Mathematical Analysis of Problems in the Natural Sciences [Paperback]

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. This book illustrates interactions of pure mathematics with other sciences, such as hydrodynamics, thermodynamics, statistical physics and information theory. It includes problems, historical remarks and Zorichs article Mathematics as Language and Method.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

From the reviews:

Vladimir Zorich has written a short and mlc§