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Advanced Calculus

Pietro-Luciano Buono
Walter de Gruyter
Publication Date: 
Number of Pages: 
De Gruyter Textbook
[Reviewed by
P. N. Ruane
, on

Intended for students with a background of two semester courses in (single variable?) calculus and with at least one semester of linear algebra, this book provides an alternative approach to multivariable calculus. It covers much of what might be contained in a course of vector analysis, but ‘advanced calculus’ is a better label.

Indeed, brief perusal of the table of contents shows that, apart from the implicit function theorem and Lagrange multipliers, most of the topics associated with advanced calculus are covered here. The approach is decidedly geometric, and one of the most appealing features is the abundance of coloured diagrams that permeate the text. Linear algebra underpins the mathematical development, and its power is most evident with the classification of critical points in terms of eigenvalues of Hessians of functions \(f:\mathbb{R}^n\to\mathbb{R}\).

An unusual aspect of the book is its early introduction to the concept of tangent spaces on a curve (as distinct from a line tangent to a curve). A key idea is that the real line coincides geometrically with its own tangent space, so that a tangent vector \(v\in T_{t_0}\mathbb{R}\) is depicted by an arrow on the real line with base point at \(t_0\in\mathbb{R}\). This is the basis for the definition of differentials, 1- forms and pullbacks of real-valued functions — initially via discussion of line integrals. Moreover, integration by substitution and reparameterization of curves are both interpreted via the notion of the pull-back of 1-forms. The wedge product arises in the last quarter of the book with the introduction of 2-forms and 3-forms.

In keeping with the book’s subtitle, a major aim is to provide a unified treatment of Green’s, Stokes’ and Gauss’ theorems. These are expressed in classical (vector) notation and more succinctly in terms of 1-forms and their exterior derivatives. By this stage, readers will have gained good insight into some of the underlying concepts of differential geometry. Curvature is defined for parametrically-defined space curves and orientability and tangent planes for surfaces emerge clearly.

In general, the mathematical content is presented with great clarity and there is an abundance of worked examples and diagrams. However, there are a few obvious typos scattered throughout the book and some questionable use of notation in the early chapters. For example, the tangent line to a curve \(\mathbf{r}:[a,b]\to\mathbb{R}^2\) is expressed with only partial use of vector notation as \(l(s)=\mathbf{r}(t_0) + s\mathbf{r}'(t_0)\). An average student will spot most of these anomalies; but notational aspects of the important Proposition 3.1.9 are more significant. Specifically, the image of \(v\in T_{t_0}\mathbb{R}\) under the mapping \(\mathbf{r}'(t_0): T_{t_0}\mathbb{R}\to T_pC\) should be a tangent vector at \(p = \mathbf{r}(t_0)\) on the smooth curve \(C\). In fact, it should be a scalar multiple of \(\mathbf{r}'(t_0)\). Instead, it is denoted by \(v\mathbf{r}'(t_0)\), meaning that the vector \(v\in T_{t_0}\mathbb{R}\) has been magically converted to a scalar.

Quibbles aside, I’m greatly impressed by this book. There is a very nice section on differential operators, and the treatment of double and triple integrals leads naturally to the idea of integration of forms and integrals on a surface. Also, by restricting discussion to one, two and three forms in a geometric context, the book avoids many of those daunting generalities on differential forms. Such features, together with an abundant provision of practice exercises (no hints or solutions) make this an ideal teaching text.

A final comment: the author rightly claims that this book prepares students for a subsequent course in differential geometry. But I know of only one introductory text (by Barrett O’Neill) that invokes the use of differential forms — and that book provides its own introduction to them anyway. 

Peter Ruane is retired from the field of mathematics education, which involved the training of primary and secondary school teachers. His postgraduate study included of algebraic topology and differential geometry, with applications to superconductivity.

See the table of contents in the publisher's webpage.