A First Course in Dimensional Analysis

A First Course In Dimensional Analysis is an introductory undergraduate level textbook on the methods of dimensional analysis.  It covers the principle of dimensional homogeneity, Ipsen’s method for finding dimensionless groups, the Buckingham \( \Pi \) theorem, using dimensionless groups to build models, and common dimensionless groups such as the Reynolds number and the Peclet number.    

The author’s general approach is to introduce the basic concept of dimensional homogeneity and then use Ipsen’s method to eliminate dimensions from the problem successively.  He also advocates using governing equations to derive dimensionless parameters. The author is not a fan of Buckingham’s \( \Pi \) theorem but the book does include a chapter on the use of the \( \Pi \) theorem to find dimensionless groups.  In comparison with Ipsen’s method, using the \( \Pi \) theorem feels like magic, particularly when the theorem is simply stated without proof as in most other introductory books on dimensional analysis. Having taught this material many times using Buckingham’s \( \Pi \) theorem I’m convinced that the author’s approach has some merit.    

One difficulty that applied mathematics students often have is a lack of familiarity with basic concepts of fluid dynamics and heat transfer.  The examples in this book are supported by text and figures that help to explain these concepts. However, students might benefit from modeling projects and exercises in which they explore these concepts in more depth. 

I was somewhat disappointed that the author did not discuss the need to nondimensionalize a model before analyzing the asymptotic behavior of the model with respect to small or large parameters.  Perturbation methods require nondimensional parameters and students who go on to take a course in the methods of applied mathematics will need to be able to nondimensionalize models.  

Overall, the author succeeds in presenting dimensional analysis in a way that will be accessible to undergraduate students in engineering, the physical sciences, and applied mathematics.  There isn’t enough material in the book to really make a semester-long course, but the book could be very useful as a supplementary text for a larger modeling course. For those who teach modeling courses that include dimensional analysis, the book will also be a good source of examples and exercises. 

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