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Serious mathematics, written with the reader in mind.
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
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.