![]() We then review the elementary theory of differentiation for single-variable functions, mostly through exercises. First we recall some basic definitions, and then, very quickly, we go over the concepts of limits and continuity of real functions defined on the real line. The final section of the chapter is nothing more than a swift refresher on the analysis of real functions. These results will be used freely in the remainder of the book. In particular, even though our coverage will serve only as a review for most readers, we study here the monotonic sequences and subsequential limits with some care, and prove a few useful results, such as the Bolzano-Weierstrass Theorem and Dirichlet's Rearrangement Theorem. These we discuss relatively thoroughly because of the important role they play in real analysis. The related discussion is intended to be read more quickly than anywhere else in the text. After a short elaboration on ordered fields and the Completeness Axiom, we note without proof that the rational numbers form an ordered field and the real numbers form a complete ordered field. No construction for the integers is given, in particular. We assume here that the reader is familiar with the elementary properties of the real numbers and thus provide only a heuristic discussion of the basic number systems. Our coverage of abstract set theory concludes with a brief discussion of the Axiom of Choice and the proof of Szpilrajn's Theorem on the completion of a partial order. After a quick excursion to order theory (in which the only relatively advanced topic that we cover is the completion of a partial order), functions are introduced as special cases of binary relations and sequences as special cases of functions. We start with an intuitive discussion of the notion of set, and then introduce the basic operations on sets, Cartesian products, and binary relations. Supporters of Newton and Leibniz often arguing along bitter and blatantly nationalistic lines and the feud itself had a profound influence on the subsequent development of calculus and other branches of mathematical analysis in England and in Continental Europe.Ī principal objective of this largely rudimentary chapter is to introduce the basic set-theoretical nomenclature that we adopt throughout the text. The controversy regarding credit for the origin of calculus quickly became more than a simple dispute between mathematicians. Although it is clear that Newton made his discoveries regarding calculus years before Leibniz, most historians of mathematics assert that Leibniz independently developed the techniques, symbolism, and nomenclature reflected in his preemptory publications of the calculus in 16. ![]() By the end of the 18th century, calculus had proved a powerful tool that allowed mathematicians and scientists to construct accurate mathematical models of physical phenomena ranging from orbital mechanics to particle dynamics. Although the logical underpinnings of calculus were hotly debated, the techniques of calculus were immediately applied to a variety of problems in physics, astronomy, and engineering. ![]() ![]() Although the evolution of the techniques included in the calculus spanned the history of mathematics, calculus was formally developed during the last decades of the 17th century by English mathematician and physicist Sir Isaac Newton (1643-1727) and, independently, by German mathematician Gottfried Wilhelm von Leibniz (1646-1716). Many of the most influential advances in mathematics during the 18th century involved the elaboration of the calculus, a branch of mathematical analysis which describes properties of functions (curves) associated with a limit process.
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