A ProblemText in Advanced Calculus

2013

This textbook has been written for the analysis education of non-mathematics students at the Eötvös Loránd University, Faculty of Science, but it can also be used as a supplementary material by students of mathematics. All subjects are presented at beginners' level, where mainly methods are taught. The book is strongly application-oriented. For example, vector calculus is included for students of geophysics, and contour and surface integrals are presented for physics student.

HAL (Le Centre pour la Communication Scientifique Directe), 2020

2003

Synthesis Lectures on Mathematics & Statistics (SLMS), 2025

The notes on which this book is based were originally created and designed for mathematics teachers, returning to the university to learn, refresh or relearn more advanced calculus and mathematical analysis, not necessarily in a formal, semester long format. Without claiming to start from the beginning, and without being exhaustive, comprehensive or completely self-contained, this book attempts to serve as a short, concise guide or crash course to review and study the ideas of calculus, of limit and convergence of sequences of functions, and to illustrate the limitations of the Riemann integral and motivate the need for the Lebesgue integral. A long road is traveled without deviating much from the main route. There are many excellent, comprehensive advanced calculus texts, some of which are listed in the future readings section. These texts contain an abundance of material, and many storylines can be weaved and followed. This is not such a textbook. Here, a single, efficient storyline is proposed: start at the beginning and finish at the end. Many of the proofs of the presented results are not provided; they are suggested as activities, with occasional hints thrown in to point in the direction of a proof. No lists of exercises are included; the idea is to give a succinct and compact presentation. The tone has been kept informal, even as rigorous arguments are presented and expected. Results, properties, and demonstrations are conveyed in a relaxed prose which hopefully will make reading easier. The included activities frequently use graphing software available online: Desmos Calculator. Some of the activities present a series of notable examples which illustrate the theory; all the suggested activities should be completed. A complementary webpage has been set up, to allow direct play and interactivity with graphs and other demonstrations in the text. Mathematics teachers in Puerto Rico have very different levels of preparation. An effort was made to address this heterogeneity, which influenced many of the thematic, style, language, depth, rigor and formality choices made. The hope is that the text will allow readers to efficiently and gently approach advanced calculus and mathematical analysis, and that in a classroom setting, it can be used by instructors to ignite fruitful discussions with their students.

International Journal of Research in Undergraduate Mathematics Education, 2016

We investigate the challenges students face in the transition from calculus courses, focusing on methods related to the analysis of real valued functions given in closed form, to more advanced courses on analysis where focus is on theoretical structure, including proof. We do so based on task design aiming for a number of generic potentials for student learning, developed from and within the theory of didactic situations: adidactic potential, linkage potential, deepening potential and research potential. The context of investigation is a first year course on analysis in which the tasks thus constructed were considered relevant to solve a number of operational problems. The experimental method involves careful a priori analysis of each task in terms of the potentials, specifically related to the knowledge at stake; this analysis in confronted with a posteriori analyses of observations of student work before and in class sessions. Two cases are analyzed in detail. While some of the potentials were partly realized, we also identified clear limitations resulting from a variety of factors, including teaching assistants' management of the class sessions and students' perception of the importance, difficulty and meaning of the tasks.

Undergraduate Texts in Mathematics are generally aimed at third-and fourthyear undergraduate mathematics students at North American universities. These texts strive to provide students and teachers with new perspectives and novel approaches. The books include motivation that guides the reader to an appreciation of interrelations among different aspects of the subject. They feature examples that illustrate key concepts as well as exercises that strengthen understanding.

2013

Problem Based Learning Active (PBL) is usually a methodology applied to small class. This article presents adapted Problem Based Learning Active (PBL) activities for large class applied on Multivariable Differential Calculus in Classical Calculus II courses for second year engineering students.

PRIMUS