Synopses & Reviews
With a fresh geometric approach that incorporates more than 250 illustrations, this textbook sets itself apart from all others in advanced calculus. Besides the classical capstones--the change of variables formula, implicit and inverse function theorems, the integral theorems of Gauss and Stokes--the text treats other important topics in differential analysis, such as Morse's lemma and the Poincaré lemma. The ideas behind most topics can be understood with just two or three variables. This invites geometric visualization; the book incorporates modern computational tools to give visualization real power. Using 2D and 3D graphics, the book offers new insights into fundamental elements of the calculus of differentiable maps, such as the role of the derivative as the local linear approximation to a map and its role in the change of variables formula for multiple integrals. The geometric theme continues with an analysis of the physical meaning of the divergence and the curl at a level of detail not found in other advanced calculus books. Advanced Calculus: A Geometric View is a textbook for undergraduates and graduate students in mathematics, the physical sciences, and economics. Prerequisites are an introduction to linear algebra and multivariable calculus. There is enough material for a year-long course on advanced calculus and for a variety of semester courses--including topics in geometry. It avoids duplicating the material of real analysis. The measured pace of the book, with its extensive examples and illustrations, make it especially suitable for independent study.
Review
From the reviews: "Many concepts in calculus and linear algebra have obvious geometric interpretations. ... This book differs from other advanced calculus works ... it can serve as a useful reference for professors. ... it is the adopted course resource, its inclusion in a college library's collection should be determined by the size and interests of the mathematics faculty. Summing Up ... . Upper-division undergraduate through professional collections." (C. Bauer, Choice, Vol. 48 (8), April, 2011)
Synopsis
With a fresh geometric approach accompanied by approximately 300 figures, this textbook sets itself apart from all others in advanced calculus. Although this book presents classical capstones of advanced calculus, such as the divergence theorem for 3-dimensional regions and Stokes' theorem for 2-dimensional surfaces in space, it devotes most of the book to the physical meaning of the theorems and their underlying concepts. All relevant analytic topics, such as convergence of sequences of functions, are purposefully left out of this book because the analysis aspect is typically covered in advanced calculus textbooks while the geometric aspects are rarely represented.
The pace of Advanced Calculus: A Geometric View is designed deliberately for independent study, as it is abundant in figures, exercises, and visual examples that introduce ideas and topics. This book can be divided into two parts: Part 1: mappings; Part 2: integrals. There are many chapters in both parts, which go into extreme detail on the many different topics in advanced calculus, such as the geometry of linear maps, approximations, derivatives, implicit functions, surface integrals, and Stoke's theorem.
Containing both calculus and geometry, this textbook is intended to be used by undergraduates in mathematics and the physical sciences who are taking a course in advanced calculus and/or geometry, or engaged in independent study. The prerequisites include linear algebra and multivariable calculus.
Synopsis
A half-century ago, advanced calculus was a well-de?ned subject at the core of the undergraduate mathematics curriulum. The classic texts of Taylor 19], Buck 1], Widder 21], and Kaplan 9], for example, show some of the ways it was approached. Over time, certain aspects of the course came to be seen as more signi?cant those seen as giving a rigorous foundation to calculus and they - came the basis for a new course, an introduction to real analysis, that eventually supplanted advanced calculus in the core. Advanced calculus did not, in the process, become less important, but its role in the curriculum changed. In fact, a bifurcation occurred. In one direction we got c- culus on n-manifolds, a course beyond the practical reach of many undergraduates; in the other, we got calculus in two and three dimensions but still with the theorems of Stokes and Gauss as the goal. The latter course is intended for everyone who has had a year-long introduction to calculus; it often has a name like Calculus III. In my experience, though, it does not manage to accomplish what the old advancedcalculus course did. Multivariable calculusnaturallysplits intothreeparts: (1)severalfunctionsofonevariable, (2)one function of several variables, and (3) several functions of several variables. The ?rst two are well-developed in Calculus III, but the third is really too large and varied to be treated satisfactorily in the time remaining at the end of a semester. To put it another way: Green s theorem ?ts comfortably; Stokes and Gauss do not."
About the Author
James J. Callahan is currently a professor of mathematics at Smith College. His previous Springer book is entitled The Geometry of Spacetime: An Introduction to Special and General Relativity. He was director of the NSF-funded Five College Calculus Project and a coauthor of Calculus in Context.
Table of Contents
1 Starting Points.-1.1 Substitution.- Exercises.- 1.2 Work and path integrals.- Exercises.- 1.3 Polar coordinates.- Exercises.- 2 Geometry of Linear Maps.- 2.1 Maps from R2 to R2.- Exercises.- 2.2 Maps from Rn to Rn.- Exercises.- 2.3 Maps from Rn to Rp, n 6= p.- Exercises.- 3 Approximations.- 3.1 Mean-value theorems.- Exercises.- 3.2 Taylor polynomials in one variable.- Exercises.- 3.3 Taylor polynomials in several variables.- Exercises.- 4 The Derivative.- 4.1 Differentiability.- Exercises.- 4.2 Maps of the plane.- Exercises.- 4.3 Parametrized surfaces.- Exercises.- 4.4 The chain rule.- Exercises.- 5 Inverses.- 5.1 Solving equations.- Exercises.- 5.2 Coordinate Changes.- Exercises.- 5.3 The Inverse Function Theorem.- Exercises.- 6 Implicit Functions.- 6.1 A single equation.- Exercises.- 6.2 A pair of equations.- Exercises.- 6.3 The general case.- Exercises.- 7 Critical Points.- 7.1 Functions of one variable.- Exercises.- 7.2 Functions of two variables.- Exercises.- 7.3 Morse's lemma.- Exercises.- 8 Double Integrals.- 8.1 Example: gravitational attraction.- Exercises.- 8.2 Area and Jordan content.- Exercises.- 8.3 Riemann and Darboux integrals.- Exercises.- 9 Evaluating Double Integrals.- 9.1 Iterated integrals.- Exercises.- 9.2 Improper integrals.- Exercises.- 9.3 The change of variables formula.- 9.4 Orientation.- Exercises.- 9.5 Green's Theorem.- Exercises.- 10 Surface Integrals.- 10.1 Measuring flux.- Exercises.- 10.2 Surface area and scalar integrals.- Exercises.- 10.3 Differential forms.- Exercises.- 11 Stokes' Theorem.- 11.1 Divergence.- Exercises.- 11.2 Circulation and Vorticity.- Exercises.- 11.3 Stokes' Theorem.- 11.4 Closed and Exact Forms.- Exercises