Awards for exemplary writing in MAA journals and magazines were handed out in August at MAA MathFest in Portland, Oregon. The winning articles (PDF) are now available online, along with the award citations explaining why each article merited an award.
Adam E. Parker
Who Solved the Bernoulli Differential Equation and How Did They Do It?
The College Mathematics Journal, Vol. 44, no. 2, March 2013, pp. 89-97
Citation: We mathematicians are so focused on the theory and techniques of our subject that we virtually ignore its history. Even when we attach a name to a topic, such as the Bernoulli Differential Equation, the modern reader cannot be confident that it is the correct name. Adam Parker takes us on a mystery tour to seek to identify who was the first to actually solve Bernoulli’s Equation.
You might think this entails nothing more than a careful search of very old publications seeking to find the first published solution, but there is much more to the story than that. Parker transports us to the 1690s, a world very different than the one we know today. Leibnitz and the Bernoulli brothers, Jacob and Johann, relate to one another as mentors, mentees, friends, and sometimes fierce competitors. Jacob poses the equation in print as a challenge problem. All three present solutions using a variety of techniques including reduction to a linear differential equation, separation of variables, and variation of parameters. Leibnitz’s claim is described deliberately vaguely, apparently to hide special insights and to hold on to the competitive advantage.
Does it really matter who was first? Do we want to judge who solved it best? Parker’s engaging article, like any good math article, raises more questions than it answers.
But Parker doesn’t stop with reporting the early history. He observes that all the modern textbook authors follow like sheep, presenting Jacob’s substitution. This solves the problem easily enough, but leaves the student to memorize a specific substitution that doesn’t generalize to other problems.
Is this the best pedagogy? Parker lobbies for using this problem to introduce variation of parameters. Whether you agree or not with the teaching philosophy, Adam Parker has written an article that is enjoyable to read, informative, and thought provoking.
How Inge Lehmann Discovered the Inner Core of the Earth
The College Mathematics Journal, Vol. 44, no. 5, November 2013, pp. 398-408
Citation: How can we determine what lies deep within the Earth when we cannot travel there ourselves and cannot obtain direct measurements beyond a certain depth? As Christiane Rousseau points out, we can use our “mathematical eyes” to see. Her paper in the Mathematics of Planet Earth issue of The College Mathematics Journal provides a lesson in such mathematical sight of otherwise invisible geological features.
Rousseau starts by exploring what we can learn about the Earth using just a few concepts, including Newton’s gravitational law and a spherical model of Earth. Quickly, she shows that assuming a homogeneous interior for this simplified Earth leads to a mismatch with data on seismic waves following earthquakes. She then uses geometric and trigonometric ideas to describe refraction and reflection. These concepts become necessary when we change the model of a homogeneous Earth to consider an Earth with concentric layers with differing properties. At each stage of extending the model of Earth’s interior, Rousseau connects scientific principles, mathematical properties, and known data. Diagrams and explanations make clear how each model works, yet there remain exercises for readers who would like to try for themselves to complete some of the computations.
Interspersed with the modeling, we meet Inge Lehmann, the Danish seismologist who first determined that the Earth’s core had more than one layer. Lehmann lived and worked in Denmark. As a child, she experienced an education that saw girls as equal to boys, yet when she was an adult, her workplace did not credit the ideas of women as readily as those of men. Lehmann used modeling similar to that described by Rousseau to support her claim of the Earth’s inner core. Her claim directly contradicted the conventional wisdom of the time, that the Earth’s core was entirely liquid, and took a few years to gain acceptance.
This article combines history, the thrill of discovery, and explication of how to build a meaningful model from basic principles. It engages the reader and epitomizes the theme of Mathematics of Planet Earth.