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R.G. Bartle - The Elements of Integration

Название: The Elements of Integration
Автор: R.G. Bartle
Категория: Математика
Тип: Книга
Дата: 05.01.2009 13:09:01
Скачано: 38
Описание: There was a time when an undergraduate student of mathematics was expected to develop technique in solving problems that involved considerable computation; however, he was not expected to master theoretical subtleties such as uniform convergence or uniform continuity. The student was expected to be able to use the Implicit Function Theorem, but was not expected to know its hypotheses. The situation has changed. Now it is generally agreed that it is important for all students — whether future mathematicians, physicists, engineers, or economists — to grasp the basic theoretical nature of the subject. For, having done so, they will understand both the power and the limitation of the general theory and they will be better equipped to devise specific techniques to attack particular problems as they arise. This text has developed from my experience in teaching courses in elementary real analysis at the University of Illinois since 1955. My audience has ranged from well-prepared freshman students to graduate students; the majority in these classes are usually not mathematics majors. Generally they have taken at least the equivalent of three semesters of non-rigorous calculus, including multiple integrals, vector calculus, line integrals, infinite series, and the like. It would be desirable to have the students take a semester either in linear or modern algebra before this analysis course, for such a background facilitates the study of rigorous analysis. However, since the students I encounter do not all have this background, I purposely delay the study of analysis and first explore the notion of an ordered field to provide practice in giving proofs. Thus the first six sections of this text are mostly preparatory in nature; they can be covered in about three weeks in a normal class and more rapidly in a well-prepared one.
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