What I Learnt During My Teaching Mathematical Analysis -chen

What I Learnt During My Teaching Mathematical Analysis Tian-quan Chen Department of Mathematical Sciences Tsinghua University March 6, 2006

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I have taught Mathematical Analysis to undergraduates majoring in mathematics for about twenty years. During my teaching mathematical analysis I have consulted many books and articles on mathematical analysis and mathematical education. The following two points are what I learnt from the books and the articles I have consulted: 1. I have to teach more mathematics to my students than my teacher taught me 50 years ago. In 1954 French mathematician A.Weil, who worked at University of Chicago after the second world war, wrote an article entitled “ The Mathematics Curriculum, a short guide for students ”. The following passage is extracted from his article: The organization of the traditional curriculum (i.e., the curriculum at the beginning of 20th century) was simple. At an elementary level, it consisted of analytic geometry, in 2 and 3 dimensions, and so-called “ college algebra ”, viz. the elementary theory of equations, with the professed aim of teaching the numerical solution of equations with real coefficients, in one unknown; analytic geometry was presented in the form which it had reached in the XVIII century, with Clairaut, Euler and Lagrange; the algebra was still essentially that of Descartes, as improved by Newton. Then followed “ The Calculus ”, and its applications to the theory of curves and surfaces, following roughly the pattern laid down by Euler, and so-called “ Applied Mathematics ”, i.e., elementary theoretical dynamics on Newtonian lines. “ The Calculus ” culminated in “Functions of a Complex Variable”, a strongly bowdlerized version of some of the work of Cauchy, Riemann and more especially Weierstrass, on that subject. If, at the end of this, the student had learnt the definition of elliptic functions, and a few formulas about these, then he was deemed

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an accomplished mathematician, fit for higher work in his subject. Unfortunately, teachers and students of mathematics have less easy life nowadays: the above topics have not ceased to be basic, but they are so far from being sufficient that all devices have to be sought by which more can be accomplished within a briefer time. Also, the development of so-called “ abstract ” mathematics, and of the “ axiomatic ” method, during approximately half-century has led to an ever clearer realization of the fact that mathematics is, in part, a language, that this language must keep pace with the needs it has to serve, that it has a grammar and vocabulary of its own, and that these have eventually to be learnt. The grammar and vocabulary of modern mathematics are chiefly supplied, in the first place, by so-called “ abstract set theory ” and, beyond this, by general topology and by algebra: these are, essentially, auxiliary branches of mathematics, with, however, this notable difference between them, that abstract set theory was created less than 100 years, and general topology less than 50 years ago, and both can already be considered as completed, so far as the needs of presentday mathematics are concerned; while algebra goes back to babylonians, and is still vigorously growing. However that may be, these topics had been infiltrating the more traditional ones in the curriculum, calculus and geometry, for a long time, before the wastefulness of studying them piecewise in many contexts became recognized. For example, the process of reducing a quadratic form to a sum of squares is nothing else than the method of solving the quadratic equations by “ completing squares ”, already known to the Babylonians; it is basic for the study of conics and quadrics in plane and solid analytic geometry, for the study of the same in projective geometry, the natural extension of these subjects to higher

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dimensions, for the study of maxima and minima in thew calculus, for the “ orthogonalizing process ” in Hilbert space and many special cases of this process which had preceded the introduction of Hilbert space into mathematics. There is an obvious advantage in dealing at one stroke with the underlying idea of all such topics, in the way most suitable to the various applications which have to be made of it. Before concluding his article, A.Weil wrote down the basic knowledge of the main branches of pure mathematics all students majoring in mathematics should acquire. The basic knowledge of analysis comprise “ some knowledge of modern integration-theory (preferably not confined to the traditional Lebesgue integral of functions of one or more real variables, but extending to compact and locally compact spaces ), of Hilbert space, of differential equations ( ordinary and partial ), and of functions of one complex variable”. Fifty years have elapsed after A.Weil published his article. I think, the following two topics should be added to the basic knowledge of analysis the undergraduates majoring in mathematics should acquire: (a) manifolds, at least manifolds in Euclidean space, and the differential forms on it; (b) the theory of distributions ( of Laurent Schwartz ) and Fourier transforms of distributions. Many other topics could be included in a course of mathematical analysis, e.g., asymptotic analysis, which is useful in both pure and applied mathematics ( theory of numbers, theory of probability, differential equations, fluid dynamics, statistical mechanics, and so on). Whether it would be included in a course of mathematical analysis depends on the tastes of the teacher and students.

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I think, many topics in analysis can be arranged in one course on analysis to avoid the wastefulness of studying them piecewise in many contexts. There are many nice textbooks on mathematical analysis written with a point of view close to Andre Weil’s. The following list, far from being complete, might be useful for reference: (1a) V.A.Zorich, Mathematical Analysis, I,II ( Berlin, Springer, 2004). (1b) H.Amann und J.Escher, Analysis, I,II,III ( Basel, Birkha¨ user, 1998). (1c) H.Heuser, Lehrbuch der Analysis, I,II (Stuttgart, Teubner, 1980). (1d) A.Browder, Mathematical Analysis, An Introduction, (New York, Springer, 1996). (1e) C.C.Pugh, Real Mathematical Analysis, (New York, Springer, 2002). (1f) J.Jost, Postmodern Analysis, second edition, (Berlin, Springer, 2003). (1g) J.J.Duistermaat and J.A.C.Kolk, Multidimensional Real Analysis, I,II,(New York, Cambridge University Press, 2004). (1h) K.Maurin, Analysis, I,II (Dordrecht, D.Reidel Publishing Co. and Warsaw, Polish Scientific Publishers, 1979). (1i) H.Grauert und I.Lieb, Differential- und Integralrechnung, I,II,III, (Berlin, Springer, 1976). 2. It is desirable to use mathematical language in teaching physics to students majoring in mathematics. Learning applications of mathematics is essential for students to understand mathematics. The most important applications of mathematics are those in physics. The following extract is taken from the Josiah

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Willard Gibbs Lecture, “ Missed Opportunities ”, given by F.Dyson, Professor of physics at Princeton University, under the auspices of the American Mathematical Society in 1972. After Newton’s laws of gravitational dynamics had been promulgated in 1687, the mathematicians of the eighteenth century seized hold of these laws and generalized them into powerful mathematical theory of analytical mechanics. Through the work of Euler, Lagrange and Hamilton, the equations of Newton were analyzed and understood in depth, new branches of pure mathematics ultimately emerged. Lagrange distilled from the extremal properties of dynamical integrals the general principles of the calculus of variations. Fifty years later the work of Euler on geodesic motion led Gauss to the creation of differential geometry. Another fifty years later, the generalization of the Hamilton-Jacobi formulation of dynamics led Sophus Lie to the invention of Lie groups. And finally, the last gift of Newtonian physics to pure mathematics was the work of Poincar´e on the qualitative behavior of orbits which led to the birth of modern topology. But the mathematicians of the nineteenth century failed miserably to grasp the equally great opportunity offered to them in 1865 by Maxwell.If they had taken Maxwell,s equations to heart as Euler took Newton’s, they would have discovered, among other things, Einstein’s theory of special relativity, the theory of topological groups and their linear representations, and probably large pieces of the theory of hyperbolic differential equations and functional analysis. A great part of twentieth century physics and mathematics could have been created in the nineteenth century, simply by exploring to the end the mathematical concepts to which Maxwell’s equations naturally lead.

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I guess, quantum physics and statistical physics (including a large part of physics of condensed matter) will offer great opportunities to twenty first century mathematicians to develop new mathematics, e.g., new techniques in asymptotic analysis. In China, every student majoring in mathematics should accomplish a course in physics. But it is really miserable that most students would say goodbye to physics after passing the examination of physics. There are many causes for this miserable phenomenon. I think, one of the important causes is that we have used the vague language, not the mathematical language, in teaching physics to students majoring in mathematics. For example, most students of mathematics would be puzzled by the following confusing statement: “ An electron is a particle, and simultaneously, a wave.” In a book review on Tait’s “ Thermodynamics ”, the great British physicist J.C.Maxwell wrote: In the popular treatise, whatever shreds of the science are allowed to appear, are exhibited in an exceedingly diffuse and attenuated form, apparently with the hope that the mental faculties of the reader, though they would reject any stronger food, may insensibly become saturated with scientific phraseology, provided it is diluted with a sufficient quantity of more familiar language. In this way, by simple reading, the student may become possessed of the phrases of the science without having been put to the trouble of thinking a single thought about it. The loss implied in such an acquisition can be estimated only by those who have been compelled to unlearn a science that they might at length begin to learn it. The technical treatise do less harm, for no one ever reads them except under compulsion. From the establishment of the general equations to the

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end of the book, every page is full of symbols with indices and suffixes, so that there is not a paragraph of plain English on which the eye may rest.. Nowadays, there are some textbooks on physics for students of mathematics, which use mathematical language to express the physical concepts in an exact way. The following textbooks on physics, or on mathematics with strong physical background, are good examples: (2a) L.D.Fadeev and O.A.Yakubovskii, Lectures on Quantum Mechanics for Students of Mathematics, (In Russian), (Peterburg, Izdatel’stvo Leningrad University, 1980). (2b) W.Thirring, A Course in Mathematical Physics, I,II,III,IV, (Wien, Springer, 1992). (2c) D.M.Bressoud, Second Year Calculus, (New York, Springer, 1991). (2d) P.Bamberg and S.Sternberg, A Course in Mathematics for Students of Physics, (New York, Cambridge University Press, 1990). (2e) A.Sudbery, Quantum Mechanics and the Particles of Nature, an outline for mathematicians, (London, Cambridge University Press, 1986). The two points put forward above are controversial among my colleagues. I would like to hear criticisms from you.

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