This book, as usual by the excellent Springer publishers, continues the trend launched by the Clifford algebra people (Lounesto, Chisholm, Baylis, Pezzaglia, Okubo, Benn, etc. - see reviews of some of them), namely, to SIMPLIFY the mathematics of physics by using appropriate ALGEBRAIC techniques rather than geometry or calculus or other techniques. Both this book by Vein and Dale and the Clifford algebra books and papers use algebra in physics largely to replace hard to manipulate geometry and unwieldly matrices. A matrix is an algebraic quantity, but it is very hard to handle: it is essentially a table of numbers, for example a table of people's heights, or people's heights by weights. You add tables by adding corresponding positions in each table, and likewise for subtracting, while multiplication is much more complicated. However, as Vein and Dale show, you can replace many results in physics which involve matrices by DETERMINANTS. A determinant is a single number, typically, which is gotten by combining the numbers of the matrix table in a certain way given by a formula. Thus, replacing a matrix by a determinant means replacing a table by a single number. It turns out that the Einstein Equation(s) of general relativity can be solved in this way (for the axially symmetric field), and likewise for equations involving solitary waves (Kadomtsev-Petashvili equation), waves in a rotating fluid (Benjamin-Ono equation), etc. An important tool in this process is Backlund transformations, which are described in the appendix but are more thoroughly described in the 1989 book of Bluman and Kumei which (together with their journal publications) initiated much of the simplification of differential equations of the modern era. That book, as you may guess, was also published by Springer/Springer-Verlag.

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