An introduction to applied mathematics

Description

The first edition of this book was an attempt to give an introduction to the basic mathematics needed in physics and engineering. Only a knowledge of the principles of statics and dynamics and of the calculus was assumed; other techniques were developed ah initio, but as rigorously as possible, and an attempt was made to develop mathematical skill through the solution of a large number of problems of technical significance. To this end, in addition to conventional statics and dynamics, the book covered much of what is now described as ‘mathematical methods’, including vector analysis, numerical analysis, ordiniary and partial differential equations, special functions, Fourier series, and Fourier and Laplace Transforms. In dynamics it included a long chapter on mechanical vibrations and another on electric circuit theory. Boundary value problems were intro¬ duced through the theory of bending of beams. This new edition has been prepared largely by Professor Starfield. It has been found that most of the old material is still needed. In some places the examples have been modernized, for example solid-state devices replace vacuum tubes. Other areas have been extended; for instance servomechanisms are discussed in terms of the transfer function, and the applications of Fourier transforms have been widened to include ideas in communication theory. Also, over the last few years, the appli¬ cation of mathematics to biological and economic problems has greatly increased, and discussion and examples relating to both of these topics have been added. The great change over the past two decades has been the development of computer techniques which have not only enormously widened the range of problems which can be solved, but have changed habits of thought in the formulation of problems and demanded new approaches in their solution. The emphasis has to some extent changed from problems leading to differential equations to those involving difference equations or an algorithmic approach. Moreover, the numerical solution ofPREFACE VI differential equations involves their expression in terms of difference equations and creates a new concern with questions of accuracy and the stability of solution. In more complicated cases, such as systems of equations and the numerical solution of partial differential equations, matrices have become an essential computing tool. To cope with these changes two new chapters have been added; one on difference equations and the numerical solution of differential equations, and the other on matrices. The last two chapters on partial differential equations have also been rewritten and extended and a number of examples have been added for numerical solution on a computer.