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Presenting a classical approach to the foundations and development of the geometry of vector fields, this volume describes vector fields in three-dimensional Euclidean space, three orthogonal systems, and applications in mechanics. Other topics, including vector fields, Pfaff forms and systems in n-dimensional space, foliations and Godbillon-Vey invariant, are also considered. There is much interest in the study of geometrical objects in n-dimensional Euclidean space, and this volume provides a useful and comprehensive presentation.


Summary: vector fields in the 21'st century
Rating: 5

This well conceived and lucidly written book demonstrates once again that three dimensional Euclidean space is even in this age the source of a wealth of deep and useful understanding of that most fundamental, to all the physical sciences, object, the vector field. Vector fields appear in all the physical sciences and a renewed interest in their behaviour and the interaction between their local and global properties, has arisen in recent years. Fields of application are many, including fluid dynamics and plasma physics. For example, the plasma physicist Newcomb, in a ground breaking article in the Annals of Physics in the 60's discussed the questions of holonomy so incisively analyzed here by Aminov, and their relation to frozen in vector fields and Lie derivatives. The article by Ericksen on tensor fields in the Handbuch der Physik is another example of a masterful discussion of vector and tensor fields and their applications in continuum mechanics.

It is clear from the beginning of the book, that the author has his own fresh and original point of view on this subject. The exposition is clear and avoids the use of abstract machinery as much as possible, thus making it more accessible to a wide audience. Later chapters deal with higher dimensional Euclidean spaces. I highly recommend this book.

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