Course Notes for Mathematics 211212: Multivariable Calculus(Copyright 2008, 2009, 2010, 2011, 2012, 2013 by Jerry Shurman. Any part of the material protected by this copyright notice may be reproduced in any form for any purpose without the permission of the copyright owner, but only the reasonable costs of reproduction may be charged. Reproduction for profit is prohibited.)
This is the text for a twosemester multivariable calculus course. The setting is ndimensional Euclidean space, with the material on differentiation culminating in the Inverse Function Theorem and its consequences, and the material on integration culminating in the Generalized Fundamental Theorem of Integral Calculus (often called Stokes's Theorem) and some of its consequences in turn. The prerequisite is a proofbased course in onevariable calculus. Some familiarity with the complex number system and complex mappings is occasionally assumed as well, but the reader can get by without it. The book's aim is to use multivariable calculus to teach mathematics as a blend of reasoning, computing, and problemsolving, doing justice to the structure, the details, and the scope of the ideas. To this end, I have tried to write in a style that communicates intent early in the discussion of each topic rather than proceeding coyly from opaque definitions. Also, I have tried occasionally to speak to the pedagogy of mathematics and its effect on the process of learning the subject. Most importantly, I have tried to spread the weight of exposition among diagrams, formulas, and words. The premise is that the reader is ready to do mathematics resourcefully by marshaling the complementary skills of
In my own student days, I learned this material from books by Apostol, Buck, Rudin, and Spivak, books that thrilled me. My debt to those sources pervades these pages. There are many other fine books on the subject as well, such as the more recent one by Hubbard and Hubbard. Indeed, nothing in these notes is claimed as new, not even their neuroses. By way of a warmup, chapter 1 reviews some ideas from onevariable calculus, and then covers the onevariable Taylor's Theorem in detail. Chapters 2 and 3 cover what might be called multivariable precalculus, introducing the requisite algebra, geometry, analysis, and topology of Euclidean space, and the requisite linear algebra, for the calculus to follow. A pedagogical theme of these chapters is that mathematical objects can be better understood from their characterizations than from their constructions. Vector geometry follows from the intrinsic (coordinatefree) algebraic properties of the vector inner product, with no reference to the inner product formula. The fact that passing a closed and bounded subset of Euclidean space through a continuous mapping gives another such set is clear once such sets are characterized in terms of sequences. The multiplicativity of the determinant, and the fact that that the determinant indicates whether a linear mapping is invertible, are consequences of the determinant's characterizing properties. The geometry of the cross product follows from its intrinsic algebraic characterization. Furthermore, the only possible formula for the (suitably normalized) inner product, or for the determinant, or for the cross product, is dictated by the relevant properties. As far as the theory is concerned, the only role of the formula to show that an object with the desired properties exists at all. The intent here is that the student who is introduced to mathematical objects via their characterizations will see quickly how the objects work, and that how they work makes their constructions inevitable. In the same vein, chapter 4 characterizes the multivariable derivative as a well approximating linear mapping. The chapter then solves some multivariable problems that have onevariable counterparts. Specifically, the multivariable chain rule helps with change of variable in partial differential equations, a multivariable analogue of the max/min test helps with optimization, and the multivariable derivative of a scalarvalued function helps to find tangent planes and trajectories. Chapter 5 uses the results of the three chapters preceding it to prove the Inverse Function Theorem, then the Implicit Function Theorem as a corollary, and finally the Lagrange Multiplier Criterion as a consequence of the Implicit Function Theorem. Lagrange multipliers help with a type of multivariable optimization problem that has no onevariable analogue, optimization with constraints. For example, given two curves in space, what pair of pointsone on each curveis closest to each other? Not only does this problem have six variables (the three coordinates of each point), but furthermore they are not fully independent: the first three variables must specify a point on the first curve, and similarly for the second three. In this problem, the coordinates can be conceived of as varying though a subset of sixdimensional space, conceptually a twodimensional subset (one degree of freedom for each curve) that is bending around in the ambient six dimensions. That is, optimization with constraints can be viewed as a beginning example of calculus on curved spaces. Moving to integral calculus, chapter 6 introduces the integral of a scalarvalued function of many variables, taken over a domain of its inputs. When the domain is a box, the definitions and the basic results are essentially the same as for one variable. However, in multivariable calculus we want to integrate over regions other than boxes, and ensuring that we can do so takes a little work. After this is done, the chapter proceeds to two main tools for multivariable integration, Fubini's Theorem and the Change of Variable Theorem. Fubini's Theorem reduces one ndimensional integral to n onedimensional integrals, and the Change of Variable Theorem replaces one ndimensional integral with another that may be easier to evaluate. Chapter 7 discusses the fact that continuous functions, or oncedifferentiable functions, or twicedifferentiable functions, are well approximated by smooth functions, meaning functions that can be differentiated endlessly. The approximation technology is an integral called the convolution. With approximation by convolution in hand, we feel free to assume in the sequel that functions are smooth. Chapter 8 introduces parameterized curves as a warmup for chapter 9 to follow. The subject of chapter 9 is integration over kdimensional parameterized surfaces in ndimensional space, and parameterized curves are the special case k=1. Aside from being onedimensional surfaces, parameterized curves are interesting in their own right. Chapter 9 presents the integration of differential forms. This subject poses the pedagogical dilemma that fully describing its structure requires an investment in machinery untenable for students who are seeing it for the first time, whereas describing it purely operationally is unmotivated. The approach here begins with the integration of functions over kdimensional surfaces in ndimensional space, a natural thing to want to do, with a natural definition of how to do it suggesting itself. For certain such integrals, called flow and flux integrals, the integrand takes a particularly workable form consisting of sums of determinants of derivatives. It is easy to see what other integrandsincluding integrands suitable for ndimensional integration in the sense of chapter 6, and including functions in the usual sensehave similar features. These integrands can be uniformly described in algebraic terms as objects called differential forms. That is, differential forms assemble the smallest coherent algebraic structure encompassing the various integrands of interest to us. The fact that differential forms are algebraic makes them easy to study without thinking directly about integration. The algebra leads to a general version of the Fundamental Theorem of Integral Calculus that is rich in geometry. The theorem subsumes the three classical vector integration theorems, Green's Theorem, Stokes's Theorem, and Gauss's Theorem, also called the Divergence Theorem.
Comments and corrections should be sent to jerry@reed.edu.
