Marketing research class lectures_136603365-Lectures

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Lectures on the Geometry of ManifoldsLiviu I. NicolaescuJanuary 23, 2013IntroductionShape is a fascinating and intriguing subject which has stimulated the imagination of many people.It suffices to look around to become curious. Euclid did just that and came up with the first purecreation. Relying on the common experience, he created an abstract world that had a life of its own.As the human knowledge progressed so did the ability of formulating and answering penetratingquestions. In particular, mathematicians started wondering whether Euclids obvious absolutepostulates were indeed obvious and/or absolute. Scientists realized that Shape and Space are twoclosely related concepts and asked whether they really look the way our senses tell us. As FelixKlein pointed out in his Erlangen Program, there are many ways of looking at Shape and Spaceso that various points of view may produce different images. In particular, the most basic issueof measuring the Shape cannot have a clear cut answer. This is a book about Shape, Space andsome particular ways of studying them.Since its inception, the differential and integral calculus proved to be a very versatile tool indealing with previously untouchable problems. It did not take long until it found uses in geometryin the hands of the Great Masters. This is the path we want to follow in the present book.In the early days of geometry nobody worried about the natural context in which the methodsof calculus

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