Barnett S. Some Modern Applications of Mathematics 1995

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Barnett S. Some Modern Applications of Mathematics 1995

Textbook in PDF format For too long ‘applied mathematics’ in schools and universities has meant ‘mechanics’. Although this area still has an important role to play, many students are turning away from it and are showing a growing interest in newer areas of applications of mathematics, such as those covered by the various syllabuses in A-level ‘Decision mathematics’. These topics appeal to students because the applications are to problems arising in commerce, information technology and the environment, and generally do not involve a knowledge of physical principles. The objective of this book is to communicate the flavour of some of these areas of recent applications of mathematics. It is intended to be read by students and their teachers on A-level courses at school or college, or in the first year of undergraduate degree courses. I have taught in the classroom all the material which is covered in this book, and it is interesting that this has steadily ‘moved down’ the curriculum. For example, I gave postgraduate lectures on linear programming in the early 1960s, yet this subject is now in Level 10 of the National Curriculum, as well as forming part of the A-level ‘Decision mathematics’ syllabus. In a similar way, introduced control theory as an option for final-year mathematics undergraduates 25 years ago, whereas introductory courses are now taught at much lower levels. Part of the explanation for this moving down is due to the ready availability of computing power, enabling the use of efficient algorithms to solve problems. Throughout I have tried to emphasize discrete models using difference equations and matrix representations, and to reduce the emphasis on calculus and differential equations. This is for two reasons: first, students generally find discrete mathematics easier to grasp compared with the rather difficult concepts of calculus; and secondly, the amount of calculus being studied in schools is likely to decrease further in future. I have deliberately not included any discussion of computer software packages such as Derive or Maple. Not every student has ready access to these, and in any case software is revised or replaced so frequently that textbook treatments can quickly become out of date. Furthermore, I believe that at a beginning stage you learn and absorb concepts and techniques more readily by actually tackling problems with no more to help you than a good mathematical pocket calculator