Fundamentals of Advanced Mathematics V3

Fundamentals of Advanced Mathematics, Volume Three, begins with the study of differential and analytic infinite-dimensional manifolds, then progresses into fibered bundles, in particular, tangent and cotangent bundles. In addition, subjects covered include the tensor calculus on manifolds, differential and integral calculus on manifolds (general Stokes formula, integral curves and manifolds), an analysis on Lie groups, the Haar measure, the convolution of functions and distributions, and the harmonic analysis over a Lie group. Finally, the theory of connections is (linear connections, principal connections, and Cartan connections) covered, as is the calculus of variations in Lagrangian and Hamiltonian formulations.

This volume is the prerequisite to the analytic and geometric study of nonlinear systems.

"The present volume is the third one of a series which presents the fundamental elements of advanced mathematics that is at the basis of a number of contemporary scientific methods. More precisely, it deals with differential and integral calculus in their local and global components. The book is designed not only for mathematicians, but also for everyone who uses mathematics and needs to understand the control of nonlinear systems (in particular physicists and engineers). The ambitious goal is achieved also thanks to an excellent organization of the topics and the use of a very clear and understandable language. Interesting short historical notes introduce the different topics and help to frame the evolution of concept. The exposition is illustrated with some figures that help a lot in understanding the not easy topics. Very useful attachments are provided: a careful list of notation and term indeces, a reach bibliography, a list of cited authors with biographical notes." —ZBMath

"The book under review is the third volume in a series that lays a solid foundation for advanced mathematics, serving as a fundamental resource for various contemporary scientific methodologies. This particular volume explores the intricate realms of both differential and integral calculus, providing a comprehensive examination of their local and global components. While primarily intended for mathematicians, the book transcends disciplinary boundaries and aims to be read by individuals from diverse fields who utilize mathematics in their work, including physicists and engineers."—MathSciNet