Undergraduate Analysis - Mccluskey, Aisling

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Résumé :
Analysis underpins calculus, much as calculus underpins virtually all mathematical sciences. A sound understanding of analysis' results and techniques is therefore valuable for a wide range of disciplines both within mathematics itself and beyond its traditional boundaries. This text seeks to develop such an understanding for undergraduate students on mathematics and mathematically related programmes. Keenly aware of contemporary students' diversity of motivation, background knowledge and time pressures, it consistently strives to blend beneficial aspects of the workbook, the formal teaching text, and the informal and intuitive tutorial discussion. The authors devote ample space and time for development of confidence in handling the fundamental ideas of the topic. They also focus on learning through doing, presenting a comprehensive range of examples and exercises, some worked through in full detail, some supported by sketch solutions and hints, some left open to the reader's initiative. Without undervaluing the absolute necessity of secure logical argument, they legitimise the use of informal, heuristic, even imprecise initial explorations of problems aimed at deciding how to tackle them. In this respect they authors create an atmosphere like that of an apprenticeship, in which the trainee analyst can look over the shoulder of the experienced practitioner....

Biographie:
Brian McMaster studied at Queen's University Belfast, graduating with a PhD in 1972, and has served his alma mater department in various capacities including those of Adviser of Studies, Head of Research and Associate Director of Education. His publication profile covers over sixty refereed journal articles, mostly in the area of analytic topology but incorporating a smattering of applications in disciplines as diverse as probabilistic metric spaces and decision support theory. He has successfully supervised twelve individual postgraduate programmes including eight PhDs. He is presently formally retired but continues to deliver a full undergraduate teaching load on a voluntary basis, thus witnessing his lifelong commitment to and passion for communicating mathematics to students. His teaching interests focus around analysis (real and complex) and set theory and their development into various fields especially that of analytic topology.

Aisling McCluskey graduated from Queens University Belfast with a PhD in 1990 and subsequently was awarded a postdoctoral fellowship in Toronto. She was appointed to a permanent lectureship in Mathematics in NUI, Galway in January 1992. Since then, she has established a meaningful and rewarding academic career there, maintaining an active research profile whilst holding the teaching and learning of mathematics central to her academic endeavour.
She has enjoyed numerous international visiting researcher positions, and in like kind has hosted many eminent researchers at NUI, Galway. She has received institutional and national awards for excellence in teaching. In scholarly pursuit of her passion for and commitment to mathematics education, she completed a postgraduate certificate in teaching and learning in higher education (2009), a postgraduate diploma (2010) and a Masters degree (2011) at NUI Galway.

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Sommaire:

• 1: Preliminaries


• 2: Limit of a sequence, an idea, a definition, a tool


• 3: Interlude: different kinds of numbers


• 4: Up and down - increasing and decreasing sequences


• 5: Sampling a sequence - subsequences


• 6: Special (or specially awkward) examples


• 7: Endless sums - a first look at series


• 8: Continuous functions - the domain thinks that the graph is unbroken


• 9: Limit of a function


• 10: Epsilontics and functions


• 11: Infinity and function limits


• 12: Differentiation - the slope of the graph


• 13: The Cauchy condition - sequences whose terms pack tightly together


• 14: More about series


• 15: Uniform continuity - continuity's global cousin


• 16: Differentiation - mean value theorems, power series


• 17: Riemann integration - area under a graph


• 18: The elementary functions revisited