We picked up where we left off prior to the Exam on Tuesday. We are discussing function of several variables. I began by recapping the techniques we use for visualizing the graphs of functions of two variables, namely, surfaces and contour maps, or level curves. We used this as a springboard to move up to functions of 3 variables. Typically, any function of [tex]n[/tex] variables will need [tex]n+1[/tex] dimensions to represent the graph of the function. For example, for a function of 2 variables, [tex]f(x,y)[/tex] we define its graph to be the set [tex]\left\{ (x,y,z) \ | \ z=f(x,y) \ \mbox{s.t.} \ (x,y) \in D \right\}[/tex] where [tex]D[/tex] is the domain of [tex]f[/tex]. Thus, the graph of a function of two variables is a subset of [tex]\mathbb{R}^3[/tex].

I discussed a few ways for visualizing these graphs. The techniques I demonstrated went a little beyond what the textbook covers because I have done quite a bit of computer modeling for these types of data sets, especially in my early days at the High Performance Computing Center at Texas Tech. I showed them how to use level surfaces which are the generalization of level curves, or contour maps. I, then, demonstrated the used of animation to help get a better feel for the 4th dimension when using level surfaces. Finally I showed them examples of CT scans where slices are used to show in interior of a volume in three space. I recounted a project that I helped with as a graduate student where we were building a three dimensional computer model of a prosthetic arm using only 2D images which were “slices” of the arm. The primary challenge of that project was that the images were not properly aligned so that we had to apply an optimization technique to translate and rotate the image to minimize the change from image to the next. The result was a decent representation but a little “bumpy” due to the misaligned images.

It reminded me of the Super Bowl several years back where a researcher had been employed to set up cameras around the top of the stadium and have them filming the same point on the field. They they tried to use the cameras to rotate around a play in a given instant, producing the “Matrix” effect, where some player is frozen in mid-air and the camera rotates around them. The end result was kind of cool but still real jumpy. The reason was there was too much computation needed to align the images for rotation. It required too much processing power to do in real time. I wonder if the progress in computer hardware would make that more realistic today. I know they didn’t bring it back after that because it just looked hokey.

Next time, we will finish up the study of limits of functions of several variables and get to start doing partial derivatives. Yeehoo!