The Miracle of Applied Mathematics

Abstract

(ProQuest: ... denotes formulae omitted.)1 IntroductionIn the teaching profession, as in many other professions, an initially bright aspiration is all too often overcome by chronic, nagging frustration. The original enthusiasm that once inspired a person's choice of vocation will tend to dissipate as so-called real-world pressures take their toll, depleting mental energy and putting tight constraints on free creative acts.Ideally, a college classroom ought to be a place where genuine communication engages minds and changes lives by stimulating joyful curiosity and honest exploration. But in an academic environment in which the publish-or-perish mentality is prevalent and extreme over-specialization corrupts the educational mission, these worthy goals will often seem remote or even wholly out of reach. One way to counter such adversities is to be very intentional in setting aside some time for topical excursions that broaden the view and thereby re-energize the minds of students and teachers alike. This is not to say, of course, that large chunks of course content ought to be sacrificed to random pursuits of topical tangents. But it is to say that the cultivation of a certain versatility in subject matter selection may help to re-invigorate a teaching effort which, perhaps, has come to be burdened over time by a general loss of creative momentum.To give the reader a concrete example of how such versatility can be practiced without straying all too far from standard course material, I would like to provide an outline of a lecture that I have given in each of the past three years in my course on differential equations at John Brown University. The students taking this course are mostly engineering majors in their second year and have successfully completed Calculus I and II. They are used to placing mathematics in an application-oriented problem-solving context but have never been exposed to the full theoretical rigor that mathematics majors can be expected to practice. Consequently, some of the steps in the calculations that follow will be skipped over relatively lightly without providing concise analytical proofs. This lack of rigor, though, can easily be overcome, as any reader familiar with the fundamental principles of real analysis can simply look up the relevant arguments in standard analysis textbooks.In order not to cause any misunderstandings, it also needs to be emphasized that-at the time when the lecture is given-the students in attendance are already well familiar with the theory of linear systems of differential equations and have studied in particular the representation of solutions by means of exponential matrices. So the purpose here is not to communicate new mathematical content but rather to offer a hopefully interesting alternative view.2 Outline of the LectureOne of the most important problems in the study of differential equations is the computation of solutions of linear systems of the form...(1)where A is a constant n x n-matrix and f a continuous vector-valued function. The range of applications to which such systems are relevant is very wide indeed. From electric networks, multiple overflow systems, and mechanical linkages of masses pulled by springs, the spectrum readily extends to chemical bonds in solid-state matter, to the entire classical theory of quantum mechanics by way of the Schrodinger equation, and even more generally, to just about any theory or field that is described by linear partial differential equations.Given this very broad applicability, it is truly amazing to realize that the problem of solving equation (1) can be reduced to the following three elementary tasks:Task 1: drawing tangents to a graph.Task 2: rearranging beads of two different colors. (2)Task 3: drawing balls at random from two bowls.How is it possible that these three basic tasks combine to give us access to such a vast array of physical phenomena? …