Generalized Calculus with Applications to Matter and Forces

How does one differentiate the non-differentiable? The question is not as meaningless as it may sound — the subject is historically important, as it first arose in arguments between Delambert, Euler and others over solutions of the form \(f(x-ct) +g(x+ct)\) to the classical wave equation in one dimension. The full and rigorous solution had to wait a hundred years for Laurent Schwartz and his invention of spaces of distributions (sometimes known as generalized functions). Such entities include famous examples like the Dirac Delta function and its various derivatives. The use of these strange beasties has been a boon to physicists for some time; they are often used to represent “unit impulses” and other non-continuous phenomena. The book under review is a rather encyclopedic review of the Theory of Distributions. It includes a very broad array of applications (including loaded strings, bending of beams and axisymmetric flows of liquids, and electromagnetic images).

Campos takes a rather concrete approach to these ideas, introducing distributions as limits of integrals with parameters. This method highlights that such limits may not vary continuously with their parameters but, in fact, may be strange combinations of piecewise continuous functions (such as the Heaviside function \(H(x)=\begin{cases}0 & \text{if \(x<0\)}\\1 & \text{if \(x\geq 0\)}\end{cases}\) and its translations). This approach is a typical one for many engineering courses. Gaussian functions play a distinct role as test functions for these functionals. The unfamiliar reader must wait until Chapter 3 to discover that the more general and “official” definition of a distribution is as a continuous linear functional defined on various spaces of “test functions”. Growth and decay conditions on the elements of these function spaces determine just how generalized such functionals can be.

Along the way, the reader is also introduced to Fourier transforms of such functionals, tensor products of distributions, and convolutions. Campos’s approach throughout this very dense book is to showcase settings where distributions are used: the mathematics developed is always done in service to applications. And the applications are very long and very extensive. I have never seen any text working out so many special cases of multipoles in potential theory or utilizing hyperspherical Legendre polynomials in the act. There is some heavy machinery here.

Most mathematicians will probably be infuriated by the methodology used in presenting this material. The presentation is dense and seems to consist of a “worked problem” approach very different from the theorem-proof style used in (our) graduate textbooks. This might not have been a bad thing if Campos had chosen to leaven the verbiage with motivational text that breaks the relentless flow of writing that has the form “Put result (6.1.23x) into (6.2.11y) to obtain…” Although there are helpful notes written for those unfamiliar with Fourier series or \(L^p\) inequalities, these are also written in the same form and have the unfortunate effect of making an already dense presentation seem even more taxing. Additionally, the use of unusual names for standard results (instead of derivatives we sometimes have “Derivates”, the Cauchy-Schwarz inequality is called the “Schwarz projective inequality”, some spaces of test functions are called “Excellent” functions and others are “Superlative” functions). Shouldn’t engineers learn more typical terminology so that they can more easily communicate with their mathematical brethren?

Engineers looking for detailed approaches to the use of distributions in solving problems will certainly want to dip into this text. It certainly satisfies its stated aim of placing “The emphasis … on the application, including formulation of the problem, detailed solution, and interpretation of results (xxvii)”. On the other hand, those looking to learn the theory for the first time may well wish to look to lighter and better motivated resources.

Jeff Ibbotson is the Smith Teaching Chair at Phillips Exeter Academy. He spends much of his time reading, playing ping pong and raising beagles.