Looking for Applications: Notes from a Foreign Correspondent

When we began work on the new editions of the CCH single and multivariable calculus books, the call went out for new problems, especially problems involving interesting, current applications. For many mathematicians, for many years, "calculus applications" have automatically been identified with physics. For teaching applications, some of us may even further identify physics applications with mechanics. Some of the other basic sciences may sneak in for a problem or two, and economics gets some play, but it's mostly physics that a calculus teacher has looked to when trying to seem relevant and connected to what's happening elsewhere.

If not physics, where do you look for applications? How do you come up with examples and problems that your students will recognize as interesting and will want to know how to analyze? With all due respect to the fast-paced world of inclined planes, it's worth the effort to browse more broadly. Here are a few areas for consideration.

Data Compression
Want a commercially important example of continuity? Consider data compression. Way before MP3, there was continuity. When audio signals such as music are processed for recording on CDs, the intensity of the sound is measured every 1/44,100 second and each measurement is assigned a 16 bit number from 0 to 65,535 (or 2¹⁶-1, the largest 16 bit number). Thus every second of music can require 705,600 (44,100 x 16) zeros and ones. For even a short song this amounts to millions and millions of numbers, and it becomes a crucial question whether this information can be compressed in some way. This is where continuity comes in.

Acoustically, it is likely that there is very little change in intensity in the 1/44,100th of a second from one measurement to the next. Mathematically, this is a statement about the continuity of the intensity function I(t): the difference in intensities at successive measurements is a very small number. Furthermore, a series of measurements; I(t₀), I(t₁), ..., I(t_n) will not differ much from the starting intensity if the total elapsed time t_n-t₀ is not too large; that is, the differences I(t₁)-I(t₀), I(t₂)-I(t₀), …, I(t_n)-I(t₀) will all be small.

How large do we allow the differences to be and how long do we take the time interval? That depends on what sort of signal is being sampled, but you can imagine that engineers--and stockholders in their companies--take their epsilons and deltas seriously (and to the bank). The practical consequence is that a small difference in intensities can be expressed in a binary number much shorter than 16 digits. For each specified time interval of sound, instead of storing the individual intensities I(t₀₎, I(t₁), ..., I(t_n),  each of which may be a large binary number, one can store the starting intensity I(t₀) and the much smaller differences, I(t₁)-I(t₀), I(t₂)-I(t₀), ..., I(t_n)-I(t₀). This can result in a significant savings in storage, typically 50% or greater. When the sound is played back, the individual intensities are recovered by adding the differences to the starting intensity.

This process is called Differential Pulse Code Modulation, and is abbreviated DPCM. It is a commonly used technique and is frequently combined with other methods of digital data compression. Image Processing Take image processing for another source of different applications and use any popular, commercial software package for a classroom demo. Version 1.0 of Adobe Photoshop was released in February 1990, the current version is 6.0, and I don't know how many units have sold in between. I do know that many of your students use Photoshop, or a similar package, or will at some point. (Incidentally, 1990 was also the year that the TI-81 graphing calculator was released.) Electronic image processing involves plenty of nice math to talk about.

For example, most packages will display a histogram of the grayscale levels of an image. (Most will also do histograms for colors.) Why do the packages include a command to show histograms? Because operating on a histogram to modify an image is a standard technique in image processing, for example histogram equalization or histogram sliding. As an exercise, ask your students to sketch the histogram that would result if an image was brightened, darkened, had more contrast, or less contrast. Or investigate how first and second derivatives of intensity are used by such programs to find edges, and check out "watershed edge detection" as an example of the usefulness of locating critical points.

3-D Computer Graphics
Tired of the usual menu of quadric surfaces? Sample some of what's offered in computer graphics. Puff up your ellipsoids to "superellipsoids," or surfaces of the form

A silly generalization? Not to people who want to put pillows into their 3-D scenes. Applications of parametrized curves don't always have to be moving projectiles and the like, either. Students, and you, might find design problems interesting, as in a discussion of the mathematics of Bezier curves used in vector-based drawing programs like Adobe Illustrator, which premiered in 1987. 

As you choose examples that will be effective and memorable, remember where your students are likely headed. They may take physics, but most don't stay long. Consider this data from the Fall 1999 edition of Engineering & Technology Enrollments, published by the Engineering Workforce Commission of the American Association of Engineering Societies. The number of full time undergraduate students in Chemical Engineering was 26,027, in Civil Engineering 32,396, in Electrical Engineering 56,969, and in Computer Science 48,239. According to the American Institute of Physics' Enrollment and Degrees Report, 2000, the total number of declared undergraduate majors in Physics in 1998/99 was 10,619.