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Hot Molecules, Cold Electrons: From the Mathematics of Heat to the Development of the Trans-Atlantic Telegraph Cable

Paul J. Nahin
Princeton University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Brian Borchers
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In Hot Molecules, Cold Electrons, Paul Nahin uses the story of William Thompson’s analysis of the transatlantic telegraph cable as an example of the application of Fourier series and the method of separation of variables to solve the heat equation.  Like many of his earlier books, this book is written at a level that should be accessible to undergraduate mathematics, science, and engineering students with a solid background in calculus.  

In the first two-thirds of the book, the author gives a brief history of Fourier’s work on the heat equation and the use of Fourier series in their solution.  The presentation is quite readable but not particularly mathematically rigorous. The author then goes on to tell the story of how William Thompson used the heat equation to model the electrical behavior of the transatlantic telegraph cable.  In the final chapter, the author explains how Duhamel’s integral can be used to derive the impulse response of the cable.   

The mathematics discussed in this book is typically treated in an undergraduate course on differential equations and boundary value problems.  For example, the material is thoroughly covered in Boyce and DiPrima’s Elementary Differential Equations and Boundary Value Problems where Fourier series and the method of separation of variables are used to solve not only the heat equation but also the wave equation and Laplace’s equation.  Although Nahin does a good job of explaining the mathematics he has selected, this would not be suitable for use as a textbook due to the lack of breadth and the lack of exercises. The book might be of some interest to instructors for the historical discussion of the work of Fourier and Thompson. 


Brian Borchers is a professor of mathematics at New Mexico Tech and the editor of MAA Reviews.