Calculus IV: Functions of Several Variables
Having just finished a very short chapter on vector functions, we began the chapter that will cover Partial Derivatives. Before we started that lecture, I took a little class time to answer a homework question. It was one of my favorites. Having covered the concept of curvature, the students were asked to find a polynomial, 
such that the function
![\[
f(x) = \left\{ {\begin{array}{*{20}c}
0 \hfill & {,\,x \le 0} \hfill \\
{P(x)} \hfill & {,\,0 < x < 1} \hfill \\
1 \hfill & {,\,x \ge 1} \hfill \\
\end{array}} \right.
\]](/latexrender/pictures/b3998c39aab429fb16b7f12a9bcb26cc.png "\[
f(x) = \left\{ {\begin{array}{*{20}c}
0 \hfill & {,\,x \le 0} \hfill \\
{P(x)} \hfill & {,\,0 < x < 1} \hfill \\
1 \hfill & {,\,x \ge 1} \hfill \\
\end{array}} \right.
\]")
is continuous, smooth and has continuous curvature. Not hard but still fun to do. Feel free to try it yourself and post the answer in the comments.
We, then, began functions of several variables talking about four ways to describe a function of two variables and then looking at examples of each. (Verbally, Numerically, Graphically, Algebraically) We plotted a few interesting surfaces in Maple 10 and then began discussing level curves (contour maps). Using examples such as weather maps (temperature, pressure, etc.) and topographical maps helped the students get a good feel for how we can look at a 3 dimensional graph in a two dimensional plane (without necessarily resorting to perspective drawings). We'll start looking at level surfaces, or iso-surfaces, to generalize to functions of three variables. For example, how might you visualize a 4D hypersphere by using iso-surfaces?
For future versions of this lecture, I could definitely improve the lecture by bringing resources to class like video clips of a weather forecast or actual topographical maps of this area. Also, if I have time I'd like to have a demo of the software Viz5d so that I can visualize some data sets with animated iso-surfaces. I used to run a demo of a weather simulation when I worked for the High Performance Computer Center of Texas Tech University. I animated iso-surfaces of water vapor levels in the atmosphere. The result was a three-dimensional model resembling cloud movement.