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Not a MyNAP member yet? Register for a free account to start saving and receiving special member only perks. D URING HIS LONG and adventurous life Courant achieved many things in mathematics: in research and the applications of research, in the exposition of mathematics and the education of students, and in administrative and organizational matters. To understand how he, essentially an outsider both in Germany and the United States, accomplished these things we have to examine his personality as well as his scientific works. Courant was born on January 8, , in the small town of Lublinitz in Upper Silesia, now part of Poland but then of Germany.
Not a MyNAP member yet? Register for a free account to start saving and receiving special member only perks. D URING HIS LONG and adventurous life Courant achieved many things in mathematics: in research and the applications of research, in the exposition of mathematics and the education of students, and in administrative and organizational matters. To understand how he, essentially an outsider both in Germany and the United States, accomplished these things we have to examine his personality as well as his scientific works.
Courant was born on January 8, , in the small town of Lublinitz in Upper Silesia, now part of Poland but then of Germany. His father, Siegmund, was an unsuccessful businessman. The family moved to Breslau, and soon the precocious Richard was beginning to support himself by tutoring. In the gymnasium he came under the influence of a charismatic teacher of mathematics, Maschke, who inspired specially selected, talented students with a love of mathematics.
Six years after Courant the young Heinz Hopf entered the gymnasium in Breslau and came under the tutelage of Maschke, who trained his special pupils by posing challenging problems. Many years later Hopf recalled that he was able to solve most of them, but was stumped every once in a while.
This no doubt was the first bond in the intimate friendship that developed later between Courant and Hopf. After the gymnasium Courant was ready to attend university lectures on mathematics and physics at the University of Breslau. Because of the weakness of the physics faculty he gravitated toward mathematics. For young Courant the shining light was Hilbert, and it was his great good fortune that in Hilbert chose him to be his assistant. The next phase in his career was the writing of a dissertation.
It is illuminating to go back more than 50 years to the dissertation of Riemann. Weierstrass challenged the validity of this proof, because the existence of a minimum cannot be taken for granted. Weierstrass even gave an example of a fourth order functional whose minimum is not assumed by any function. Back to Courant. Courant succeeded, and was awarded his Ph. The same topic served for his habilitation dissertation in He was fascinated not only by its use in theory but also by the possibility of basing numerical calculations on it, as was done by the young physicist Walther Ritz.
Courant liked to spice his lectures with remarks about the personalities of scientists, to render them more human. Thus, in a talk in Kyoto in , his last public lecture, he described the work of Walther Ritz and recalled that Ritz.
Then Courant added that Walther Ritz was a member of the Swiss family whose hotels all over the world made their name synonymous with luxury. In Courant married Nelly Neumann, a fellow student from Breslau; the marriage lasted only four years. Courant was drafted into the army; he fought on the western front and was seriously wounded.
While in the trenches, Courant had seen the need for reliable means of communication, and came to the idea of a telegraph that would use Earth as a conductor.
In the end the Earth telegraph became a resounding success; equally important, the experience taught Courant how to. On the contrary, it made both of them realize their incompatibility. They were married in They had much in common—a passionate love of music—but in many respects they were very different. Their marriage was a successfully shared life.
They had four children: two boys, who became physicists, and two girls, a biologist and a musician. The years —20 were banner years for Courant. He proved that among all plane domains with prescribed perimeter, the circle had the lowest fundamental frequency. This was followed by a max-min principle that enabled him to determine the asymptotic distribution of eigenvalues of the Laplace operator over any domain, a result of great physical interest, established previously by Weyl with the aid of a min-max principle.
The combination of the two methods is particularly effective. Courant encouraged Springer to enlarge his offering in mathematics. The latter saw Courant—correctly—. The early s were a tough time in Germany. The defeat in the First World War had demoralized large segments of society and had led to rampant inflation. Courant showed his resourcefulness by keeping things afloat, partly with the help of the far-sighted industrialist Carl Still.
The first part, based on lecture notes of Hurwitz, was written from the Weierstrass point of view; its main subject was elliptic functions. Courant supplemented this material with nine chapters on Riemann surfaces, conformal mapping, and automorphic functions.
Courant used an informal, intuitive notion of a surface that displeased some readers but pleased others. Two years later, in , the first volume of Courant-Hilbert appeared. The book starts with a page chapter on linear algebra, presented from an analytic point of view, so that generalization to infinite dimension comes naturally.
This is followed by chapters on orthogonal function systems, the Fredholm theory of integral equations, the calculus of variations, and the vibrations of continuum mechanical systems, using extensively the spectral theory of self-adjoint ordinary and partial differential operators.
Fortuitously, Courant-Hilbert Volume I contained much of the mathematics needed to understand and solve. This was a striking example of mathematics anticipating the needs of a new physical theory. In Courant, Friedrichs, and Lewy published their famous paper on the partial difference equations of mathematical physics.
The main motivation for writing it was to use finite difference approximations to prove the existence of solutions of partial differential equations.
The paper discusses elliptic, parabolic, and hyperbolic equations; it con-tains a wealth of ideas, such as the probabilistic interpreta-tion of elliptic difference equations, and the restriction that has to be imposed on the ratio of the time increment and the space increment. The latter, known as the CFL condition, became famous during the computer age. Woe to the computational scientist who ignorantly violates it. This is an outstanding example of research undertaken for purely theoretical purposes turning out to be of immense practical importance.
It has been extremely successful in every sense; its translation into English by McShane has sold 50, copies of Volume 1 and 35, copies of Volume 2 in the United States. It has shaped the minds of many who wanted and needed a deeper grasp of the calculus.
Even after 70 years it is better than most, nay all, calculus books in use today in the United States. There were many assistants and postdocs around; Courant had private sources of money to pay their stipends. This caused some confusion after the Second World War, when the German government, to its credit, decided to compensate not only faculty members who were dismissed by the Nazis but assistants as well. In concrete negotiations were started, and plans laid, for housing the institutes of mathematics and physics in a permanent building, for long a dream of Felix Klein, now enthusiastically taken up by Courant.
The building of the institute was finished and dedicated in Courant became its director. Yet this moment of triumph already contained the seed of its own destruction, and that of most civilized Western institutions. The stock market in the United States crashed a few months earlier, leading to a deep economic depression that soon became worldwide. The misery caused by this drove a sizeable part of the German voting population, already embittered by the defeat in the First World War, to support the Nazis.
In January the Hitler gang took over the government. It soon established a new age of barbarism in Germany. For a start, Jewish employees of the state, including professors, were dismissed summarily, Courant among the first. For once his grasp of reality deserted him, and he went from.
A chance encounter with a member of the Nazi party, who was a member of the university community, set him right. Things will get worse and worse for you. You had better get out while you can. Get out Courant did. After a brief stay in England he, his family, and some family friends landed in New York, where thanks to Oswald Veblen and Abraham Flexner, a position was offered to him in the department of mathematics of New York University, with the charge to develop a graduate program.
His host there was the mathematician Donald Flanders, admired by all for his saintly character. He and Courant formed a deep friendship that today extends to their children.
At NYU Courant found a mathematical desert. How he made it bloom is a fascinating story. It started in with a burst of creative energy. Courant pursued these generalizations during the next 10 years. Friedrichs, and by James J. Stoker, an American. In the s, mid-career, he decided to seek a Ph. One of the first courses he took there was by Heinz Hopf on geometry. Stoker was so charmed by the subject, and the teacher, that he switched his doctoral studies to differential geometry.
In What Is Mathematics? It ought to be compulsory reading for all who today are engaged in reforming the teaching of mathematics. To enable those who worked during the day to attend classes, graduate courses were offered in the evening, once a week, for two hours at a time.
Its head, Vannevar Bush, saw the importance of mathematics for the war effort and set up the Applied Mathematics. Panel under the direction of Warren Weaver. Courant was soon invited to be a member of this elite group. The mathematical project at NYU sponsored by the OSRD was about the flow of compressible fluids in general and the formation and propagation of shock waves in particular.
There was also money to provide stipends for graduate students, some of whom were drawn into war work. Courant insisted that graduate training continue even during the war. This is a good place to describe Courant as a classroom teacher. He seldom bothered to prepare the technical details of his lecture.
Mathematics Educators Stack Exchange is a question and answer site for those involved in the field of teaching mathematics. It only takes a minute to sign up. By hard I don't mean difficult in explanations, but with extremely challenging exercises all worked out if possible and useful insights and tricks. Also, I would like you to share similarly defined books or handouts about other disciplines in an undergraduate course that is to say, the subjects mentioned here. You might also want to look at the list of honors calculus books posted in the following math StackExchange question:. Joseph Kitchen's Calculus reference. Throughout the s there was a gradual overall trend towards more difficult texts until the end of the s, at which time there were several "downward adjustments" I'm mostly thinking of the U.
Thank you for visiting nature. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser or turn off compatibility mode in Internet Explorer. In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. THE English edition of Vol. McShane, of the original German text which was published in Berlin in It is devoted to the more advanced parts of the calculus; chiefly to functions of several variables and their physical significance.
He is best known by the general public for the book What is Mathematics? Courant was born in Lublinitz , in the Prussian Province of Silesia. Edith Stein was Richard's cousin on the paternal side. During his youth his parents moved often, including to Glatz , then to Breslau and in to Berlin. He was obliged to serve in World War I , but was wounded shortly after enlisting and therefore dismissed from the military. Courant left Germany in , earlier than many Jewish escapees.
Richard Courant's classic text Differential and Integral Calculus is an essential text for those preparing for a career in physics or applied math. Volume 1.
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These books are those designed to be the required textbook for a standard high school or university calculus course. Many of them have had a large number of revisions, which makes it likely that you can find slightly older but still perfectly useful copies on the used market. They try to have everything -- proofs, intuitive explanations, illustrations, problems -- but they aren't necessarily great at anything. These books are widely derided by math majors, though I suspect this to be in part motivated by elitism and by grad students' dislike of teaching lower-level math courses. In any case, these are useful even if you just want a cheap source of problems to practice with.
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COURANT. DIFFERENTIAL by R. COURANT simultaneous treatment of differential calculus and integral tions of one variable; a second volume will be devoted to these: (1) the English edition contains a large number of classified.
ReplyThe present volume contains the more advanced parts of the differential and integral calculus, dealing mainly with functions of several variables. As in Volume I, I.
Replyedition of my lectures on the differential and integral calculus,. I at first these: (1) the English edition contains a large number of classified examples The second volume deals in full with functions of several inde- pendent by R. Courant.
ReplyRichard Courant. Differential and Integral Calculus, Interscience Publishers,. Vol. I, second edition, ; Vol. Courant Institute of Mathematical Sciences. New York University S.1 Limits and the Number Concept a. The Rational Numbers.
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