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2011 | Book

Introduction Read chapter Read first chapter

Discovering Mathematics

A Problem-Solving Approach to Mathematical Analysis with MATHEMATICA® and Maple™

Authors: Jiří Gregor, Jaroslav Tišer

Publisher: Springer London

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

The book contains chapters of structured approach to problem solving in mathematical analysis on an intermediate level. It follows the ideas of G.Polya and others, distinguishing between exercises and problem solving in mathematics. Interrelated concepts are connected by hyperlinks, pointing toward easier or more difficult problems so as to show paths of mathematical reasoning. Basic definitions and theorems can also be found by hyperlinks from relevant places. Problems are open to alternative formulations, generalizations, simplifications, and verification of hypotheses by the reader; this is shown to be helpful in solving problems. The book presents how advanced mathematical software can aid all stages of mathematical reasoning while the mathematical content remains in foreground. The authors show how software can contribute to deeper understanding and to enlarging the scope of teaching for students and teachers of mathematics.

Table of Contents

Frontmatter
Introduction
Abstract
The introduction describes some general approaches to problem solving in mathematics , advantages and methods of use of advanced computer software. The structure of the book and ways of its use are outlined. The interactive use of the MATHEMATICA Notebooks and/or the Maple worksheets given in the Electronic Supplementary Material is described. It is shown how the embedded hyperlinks specifying interconnection of problems may help to find solutions.
Jiří Gregor, Jaroslav Tišer

Concepts

Frontmatter
1. Mappings, Composite and Inverse Functions
Abstract
The chapter deals with with significant special mappings, their inverse mappings,composition, iteration, invariants and fixed points of mappings. It includes examples of functional equations, methods of investigating special functions, examples of mappings f:R 3→R 2.
Jiří Gregor, Jaroslav Tišer
2. Infinite Sequences
Abstract
Infinite sequences and their limits are basic concepts in analysis. Examples show how to deepen understanding of this concept including special methods, order of convergence, cluster points etc. Also applications of sequences in numerical mathematics, theory of equations and other parts of mathematics are shown together with some famous classical results.
Jiří Gregor, Jaroslav Tišer
3. Periodicity
Abstract
Periodicity as the abstract concept of events occurring at regular time intervals is important in modeling nature, life and man-made objects. Its importance lies in predictability of events in science and technology. Problems with identification an analysis of periodicity, periodical solutions of differential and difference equations are included. One of the most important tools in investigating periodicity is the theory of Fourier series. Only a rather formal description is covered here without dealing with the theory of convergence of Fourier series.
Jiří Gregor, Jaroslav Tišer

Tools

Frontmatter
4. Finite Sums
Abstract
Finite sums pose a problem if the number of summands is large and/or when the evaluation of each of the summands has a common and non-simple pattern. Simplification of such sums demands special methods and skill. These methods can also be used in dealing with infinite series. Examples are given for counting objects constrained by arithmetic or geometric rules. The use of computers opened new problems in this directions.
Jiří Gregor, Jaroslav Tišer
5. Inequalities
Abstract
The chapter contains inequalities which are often used in mathematical reasoning. Moreover, positivity, i.e. a special inequality, is an important issue in applications. Methods of proving and deriving new inequalities and estimates are included. Last but not least, applications of inequalities in solving mathematical problems arising outside mathematics are shown.
Jiří Gregor, Jaroslav Tišer
6. Collocation and Least Squares Methods
Abstract
Various methods to adjust given or chosen mathematical models to actual requirements or actual data are presented. Such adjustment, if it exists, means choice of proper values of some parameters, which usually leads to solution of a set of linear or nonlinear equations. When such adjustment is not possible then least squares method gives an alternative.
Jiří Gregor, Jaroslav Tišer

Applications

Frontmatter
7. Maximal and Minimal Values
Abstract
The well known fact that differentiable functions may have local extremes only at points where their first derivatives vanish is illustrated. In some cases the corresponding equations are not directly accessible or solvable and other considerations have to be applied. Constrained extremal values for real functions of several variables and extremal problems of non-differentiable functions are included.
Jiří Gregor, Jaroslav Tišer
8. Arcs and Curves
Abstract
Various ways to describe and define arcs and considerations elucidating the concept of a curve are given. Methods of studying their local behavior, shape, length and other properties are presented. For the sake of simplicity mostly planar curves are discussed.
Jiří Gregor, Jaroslav Tišer
9. Center of Mass and Moments
Abstract
The dynamics of an isolated system of N free particles, each with the mass m i , i=1,2,…,N is completely determined by the collection of their moments (of order 1). Generalizations of this concept opened new ways in applied sciences as well as in mathematics. Interconnections between the moment problem and least squares techniques form the basic idea of orthogonal series, which became an important tool in approximation theory.
Jiří Gregor, Jaroslav Tišer
10. Miscellaneous
Abstract
Miscellaneous problems, mostly dealing with polynomials (positivity, Lucas theorem, Marden’s theorem, location of zeros). Their solution may demand some “ad hoc” methods to be applied.
Jiří Gregor, Jaroslav Tišer

Appendix

Frontmatter
11. Answers to Problems
Abstract
In this chapter solutions of the problems are given whenever appropriate, in some cases in form of MATHEMATICA® or Maple® commands.
Jiří Gregor, Jaroslav Tišer
Backmatter
Metadata
Title
Discovering Mathematics
Authors
Jiří Gregor
Jaroslav Tišer
Copyright Year
2011
Publisher
Springer London
Electronic ISBN
978-0-85729-064-9
Print ISBN
978-0-85729-054-0
DOI
https://doi.org/10.1007/978-0-85729-064-9

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