Exterior analysis. Using applications of differential forms (Q2852588)

This book represents a comprehensive and meaningful presentation of the analysis based on exterior differential forms and various applications to partial differential equations, calculus of variations and physics. It starts with an introduction to the exterior algebra emphasizing the fundamental part played by the exterior product. Next, elements of the theory of differentiable manifolds are given discussing in a clear way the underlying topology, submanifolds, tangent spaces, tangent bundle, vector fields, flows, Lie derivative, distributions and their integration through the Frobenius theorem. Afterwards, basic topics involving Lie groups, Lie algebras and tensor fields are surveyed.NEWLINENEWLINE The core of the book is formed by the chapter devoted to the exterior differential forms and exterior derivative. In that context, the Riemannian manifolds are also studied. The homotopy operator leading readily to the Poincaré lemma is investigated. Then linear connections and integration of differential forms are examined, where the fundamental result is the Stokes theorem. The cohomology of de Rham is also considered. The last three chapters contain applications of the abstract geometric setting. Here some partial differential equations are handled through exterior forms and a method of generalized characteristics is produced. The calculus of variations via exterior forms is revisited, in particular the Euler-Lagrange equations. Finally, the author provides applications to analytical mechanics, electromagnetism and thermodynamics.NEWLINENEWLINE The book is carefully written. It contains a rich material and covers an important part of differential geometry. Applications of the main abstract results can be found frequently. There are many examples and exercises as well as documented comments regarding different developments and applications. The book is useful for mathematicians, applied scientists and engineers.