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Mathematical Analysis of Problems in the Natural Sciences

  • Textbook
  • © 2011

Overview

Authors:
  1. Vladimir Zorich

    1. , Department of Mathematics, Moscow State University, Moscow, Russian Federation

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  • Short course of natural sciences Enables further developments in various directions Brilliantly manifested profound applications of fundamental mathematics
  • Includes supplementary material: sn.pub/extras

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-xi
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  2. Analysis of Dimensions of Physical Quantities

    1. Front Matter

      Pages 1-3
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    2. Elements of the theory

      • Vladimir Zorich
      Pages 5-10

    3. Examples of applications

      • Vladimir Zorich
      Pages 11-21

    4. Further applications: hydrodynamics and turbulence

      • Vladimir Zorich
      Pages 23-31

  3. Multidimensional Geometry and Functions of a Very Large Number of Variables

    1. Front Matter

      Pages 33-35
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    2. Some examples of functions of very many variables in natural science and technology

      • Vladimir Zorich
      Pages 37-44

    3. Concentration principle and its applications

      • Vladimir Zorich
      Pages 45-53

    4. Communication in the presence of noise

      • Vladimir Zorich
      Pages 55-75

  4. Classical Thermodynamics and Contact Geometry

    1. Front Matter

      Pages 77-79
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    2. Classical thermodynamics (basic ideas)

      • Vladimir Zorich
      Pages 81-89

    3. Thermodynamics and contact geometry

      • Vladimir Zorich
      Pages 91-97

    4. Thermodynamics classical and statistical

      • Vladimir Zorich
      Pages 99-123

  5. Back Matter

    Pages 125-135
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Keywords

About this book

Based on a two-semester course aimed at illustrating various interactions of "pure mathematics" with other sciences, such as hydrodynamics, thermodynamics, statistical physics and information theory, this text unifies three general topics of analysis and physics, which are as follows: the dimensional analysis of physical quantities, which contains various applications including Kolmogorov's model for turbulence; functions of very large number of variables and the principle of concentration along with the non-linear law of large numbers, the geometric meaning of the Gauss and Maxwell distributions, and the Kotelnikov-Shannon theorem; and, finally, classical thermodynamics and contact geometry, which covers two main principles of thermodynamics in the language of differential forms, contact distributions, the Frobenius theorem and the Carnot-Caratheodory metric. It includes problems, historical remarks, and Zorich's popular article, "Mathematics as language and method."

Reviews

From the reviews:

“Vladimir Zorich has written a short and mathematically advanced text on the natural sciences as seen through mathematics. … The text touches on many ideas: the dimension of a television signal, the molecular theory of matter, transmission line capacity, to name a few. … 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, The Mathematical Association of America, August, 2011)

Authors and Affiliations

  • , Department of Mathematics, Moscow State University, Moscow, Russian Federation

    Vladimir Zorich

About the author

Vladimir A. Zorich is a distinguished Professor of Mathematics at the University of Moscow who solved the problem of global homeomorphism for space quasi-conformal mappings and provided its far-reaching generalizations.

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