Vectors
Calculus in 3D Geometry, Vectors, and Multivariate Calculus Zbigniew H. Nitecki
Tufts University
August 19, 2012
ii
This work is subject to copyright. It may be copied for non-commercial purposes.
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
Tufts University
August 19, 2012
ii
This work is subject to copyright. It may be copied for non-commercial purposes.
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
Bibliography: INDEX JF , see Jacobianpage:partialdet420 Jordan, Camille Marie Ennemond (1838-1922), 487, 610 811 INDEX 791, 793 Geometric Calculus (1888), 22 surface area, 522, 791 permutation cyclic, 74 planar loci, 115 planes angle between, 62 intersection of, 100–102 parallel, 60–62 parametrized, 64–69 Pappus of Alexandria (ca