The book "Topological Methods in Hydrodynamics" by Vladimir I. Arnold and Boris A. Khesin explores the intersection of topology and hydrodynamics, providing a comprehensive overview of the mathematical theories underlying fluid dynamics. It covers topics such as the geometric structure of fluid flows, the topology of steady fluid flows, and the properties of magnetic and vorticity fields. The authors discuss the application of Riemannian geometry to the study of fluid dynamics, particularly through the lens of geodesics on diffeomorphism groups. They also examine the stability of fluid flows, the classification of three-dimensional steady flows, and the role of topological invariants in hydrodynamics. The book includes detailed discussions on the Euler equations, the Navier-Stokes equation, and the dynamics of magnetic fields. It also addresses the challenges in proving the existence of smooth solutions to the Navier-Stokes equations and the instability of ideal fluid flows. The authors highlight the importance of topological methods in understanding the behavior of fluid systems, particularly in the context of turbulence and the unpredictability of long-term weather forecasts. The book is structured into several chapters, each focusing on different aspects of hydrodynamics, and includes a variety of mathematical tools and concepts from differential geometry, topology, and dynamical systems. The authors also provide a detailed bibliography and index, making the book a valuable resource for researchers and students in the field of applied mathematics and fluid dynamics.The book "Topological Methods in Hydrodynamics" by Vladimir I. Arnold and Boris A. Khesin explores the intersection of topology and hydrodynamics, providing a comprehensive overview of the mathematical theories underlying fluid dynamics. It covers topics such as the geometric structure of fluid flows, the topology of steady fluid flows, and the properties of magnetic and vorticity fields. The authors discuss the application of Riemannian geometry to the study of fluid dynamics, particularly through the lens of geodesics on diffeomorphism groups. They also examine the stability of fluid flows, the classification of three-dimensional steady flows, and the role of topological invariants in hydrodynamics. The book includes detailed discussions on the Euler equations, the Navier-Stokes equation, and the dynamics of magnetic fields. It also addresses the challenges in proving the existence of smooth solutions to the Navier-Stokes equations and the instability of ideal fluid flows. The authors highlight the importance of topological methods in understanding the behavior of fluid systems, particularly in the context of turbulence and the unpredictability of long-term weather forecasts. The book is structured into several chapters, each focusing on different aspects of hydrodynamics, and includes a variety of mathematical tools and concepts from differential geometry, topology, and dynamical systems. The authors also provide a detailed bibliography and index, making the book a valuable resource for researchers and students in the field of applied mathematics and fluid dynamics.