[.ca] General Vector and Dyadic Analysis: Applied Mathematics ... (ISBN 0780334132)

long awaited book:
If you ever came across the need of expressing vector operators in curvilinear coordinates, while writing numerical codes or solving PDEs in generalised coordinates, then you might have felt the need of a rigourous presentation of the subject. Rigorous yet usable, I would say. This is not a book for pure mathematicians (the theorem you need is at pag. 435 consisting of one line saying the proof follows from the 2000 previous Lemma-theorem-corollary scattered all around...). The author has the physical problems well in his mind (he's Emeritus at Univ of Michigan, dept of Electrical Engineering) and makes frequent references to where in physics or geometry you will encounter such mathematical entities. Chapter 8, devoted to the history of vector analysis, makes you clear why you haven't been able to find a neat rigorous presentation so far. This digression (I'm reluctant to use this expression for it's not a minor chapter in my opinion) gives you an even sharper feeling of what's the core of the matter by comparing critically different approaches. The first introductory chapters on geometry, coordinate systems, dyadics provide a sound general background, presenting derivatives of vectors in generalised coordinate system from the beginning, resulting in a powerful and unifying exposition of the matter. All is extremely clear, a characteristic of the book in its whole. The author always stress upon the transformation properties of the objects which have been defined, thus clarifying their meaning and mathematical character. The appendix on vector analysis in the special theory of relativity is enjoyable and makes you clear how E and B can be treated as vectors even if they transform like the components of a tensor (OK, you already knew that, but you were using tensor analysis from the start, didn't you?). There is much more in this book, which I would suggest to keep as a reference to any physicist, applied mathematician, biologist, engineer, numerical people and (even!) mathematician (developing communication skills...). First years students, which are always a bit confused on why one should use vectors at all and having accepted that think that everything is a vector, which is not, should definitely start with this book. Next time I will be asked to teach anything related to mechanics, I will adopt this book for the fundamental mathematics (but also check the excellent Biscari-Poggi-Virga Mechanics Notebook, 1999 Liguori ed. Napoli (IT) ). It's also an excellent way to approach what you want next, which is likely to be tensor analysis which this book, unfortunately, cannot cover (can you hear me, Professor Tai?...).