short note that my dad published in the American Mathematical Monthly, long, long ago, we discover that the multivariate case is more interesting.
I have since used Mathematica to recreate his work.
Today I'd like to take a look at some of the problems in Section 14.1.
Since some of us don't draw as well as we'd like, we should use technology -- e.g. Mathematica -- to help us to visualize multivariate functions.
Discussion/Problems:
- Contours: "isobars"; "isotherms"; "isochills"? Topo maps are examples of contour maps (e.g. p. 907, Lonesome Mountain).
  When you see a contour map, think of it as a landscape. This is a very useful analogy. Just like you might think of every univariate graph as a sledding hill...
- p. 913, #32
- p. 913, #34-36
- An example from a cholera outbreak in London, 1849: John Snow's famous "Pump Handle" intervention.
  - the data
  - the method
  - rectangular grid
  - Now, before moving on to the visualizations, I'd like you to see if you can come up with a guess for the contour lines of 50 and 100. Work with a partner.
  - Some visualization techniques
  - Some models
  - a recent paper (2003) about the case
  - John Snow's case data (time series) from 1854.

Next time: 14.2, and exercises

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