Exploratory Examples For Real Analysis
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Exploratory Examples for Real Analysis
Author: Joanne E. Snow
language: en
Publisher: American Mathematical Soc.
Release Date: 2003-12-31
This text supplement contains 12 exploratory exercises designed to facilitate students' understanding of the most elemental concepts encountered in a first real analysis course: notions of boundedness, supremum/infimum, sequences, continuity and limits, limit suprema/infima, and pointwise and uniform convergence. In designing the exercises, the [Author];s ask students to formulate definitions, make connections between different concepts, derive conjectures, or complete a sequence of guided tasks designed to facilitate concept acquisition. Each exercise has three basic components: making observations and generating ideas from hands-on work with examples, thinking critically about the examples, and answering additional questions for reflection. The exercises can be used in a variety of ways: to motivate a lecture, to serve as a basis for in-class activities, or to be used for lab sessions, where students work in small groups and submit reports of their investigations. While the exercises have been useful for real analysis students of all ability levels, the [Author];s believe this resource might prove most beneficial in the following scenarios: A two-semester sequence in which the following topics are covered: properties of the real numbers, sequences, continuity, sequences and series of functions, differentiation, and integration. A class of students for whom analysis is their first upper division course. A group of students with a wide range of abilities for whom a cooperative approach focusing upon fundamental concepts could help to close the gap in skill development and concept acquisition. An independent study or private tutorial in which the student receives a minimal level of instruction. A resource for an instructor developing a cooperative, interactive course that does not involve the use of a standard text. Ancillary materials, including Visual Guide Sheets for those exercises that involve the use of technology and Report Guides for a lab session approach are provided online at: http:www.saintmarys.edu/~jsnow. In designing the exercise, the [Author];s were inspired by Ellen Parker's book, Laboratory Experiences in Group Theory, also published by the MAA.
Exploratory Examples for Real Analysis
Author: Joanne E. Snow
language: en
Publisher: Cambridge University Press
Release Date: 2003
Every mathematician must make the transition from the calculations of high school to the structural and theoretical approaches of graduate school. Essentials of Mathematics provides the knowledge needed to move onto advanced mathematical work, and a glimpse of what being a mathematician might be like. No other book takes this particular holistic approach to the task. The content is of two types. There is material for a ""transitions"" course at the sophomore level; introductions to logic and set theory, discussions of proof writing and proof discovery, and introductions to the number systems (natural, rational, real, and complex). The material is presented in a fashion suitable for a Moore Method course, although such an approach is not necessary. An accompanying Instructor's Manual provides support for all flavors of teaching styles. In addition to presenting the important results for student proof, each area provides warm-up and follow-up exercises to help students internalize the material. The second type of content is an introduction to the professional culture of mathematics. There are many things that mathematicians know but weren't exactly taught. To give college students a sense of the mathematical universe, the book includes narratives on this kind of information. There are sections on pure and applied mathematics, the philosophy of mathematics, ethics in mathematical work, professional (including student) organizations, famous theorems, famous unsolved problems, famous mathematicians, discussions of the nature of mathematics research, and more. The prerequisites for a course based on this book include the content of high school mathematics and a certain level of mathematical maturity. The student must be willing to think on an abstract level. Two semesters of calculus indicates a readiness for this material.
Real Analysis Through Modern Infinitesimals
Author: Nader Vakil
language: en
Publisher: Cambridge University Press
Release Date: 2011-02-17
A coherent, self-contained treatment of the central topics of real analysis employing modern infinitesimals.