Differential Calculus

The beauty of applied mathematics is that there is always a reason for
everything. When students take calculus they want to know *why*
they should be learning what they are taught. For example, why study
related rates? I tell students about the Engineer I know who studies
bubble dynamics. I instruct them to bring bubble blowing aparatus to class.
We spend a few minutes blowing bubbles.
The following questions are written on the board and the students record
the discussed responses in their lecture notes:

- How do you describe the interface between bubbles? (I introduce/remind the class of the
mathematical meaning of ``polygonal''.
- How would you write an equation describing the surface area
of two bubbles joined together?
- How would you describe the related rates problem of one bubble being
absorbed into another?

Understanding how foam (which is made up of millions of bubbles) will behave under a given set of conditions is a very important research area. This is what my engineer acqaintance studies. Of course the students ask: ``Why?!?'' I recall for my students the events at Chernobyl on the 26 of April, 1986. Many meters below the damaged nuclear power plant at Chernobyl lies the water table from which issues the water supply for Kiev, a city of over two million people. If nothing is placed between the leaking radiation of the plant and the water table, the citizens of Kiev will become contaminated with radiation. All the products their factories produce and all the crops their farmers harvest will be unsellable because they will have been toxically irradiated. The city will face first financial and then, later, utter physical ruin of its inhabitants.

How do we stop the leak of radiation as it seeps through the ground?

With foam.

By injecting a radiative-blocking foam into the ground between the water table and the leaking nuclear plant the millions of lives in Kiev may be saved. The challenge to scientists (and for my engineering friend) is how to ensure the foam maintains its integrity and persists in a thick enough layer to save Kiev.

The related rates at which neighboring bubbles expand and
collapse is central to the study of foam. If the foaming agent is mixed with too much air too fast, the bubbles burst. If the mixture is too low in air pressure, the foam may form but then collapse as bubbles absorb into one another. Engineers are seeking a *range* of acceptable rates of inflation and collapse. How do you represent this idea in math? Many students are utterly unaware of the role that *in*equalities play in applied mathematics, I take this opportunity to talk about of the importance of inequality. I assign the class the task of creating scenarios of absorbtion and pumping rate ranges using the related rates problems in their text.

The lesson closes with my reminder to them that the research on such foam is going on at full speed. After all, a lack of timely intervention will doom the water supply to contamination.