Author : Kenneth Kuttler, Department of Mathematics, Brigham Young University
Pages : 908 pages
Publication Date : May 11, 2007
Excerpts from the Introduction:
Calculus consists of the study of limits of various sorts and the systematic exploitation of the completeness axiom. It was developed by physicists and engineers over a period of several hundred years in order to solve problems from the physical sciences. It is the language by which precision and quantitative predictions for many complicated problems are obtained. It is used to find lengths of curves, areas and volumes of regions which are not bounded by straight lines. It is used to predict and account for the motion of satellites. It is essential in order to solve many maximization problems and it is prerequisite material in order to understand models based on differential equations. These and other applications are discussed to some extent in this book.
It is assumed the reader has a good understanding of algebra on the level of college algebra or what used to be called algebra II along with some exposure to geometry and trigonometry although the book does contain an extensive review of these things. If the optional sections and non standard sections are not included, this book is fairly short. However, there is a lot of non standard material, including the big theorems of advanced calculus.
I have also tried to give complete proofs of all theorems in one variable calculus and to at least give plausibility arguments for those in multiple dimensions with complete proofs given in appendices or optional sections. I have done this because I am sick and tired of books which do not bother to present proofs of the theorems stated. For a serious student, mathematics is not about accepting on faith unproved assertions presumably understood by someone else but "beyond the scope of this book". It has become fashionable to care nothing about such serious students and to write books for the entertainment of those who care nothing for explanations. Serious students will find complete explanations here. Physical models are derived in the usual way through the use of differentials leading to differential equations which are introduced early and used throughout the book as the basis for physical models.
I expect the reader to be able to use a calculator whenever it would be helpful to do so. Some introduction to the use of computer algebra systems is also presented and there are exercises which require the use of some form of technology. Having said this, calculus is not about using calculators or any other form of technology. I believe that when the syntax and arcane notation associated with technology are presented too prominently, these things become the topic of study rather than the concepts of calculus. This is a book on calculus.
Pictures are often helpful in seeing what is going on and there are many pictures in this book for this reason. However, calculus is not about drawing pictures and ultimately rests on logic and definitions. Algebra plays a central role in gaining the sort of understanding which generalizes to higher dimensions where pictures are not available. Therefore, I have emphasized the algebraic aspects of this subject far more than is usual, especially those parts of linear algebra which are essential to understand in order to do multi-variable calculus.
I have also featured the repeated index summation convention and the usual reduction identities which allow one to discover vector identities.
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