Preliminary table of contents.

Videos.

The text seamlessly connects to applications (oscillating systems, circuits, heat equation, hydrogen atom) and is written from the point-of-view that reading is an active task. Introductory and practice activities are designed to prepare the reader for a topic and to practice after the solution method is understood. For many of the topics, a video presentation is available.

This text gives a concise introduction to the analysis of functions of one real variable (continuity, differentiation, Riemann and Lebesgue integration) in part I, an introduction to measure spaces, metric spaces, normed spaces, manifolds and Hilbert spaces in part II and an outlook toward applications (including the finite element method) in part III.
 

Foreword and table of contents (pdf).

Sample Module (pdf).

Featured Exercises (pdf).

Classroom activities (pdf).

Instructor's manual (pdf).

Introduction, Table of Contents, Chapter 1 (locked pdf)

Link to Birkhäuser's site for the book (order can be placed there)

Remarks, errata

This text introduces the reader to the main constructions and ideas in the theory of ordered sets. The theme-based approach presents all constructions in a natural context and allows the reader to confront each new concept in a familiar setting. A multitude of exercises provides training. Open research problems are stated at the end of practically every chapter. These problems are understandable and possibly even solvable with the tools introduced in the respective chapter.

Since there are few prerequisites, the text can be used as a focused follow-up or companion to a first proof class (set theory and relations) or to a graph theory class. After covering a comparatively lean core, the text can be used to concentrate on topics such as, for example,
structure theory, enumeration or algorithmic aspects. In each of these topics the text lays a solid foundation upon which research in the area can be started by a mathematically mature reader.

Aside from introducing open problems that have served and will continue to serve as inspiration for research in ordered sets, the text covers some important topics less customary to discrete mathematics/graph theory.
Among these topics are, for example an efficient introduction of homology for graphs and the presentation of forward checking as a more efficient alternative to the standard backtracking algorithm. These topics
as well as the many fundamental results presented give the text lasting value as a reference.