Vectors

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Calculus in 3D Geometry, Vectors, and Multivariate Calculus Zbigniew H. Nitecki Tufts University
August 19, 2012

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Preface
The present volume is a sequel to my earlier book, Calculus Deconstructed: A Second Course in First-Year Calculus, published by the Mathematical Association in 2009. I have used versions of this pair of books for severel years in the Honors Calculus course at Tufts, a two-semester “boot camp” intended for mathematically inclined freshmen who have been exposed to calculus in high school. The first semester of this course, using the earlier book, covers single-variable calculus, while the second semester, using the present text, covers multivariate calculus. However, the present book is designed to be able to stand alone as a text in multivariate calculus. The treatment here continues the basic stance of its predecessor, combining hands-on drill in techniques of calculation with rigorous mathematical arguments. However, there are some differences in emphasis. On one hand, the present text assumes a higher level of mathematical sophistication on the part of the reader: there is no explicit guidance in the rhetorical practices of mathematicians, and the theorem-proof format is followed a little more brusquely than before. On the other hand, the material being developed here is unfamiliar territory, for the intended audience, to a far greater degree than in the previous text, so more effort is expended on motivating various approaches and procedures. Where possible, I have followed my own predilection for geometric arguments over formal ones, although the two perspectives are naturally intertwined. At times, this may feel like an analysis text, but I have studiously avoided the temptation to give the general, n-dimensional versions of arguments and results that would seem natural to a mature mathematician: the book is, after all, aimed at the mathematical novice, and I have taken seriously the limitation implied by the “3D” in my title. This has the advantage, however, that many ideas can be motivated by natural geometric arguments. I hope that this approach lays a good intuitive foundation for further generalization that the reader will see in later courses. Perhaps the fundamental subtext of my treatment is the way that the theory developed for functions of one variable interacts with geometry to iii

iv handle higher-dimension situations. The progression here, after an initial chapter developing the tools of vector algebra in the plane and in space (including dot products and cross products), is to first view vector-valued functions of a single real variable in terms of parametrized curves—here, much of the theory translates very simply in a coordinate-wise way—then to consider real-valued functions of several variables both as functions with a vector input and in terms of surfaces in space (and level curves in the plane), and finally to vector fields as vector-valued functions of vector variables. This progression is not followed perfectly, as Chapter 4 intrudes between Chapter 3, the differential and Chapter 5, the integral calculus of real-valued functions of several variables, to establish the change-of-variables formula for multiple integrals.

Idiosyncracies
There are a number of ways, some apparent, some perhaps more subtle, in which this treatment differs from the standard ones: Parametrization: I have stressed the parametric representation of curves and surfaces far more, and beginning somewhat earlier, than many multivariate texts. This approach is essential for applying calculus to geometric objects, and it is also a beautiful and satisfying interplay between the geometric and analytic points of view. While Chapter 2 begins with a treatment of the conic sections from a classical point of view, this is followed by a catalogue of parametrizations of these curves, and in § 2.4 a consideration of what...
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