tailieunhanh - Ebook Math advanced calculus: Part 1

(BQ) Part 1 book "Math advanced calculus" has contents: Numbers, functions, the derivative, the riemann integral, the euclidean n space, vector valued functions of a vector variable, sequences of functions, linear functions. | Hans Sagan North Carolina State University Advanced Calculus of Real-Valued Functions of a Real Variable and Vector-Valued Functions of a Vector Variable Houghton Mifflin Company Boston Atlanta Dallas Geneva Illinois Hopewell New Jersey Palo Alto London Numbers Set Language A Mathematical Shorthand This section is devoted to a discussion of certain abstract concepts namely sets and some manipulations with these concepts namely the elementary set operations. It is not intended as an introduction to set theory. It merely serves to introduce some terminology the language of sets that will prove to be helpful in our studies. A command of this language is not absolutely essential for a study of analysis but is of considerable assistance in such a study as it provides for short and concise formulations as well as for an abstract structure for arguments which will occur repeatedly in the study of analysis. Such arguments would be awkward and overburdened with imprecise and excessive verbiage were it not for the language of sets. We shall take a naive approach and say in the spirit of Georg Cantor f that a set is a collection of well-distinguished objects of our conception or imagination. We note that this idea of a set when carried too far leads to grave difficulties as we shall point out in section 3. The collection of coins in your pocket is a set it may be the empty set. Definition The empty set 0 is the set which contains no elements. The collection of all matter in the universe is a set for those who are materialistically inclined this may well be the universal set. The universal set is the set of all objects that are under consideration in a specific investigation. For a number theorist the universal set may be the set of all integers for an analyst the universal set may be the set of all real numbers or the set of all functions for a geometer the universal set may be the set of all points on a line or the set of all lines in a plane or what have you. t .

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