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Mathematical Analysis of Problems in the Natural Sciences

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Vladimir Zorich has written a short and mathematically advanced text on the natural sciences as seen through mathematics. Here is a book that is terse throughout, yet extensive in its breadth of coverage of science. If Zorich’s aim was to show you how mathematics applies and is used in a vast multitude of fields, he has succeeded admirably. If you have seen the physics before, the text will bring back memories and formulas and remind you of just how rich each area really is mathematically. The text will not, however, teach you the material. It will show you the mathematics and in that sense it takes you on a journey of science through the equations that dictate its behavior.

Part I begins with analysis of dimensions. It seems, at first, close to a beginning physics class, but then we see how to apply this analytical technique to orbital mechanics, pendulum oscillations, motion in a viscous fluid, fluid mechanics and Reynolds numbers. Whole books are written on any one of these topics, but here we have a concise explanation of the physics and equations that describe the behaviors.

Part II looks at technology. The reader will find the familiar Fourier Transform and Shannon’s sampling theorem. I found something else, too. Shannon was not the first to discover his famous “named-for” theorem. Rather, Kotel’nikov discovered it in 1933; Shannon actually rediscovered it in 1949.

As an aside, I have a Russian friend who is a telecommunications engineer. When I read of Kotel’nikov I asked my friend, “Who is this person, I never heard of him.” He told me that Kotel’nikov discovered the sampling theorem. I told him no, that can’t be. I only heard of Shannon. He replied “And I only heard of Kotel’nikov. Until I came to the States!”

The text touches on many ideas: the dimension of a television signal, the molecular theory of matter, transmission line capacity, to name a few. It spends a good deal of time on information theory and entropy but the treatment is still too short to be of great help to a beginner.

The last part of the book is about thermodynamics, the Frobenius theorem, entropy and ergodicity.

Overall, I would not recommend this book to a non-expert, or even to an expert who wants a detailed treatment of a particular field. But, if you want to see how mathematics is intertwined in nature and physics, how mathematics describes and explains our world, then this book paints that picture.

David S. Mazel received his Ph. D. from Georgia Tech in electrical engineering and is a practicing engineer in Washington, DC. His research interests are in the dynamics of billiards, signal processing, and cellular automata.

Date Received: 
Wednesday, January 12, 2011
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Vladimir Zorich
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David S. Mazel

Part I Analysis of Dimensions of Physical Quantities: 1 Elements of the theory.- 2 Examples of applications.- 3 Further applications: hydrodynamics and turbulence.- Part II: Multidimensional Geometry and Functions of a Very Large Number of Variables: 1 Some examples of functions of very many variables in natural science and technology.- 2 Concentration principle and its applications.- 3 Communication in the presence of noise.- Part III Classical Thermodynamics and Contact Geometry: 1 Classical thermodynamics (basic ideas).- 2 Thermodynamics and contact geometry.- 3 Thermodynamics classical and statistical.- References.- Appendix. Mathematics as Language and Method

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Tuesday, August 2, 2011