Of all the scientific or mathematical books that I have reviewed or even read, I would place this book at the position of number one (1) in excellence, creativity, genius, inspiration, intuition, and usefulness. It has inspired some of my own best research and I often cite it in presenting papers at conferences and publishing papers. In my opinion, Nonsmooth Analysis is one of the 20 main research areas in mathematics of the last 5 years (others include rare events/large deviations, solutions of Navier Stokes/Einstein field equations/Schrodinger equation, fractals/chaos/entropy, fuzzy sets/fuzzy logic/multivalued logic/other logics, semigroups/Clifford algebras/spacetime algebras,etc.). Perhaps the most astonishing finding of Clarke et al., book here and in their journal papers (and those of their colleagues), is that equations become inequalities and subset relationships when one goes from smooth physics to disconnected and sharp-bend physics. The latter types of physics may seem difficult to visualize at first, but think of what happens when ice suddenly changes phase to water, or water changes phase suddenly to vapor/steam. Or think of what happens when a runner or a racecar or a plane suddenly makes a 180 degree about-face (runners might be able to do this, but planes can only do it approximately at usual speeds). Ordinary physics and mathematics cannot handle these situations. Other examples are catastrophes, sudden strokes of good fortune, etc. You can see that these are often related to rare events, which I have reviewed elsehwhere. It turns out that the usual mathematics which involves equations becomes inequalities (less than, greater than, etc.) and subset relationships (A is inside B or is a subset of B) in the new situations. Clarke et. al. prove theorems quite rigorously in this area. If you have any hesitation in reading this book because of its mathematical content, hire a reputable consultant or tutor to translate the results into an approximation to ordinary English. If you don't, you'll miss out on opportunities to apply the results to your own area and maybe even your own daily life.

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