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Review from the MAA Monthly

Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach



A review of the 2nd edition, by Warwick Tucker of Uppsala University, appeared in the October 2003 Mathematical Association of America Monthly.  We thank the Monthly and Professor Tucker for permission to publish the following excerpts. Because of limitations of html, mathematical equations are omitted.

Review:

During the academic year 2001-2 I had the pleasure to teach from this book while giving the Cornell University honors course for advanced freshmen.  These students are expected to have a 5 on the Advanced Placement BC Calculus exam prior to enrollment, and are thus already well-acquainted with single-variable calculus.  In other colleges the book would be well suited to a sophomore or junior honors course, and it could be used as an analysis text for more advanced students if the proofs in the hundred-page appendix were to be included in the syllabus.

Let me begin by stating my conclusions:  this book is a real gem.  It has a breadth and depth that is rarely seen in undergraduate texts, and it teaches real mathematics from a researcher's point of view instead of the standard off-the-shelf recipes that have little use outside the classroom.  Its definitions and theorems are carefully formulated so that the essential content of the results is clearly manifested.  Another welcome feature is its strong emphasis on computability; it presents techniques whose applicability to nontrivial situations is illustrated by many examples throughout the text.  The book is cleverly structured, and allows one to omit the most difficult topics and yet have a coherent text on a more standard level, so that it can be used successfully with students on several levels.  It has hundreds of exercises, and a separate student solution manual with solutions to all odd-numbered exercises is available from Matrix Editions ....

I will now walk you through the book, chapter by chapter, and highlight some of my favorite passages.  Of course, this is a highly subjective selection, and many omissions must be made... I strongly urge you to satisfy any resulting curiosity by reading the book!

The first chapter introduces the fundamentals of  linear algebra and multi-variable calculus.  Among the topics covered are matrix computations, linear transformations,  the dot product, determinants, and cross products (the latter two in R3 ).  The ``size''  of a matrix A  is measured by its length |A|, the square root of the sum of the squares of its components ...The point is that one can easily compute the length of a matrix, whereas computing its operator norm is usually quite hard.  Many theorems throughout the book are stated in terms of the length of a matrix in order to simplify the verification of  hypotheses.

Next, limits and continuity in Rn are treated, as are compact sets and their properties.  After stating and proving the Bolzano-Weierstrass theorem ...the authors illustrate the troubling implications of its nonconstructive proof with the following example... This example leads to some very intriguing questions about normal  numbers, a vast topic in its own right.

The last part of the chapter develops the differential calculus  in Rn.  The notion of derivative is introduced in terms of linear approximation... As a nice example, on page 141 one finds the matrix generalization of the formula for the derivative of a reciprocal.... The chapter is rounded off with a thorough treatment of the chain rule, the mean value theorem, and differentiability - all in  Rn .

The second chapter deals with solving equations, both linear and nonlinear.  The fundamental algorithm for solving systems of linear equations is row reduction, presented in a modern fashion in terms of multiplication by elementary matrices. ... Next the authors introduce orthonormal bases, kernels, images, rank, and the dimension formula for a linear map ...

The examples and exercises include applications to numerical analysis and interpolation theory...

The discussion then moves into the nonlinear world, where Newton's method is presented as the  method for solving nonlinear systems of equations.  The authors are careful to point out that, although Newton's method will fail under certain circumstances, in most situations it is possible to guarantee that it will  succeed. Sufficient conditions for the convergence of Newton's method are given by Kantorovich's theorem, a result seldom quoted in undergraduate textbooks. Assuming that the system is sufficiently smooth (e.g., C2), the conditions in Kantorovich's theorem can be checked quite easily.  As mentioned  earlier, the actual computations are greatly simplified by computing the length of a matrix rather than its operator norm.  This permits the reader to work with nontrivial examples such as the following (p. 250):
Show that Newton's method will converge to a solution of the system of equations
cos(x-y)= y
sin(x+y)=x,
if the initial point is taken to be (1,1).

Perhaps the greatest benefit of the detailed study of Newton's method lies in its theoretical applications: the implicit function theorem and the inverse function theorem.  The inverse function theorem is carefully stated so that if the conditions of the theorem are satisfied, then Kantorovich's theorem applies, and Newton's method can be used to find the inverse function.  The implicit function theorem is then proved as a corollary of the inverse function theorem.
 The third chapter is entitled ``Higher partial derivatives, quadratic forms, and manifolds''.  A smooth k-dimensional manifold in  Rn is defined to be a set that is locally the graph of a C1 mapping from Rn into Rn-k .... Several illuminating examples of smooth manifolds are given, as well as necesssary conditions for certain subsets of  Rn to be smooth manifolds.  Here the implicit function theorem plays an important role.  Also, the definition of a parametrization of a manifold is given, as is the proof showing that the defining property of a manifold is independent of the choice of coordinates.

The subject then changes to Taylor polynomials in higher dimensions, which are treated as the natural extension of the drivative... The section concludes with a brief treatment of Taylor polyinomials of implicit functions.

Next, quadratic forms are introduced are used to classify critical points, and the final topic of this chapter is basic differential geometry.   The Frenet frame of a space-curve, curvature and torsion are concisely presented, as are the Gaussian and mean curvatures of surfaces.

The fourth chapter is devoted to integration in Rn .  This chapter is probably the most original and challenging of them all.  It begins with the classical Riemann integral, defined used dyadic cubes for partitioning domains in Rn . (The dyadic cubes are advantageous from a computational point of view, but later it is shown that more general decompositions give rise to the same integral.)
This leads to the notion of n-dimensional volume...

The authors give a careful discussion of conditions for a function to be Riemann integrable.  First, they given a condition in terms of the function's oscillation that is necessary and sufficient but hard to check.  They then show that a bounded function of bounded support is integrable if it is continuous except on a set of volume zero.  Finally, they define the notion of a set of (Lebesgue) measure zero and prove that continuity except on a set of measure zero is necessary and sufficient.

At this point there is a small digression about determinants. This is an interesting section in several ways. The authors define the determinant of an n x n matrix
 A= [a1 ...an] (considered as a row of column vectors)  to be the unique real-valued function of (a1 ...an) that is linear in each of its arguments, skew-symmetric under interchange of arguments, and normalized so that dete(I)=1. (Of course it then must be proved that such a function actually exists and is unique.)  The idea of defining the determinant in terms of its operator properties has the very attractive benefit of simplifying proofs of other standard theorems, for example, that det (AB)= det (A) det(B). It also prepares the reader for the coming chapter on differential forms.

Returning to integration, the authors offer a novel treatment of the Lebesgue integral.  It avoids questions of sigma-algebras and measurable sets by adopting the following definition.... Once this is established, the classical results are proved, by arguments that are often quite original:  monotone convergence, dominated convergence, Fubini's theorem, and the change of variables formula.

Chapter 5 develops the notion of volume for submanifolds of Rn ....

The main goal of the final chapter is to develop the theory of integration of differential forms on manifolds and to generalize the fundamental theorem of calculus to higher dimensions.... The authors present such interesting examples as the set of 2 x 3 matrices of rank one, which is a nonorientable 4-manifold. They then show how to integrate differential k-forms over an oriented k-manifold. The heavy machinery needed for integrating k-forms over a parametrized domain was covered in the previous chapter, so little additional effort is required.

To reach the final goal of this chapter, two more concepts are needed.  First, the oriented boundary ...of an oriented k-parallelogram P is defined a a formal sum... Second, the authors introduce the exterior derivative in an unusual way...

At last the reader is prepared for the highlight of this last chapter, the generalized Stokes's theorem....Of course, plenty of applications of the standard integral theorems of vector calculus are given; the necessary formulas are all special cases of Stokes's theorem.

The appendix, which is over a hundred pages long, covers the more technical proofs as well as a justification of arithmetic over the real numbers.... The material covered here is more suited for an advanced analysis course, but the keen undergraduate will find several interesting topics within reach.

Closing notes.   Students will readily gain valuable knowledge from reading this book because the authors have been very careful in their selection and presentation of material.  Discussing manifolds rather than just curves and surfaces in chapter 3 enables them to state the generalized Stokes's theorem; the treatment also makes it clear that curves and surfaces are special cases of manifolds in Rn .  In chapter 4, replacing improper integrals with Lebesgue integration does not require much additional effort and leads to immeasurably better theorems.  In chapter 6, the definition of a piece with boundary gets to the  essence of what the boundary of an n-dimensional region must be if Stokes's theorem is to apply... This definition is one that can be used in real life, not just in textbook examples...

The authors compare calculus to learning how to drive, analysis to designing and building a car.  The main part of the text teaches ``how to drive"; the proofs in the appendix provide a manual for `designing and building the car."  Altogether, the book provides a valuable resource for the next generation to explore the highways of a central province of mathematics.